TSTP Solution File: BOO006-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : BOO006-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 6 17:18:37 EDT 2022
% Result : Unsatisfiable 74.72s 48.42s
% Output : Proof 74.84s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : BOO006-1 : TPTP v8.1.0. Released v1.0.0.
% 0.07/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.33 % Computer : n015.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Tue Aug 30 02:45:50 EDT 2022
% 0.13/0.33 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 74.72/48.42 % SZS status Unsatisfiable
% 74.72/48.42 % SZS output start Proof
% 74.72/48.42 tff(product_type, type, (
% 74.72/48.42 product: ( $i * $i * $i ) > $o)).
% 74.72/48.42 tff(add_type, type, (
% 74.72/48.42 add: ( $i * $i ) > $i)).
% 74.72/48.42 tff(inverse_type, type, (
% 74.72/48.42 inverse: $i > $i)).
% 74.72/48.42 tff(additive_identity_type, type, (
% 74.72/48.42 additive_identity: $i)).
% 74.72/48.42 tff(x_type, type, (
% 74.72/48.42 x: $i)).
% 74.72/48.42 tff(multiplicative_identity_type, type, (
% 74.72/48.42 multiplicative_identity: $i)).
% 74.72/48.42 tff(sum_type, type, (
% 74.72/48.42 sum: ( $i * $i * $i ) > $o)).
% 74.72/48.42 tff(multiply_type, type, (
% 74.72/48.42 multiply: ( $i * $i ) > $i)).
% 74.72/48.42 tff(1,plain,
% 74.72/48.42 (^[X: $i] : refl(sum(X, additive_identity, X) <=> sum(X, additive_identity, X))),
% 74.72/48.42 inference(bind,[status(th)],[])).
% 74.72/48.42 tff(2,plain,
% 74.72/48.42 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 74.72/48.42 inference(quant_intro,[status(thm)],[1])).
% 74.72/48.42 tff(3,plain,
% 74.72/48.42 (![X: $i] : sum(X, additive_identity, X) <=> ![X: $i] : sum(X, additive_identity, X)),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(4,axiom,(![X: $i] : sum(X, additive_identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_identity2')).
% 74.72/48.42 tff(5,plain,
% 74.72/48.42 (![X: $i] : sum(X, additive_identity, X)),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[4, 3])).
% 74.72/48.42 tff(6,plain,(
% 74.72/48.42 ![X: $i] : sum(X, additive_identity, X)),
% 74.72/48.42 inference(skolemize,[status(sab)],[5])).
% 74.72/48.42 tff(7,plain,
% 74.72/48.42 (![X: $i] : sum(X, additive_identity, X)),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[6, 2])).
% 74.72/48.42 tff(8,plain,
% 74.72/48.42 ((~![X: $i] : sum(X, additive_identity, X)) | sum(inverse(additive_identity), additive_identity, inverse(additive_identity))),
% 74.72/48.42 inference(quant_inst,[status(thm)],[])).
% 74.72/48.42 tff(9,plain,
% 74.72/48.42 (sum(inverse(additive_identity), additive_identity, inverse(additive_identity))),
% 74.72/48.42 inference(unit_resolution,[status(thm)],[8, 7])).
% 74.72/48.42 tff(10,plain,
% 74.72/48.42 (^[Y: $i, X: $i] : refl(sum(X, Y, add(X, Y)) <=> sum(X, Y, add(X, Y)))),
% 74.72/48.42 inference(bind,[status(th)],[])).
% 74.72/48.42 tff(11,plain,
% 74.72/48.42 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 74.72/48.42 inference(quant_intro,[status(thm)],[10])).
% 74.72/48.42 tff(12,plain,
% 74.72/48.42 (![Y: $i, X: $i] : sum(X, Y, add(X, Y)) <=> ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(13,axiom,(![Y: $i, X: $i] : sum(X, Y, add(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','closure_of_addition')).
% 74.72/48.42 tff(14,plain,
% 74.72/48.42 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[13, 12])).
% 74.72/48.42 tff(15,plain,(
% 74.72/48.42 ![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 74.72/48.42 inference(skolemize,[status(sab)],[14])).
% 74.72/48.42 tff(16,plain,
% 74.72/48.42 (![Y: $i, X: $i] : sum(X, Y, add(X, Y))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[15, 11])).
% 74.72/48.42 tff(17,plain,
% 74.72/48.42 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))),
% 74.72/48.42 inference(quant_inst,[status(thm)],[])).
% 74.72/48.42 tff(18,plain,
% 74.72/48.42 (sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))),
% 74.72/48.42 inference(unit_resolution,[status(thm)],[17, 16])).
% 74.72/48.42 tff(19,plain,
% 74.72/48.42 (^[V: $i, Y: $i, U: $i, X: $i] : refl(((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))) <=> ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))))),
% 74.72/48.42 inference(bind,[status(th)],[])).
% 74.72/48.42 tff(20,plain,
% 74.72/48.42 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 74.72/48.42 inference(quant_intro,[status(thm)],[19])).
% 74.72/48.42 tff(21,plain,
% 74.72/48.42 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(22,plain,
% 74.72/48.42 (^[V: $i, Y: $i, U: $i, X: $i] : rewrite((((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V)) <=> ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U))))),
% 74.72/48.42 inference(bind,[status(th)],[])).
% 74.72/48.42 tff(23,plain,
% 74.72/48.42 (![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V)) <=> ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 74.72/48.42 inference(quant_intro,[status(thm)],[22])).
% 74.72/48.42 tff(24,axiom,(![V: $i, Y: $i, U: $i, X: $i] : (((~sum(X, Y, U)) | (~sum(X, Y, V))) | (U = V))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','addition_is_well_defined')).
% 74.72/48.42 tff(25,plain,
% 74.72/48.42 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[24, 23])).
% 74.72/48.42 tff(26,plain,
% 74.72/48.42 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[25, 21])).
% 74.72/48.42 tff(27,plain,(
% 74.72/48.42 ![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 74.72/48.42 inference(skolemize,[status(sab)],[26])).
% 74.72/48.42 tff(28,plain,
% 74.72/48.42 (![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[27, 20])).
% 74.72/48.42 tff(29,plain,
% 74.72/48.42 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))) | (add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))) | (add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity))))),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(30,plain,
% 74.72/48.42 (((add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity))) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity)))) <=> ((~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))) | (add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity))))),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(31,plain,
% 74.72/48.42 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity))) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))) | (add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity)))))),
% 74.72/48.42 inference(monotonicity,[status(thm)],[30])).
% 74.72/48.42 tff(32,plain,
% 74.72/48.42 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity))) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))) | (add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity))))),
% 74.72/48.42 inference(transitivity,[status(thm)],[31, 29])).
% 74.72/48.42 tff(33,plain,
% 74.72/48.42 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity))) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))),
% 74.72/48.42 inference(quant_inst,[status(thm)],[])).
% 74.72/48.42 tff(34,plain,
% 74.72/48.42 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))) | (add(inverse(additive_identity), additive_identity) = inverse(additive_identity)) | (~sum(inverse(additive_identity), additive_identity, inverse(additive_identity)))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[33, 32])).
% 74.72/48.42 tff(35,plain,
% 74.72/48.42 (add(inverse(additive_identity), additive_identity) = inverse(additive_identity)),
% 74.72/48.42 inference(unit_resolution,[status(thm)],[34, 28, 18, 9])).
% 74.72/48.42 tff(36,plain,
% 74.72/48.42 (^[X: $i] : refl(sum(inverse(X), X, multiplicative_identity) <=> sum(inverse(X), X, multiplicative_identity))),
% 74.72/48.42 inference(bind,[status(th)],[])).
% 74.72/48.42 tff(37,plain,
% 74.72/48.42 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 74.72/48.42 inference(quant_intro,[status(thm)],[36])).
% 74.72/48.42 tff(38,plain,
% 74.72/48.42 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(39,axiom,(![X: $i] : sum(inverse(X), X, multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_inverse1')).
% 74.72/48.42 tff(40,plain,
% 74.72/48.42 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[39, 38])).
% 74.72/48.42 tff(41,plain,(
% 74.72/48.42 ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 74.72/48.42 inference(skolemize,[status(sab)],[40])).
% 74.72/48.42 tff(42,plain,
% 74.72/48.42 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[41, 37])).
% 74.72/48.42 tff(43,plain,
% 74.72/48.42 ((~![X: $i] : sum(inverse(X), X, multiplicative_identity)) | sum(inverse(additive_identity), additive_identity, multiplicative_identity)),
% 74.72/48.42 inference(quant_inst,[status(thm)],[])).
% 74.72/48.42 tff(44,plain,
% 74.72/48.42 (sum(inverse(additive_identity), additive_identity, multiplicative_identity)),
% 74.72/48.42 inference(unit_resolution,[status(thm)],[43, 42])).
% 74.72/48.42 tff(45,plain,
% 74.72/48.42 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(46,plain,
% 74.72/48.42 (((add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity)))) <=> ((~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(47,plain,
% 74.72/48.42 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity)))))),
% 74.72/48.42 inference(monotonicity,[status(thm)],[46])).
% 74.72/48.42 tff(48,plain,
% 74.72/48.42 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))),
% 74.72/48.42 inference(transitivity,[status(thm)],[47, 45])).
% 74.72/48.42 tff(49,plain,
% 74.72/48.42 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity))))),
% 74.72/48.42 inference(quant_inst,[status(thm)],[])).
% 74.72/48.42 tff(50,plain,
% 74.72/48.42 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(inverse(additive_identity), additive_identity, multiplicative_identity)) | (add(inverse(additive_identity), additive_identity) = multiplicative_identity) | (~sum(inverse(additive_identity), additive_identity, add(inverse(additive_identity), additive_identity)))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[49, 48])).
% 74.72/48.42 tff(51,plain,
% 74.72/48.42 (add(inverse(additive_identity), additive_identity) = multiplicative_identity),
% 74.72/48.42 inference(unit_resolution,[status(thm)],[50, 28, 44, 18])).
% 74.72/48.42 tff(52,plain,
% 74.72/48.42 (multiplicative_identity = add(inverse(additive_identity), additive_identity)),
% 74.72/48.42 inference(symmetry,[status(thm)],[51])).
% 74.72/48.42 tff(53,plain,
% 74.72/48.42 (multiplicative_identity = inverse(additive_identity)),
% 74.72/48.42 inference(transitivity,[status(thm)],[52, 35])).
% 74.72/48.42 tff(54,plain,
% 74.72/48.42 (product(add(inverse(x), inverse(additive_identity)), multiplicative_identity, add(inverse(x), inverse(additive_identity))) <=> product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))),
% 74.72/48.42 inference(monotonicity,[status(thm)],[53])).
% 74.72/48.42 tff(55,plain,
% 74.72/48.42 (product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity))) <=> product(add(inverse(x), inverse(additive_identity)), multiplicative_identity, add(inverse(x), inverse(additive_identity)))),
% 74.72/48.42 inference(symmetry,[status(thm)],[54])).
% 74.72/48.42 tff(56,plain,
% 74.72/48.42 ((~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) <=> (~product(add(inverse(x), inverse(additive_identity)), multiplicative_identity, add(inverse(x), inverse(additive_identity))))),
% 74.72/48.42 inference(monotonicity,[status(thm)],[55])).
% 74.72/48.42 tff(57,assumption,(~sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x))), introduced(assumption)).
% 74.72/48.42 tff(58,plain,
% 74.72/48.42 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 74.72/48.42 inference(bind,[status(th)],[])).
% 74.72/48.42 tff(59,plain,
% 74.72/48.42 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 74.72/48.42 inference(quant_intro,[status(thm)],[58])).
% 74.72/48.42 tff(60,plain,
% 74.72/48.42 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(61,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','closure_of_multiplication')).
% 74.72/48.42 tff(62,plain,
% 74.72/48.42 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[61, 60])).
% 74.72/48.42 tff(63,plain,(
% 74.72/48.42 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 74.72/48.42 inference(skolemize,[status(sab)],[62])).
% 74.72/48.42 tff(64,plain,
% 74.72/48.42 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[63, 59])).
% 74.72/48.42 tff(65,plain,
% 74.72/48.42 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(additive_identity, x, multiply(additive_identity, x))),
% 74.72/48.42 inference(quant_inst,[status(thm)],[])).
% 74.72/48.42 tff(66,plain,
% 74.72/48.42 (product(additive_identity, x, multiply(additive_identity, x))),
% 74.72/48.42 inference(unit_resolution,[status(thm)],[65, 64])).
% 74.72/48.42 tff(67,plain,
% 74.72/48.42 ((~![X: $i] : sum(X, additive_identity, X)) | sum(x, additive_identity, x)),
% 74.72/48.42 inference(quant_inst,[status(thm)],[])).
% 74.72/48.42 tff(68,plain,
% 74.72/48.42 (sum(x, additive_identity, x)),
% 74.72/48.42 inference(unit_resolution,[status(thm)],[67, 7])).
% 74.72/48.42 tff(69,plain,
% 74.72/48.42 (^[Z: $i, Y: $i, X: $i] : refl(((~sum(X, Y, Z)) | sum(Y, X, Z)) <=> ((~sum(X, Y, Z)) | sum(Y, X, Z)))),
% 74.72/48.42 inference(bind,[status(th)],[])).
% 74.72/48.42 tff(70,plain,
% 74.72/48.42 (![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))),
% 74.72/48.42 inference(quant_intro,[status(thm)],[69])).
% 74.72/48.42 tff(71,plain,
% 74.72/48.42 (![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))),
% 74.72/48.42 inference(rewrite,[status(thm)],[])).
% 74.72/48.42 tff(72,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')).
% 74.72/48.42 tff(73,plain,
% 74.72/48.42 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[72, 71])).
% 74.72/48.42 tff(74,plain,(
% 74.72/48.42 ![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 74.72/48.42 inference(skolemize,[status(sab)],[73])).
% 74.72/48.42 tff(75,plain,
% 74.72/48.42 (![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))),
% 74.72/48.42 inference(modus_ponens,[status(thm)],[74, 70])).
% 74.72/48.42 tff(76,plain,
% 74.72/48.42 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(x, additive_identity, x)) | sum(additive_identity, x, x))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(x, additive_identity, x)) | sum(additive_identity, x, x))),
% 74.72/48.43 inference(rewrite,[status(thm)],[])).
% 74.72/48.43 tff(77,plain,
% 74.72/48.43 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(x, additive_identity, x)) | sum(additive_identity, x, x))),
% 74.72/48.43 inference(quant_inst,[status(thm)],[])).
% 74.72/48.43 tff(78,plain,
% 74.72/48.43 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(x, additive_identity, x)) | sum(additive_identity, x, x)),
% 74.72/48.43 inference(modus_ponens,[status(thm)],[77, 76])).
% 74.72/48.43 tff(79,plain,
% 74.72/48.43 (sum(additive_identity, x, x)),
% 74.72/48.43 inference(unit_resolution,[status(thm)],[78, 75, 68])).
% 74.72/48.43 tff(80,plain,
% 74.72/48.43 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 74.72/48.43 inference(bind,[status(th)],[])).
% 74.72/48.43 tff(81,plain,
% 74.72/48.43 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 74.72/48.43 inference(quant_intro,[status(thm)],[80])).
% 74.72/48.43 tff(82,plain,
% 74.72/48.43 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 74.72/48.43 inference(rewrite,[status(thm)],[])).
% 74.72/48.43 tff(83,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_identity1')).
% 74.72/48.43 tff(84,plain,
% 74.72/48.43 (![X: $i] : sum(additive_identity, X, X)),
% 74.72/48.43 inference(modus_ponens,[status(thm)],[83, 82])).
% 74.72/48.43 tff(85,plain,(
% 74.72/48.43 ![X: $i] : sum(additive_identity, X, X)),
% 74.72/48.43 inference(skolemize,[status(sab)],[84])).
% 74.72/48.43 tff(86,plain,
% 74.72/48.43 (![X: $i] : sum(additive_identity, X, X)),
% 74.72/48.43 inference(modus_ponens,[status(thm)],[85, 81])).
% 74.72/48.43 tff(87,plain,
% 74.72/48.43 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, additive_identity, additive_identity)),
% 74.72/48.43 inference(quant_inst,[status(thm)],[])).
% 74.72/48.43 tff(88,plain,
% 74.72/48.43 (sum(additive_identity, additive_identity, additive_identity)),
% 74.72/48.43 inference(unit_resolution,[status(thm)],[87, 86])).
% 74.72/48.43 tff(89,plain,
% 74.72/48.43 (^[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))))),
% 74.72/48.43 inference(bind,[status(th)],[])).
% 74.72/48.43 tff(90,plain,
% 74.72/48.43 (![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)))),
% 74.72/48.43 inference(quant_intro,[status(thm)],[89])).
% 74.72/48.43 tff(91,plain,
% 74.72/48.43 (![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)))),
% 74.72/48.43 inference(rewrite,[status(thm)],[])).
% 74.72/48.43 tff(92,plain,
% 74.72/48.43 (^[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)))))),
% 74.72/48.43 inference(bind,[status(th)],[])).
% 74.72/48.43 tff(93,plain,
% 74.72/48.43 (![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)))),
% 74.72/48.43 inference(quant_intro,[status(thm)],[92])).
% 74.72/48.43 tff(94,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')).
% 74.72/48.43 tff(95,plain,
% 74.72/48.43 (![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)))),
% 74.72/48.43 inference(modus_ponens,[status(thm)],[94, 93])).
% 74.72/48.43 tff(96,plain,
% 74.72/48.43 (![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)))),
% 74.72/48.43 inference(modus_ponens,[status(thm)],[95, 91])).
% 74.72/48.43 tff(97,plain,(
% 74.72/48.43 ![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)))),
% 74.72/48.43 inference(skolemize,[status(sab)],[96])).
% 74.72/48.43 tff(98,plain,
% 74.72/48.43 (![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)))),
% 74.72/48.43 inference(modus_ponens,[status(thm)],[97, 90])).
% 74.75/48.43 tff(99,plain,
% 74.75/48.43 (((~![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(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | sum(additive_identity, multiply(additive_identity, x), multiply(additive_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(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x)))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(100,plain,
% 74.75/48.43 ((sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x)))) <=> ((~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x)))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(101,plain,
% 74.75/48.43 (((~![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(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_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(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x))))),
% 74.75/48.43 inference(monotonicity,[status(thm)],[100])).
% 74.75/48.43 tff(102,plain,
% 74.75/48.43 (((~![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(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_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(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x)))),
% 74.75/48.43 inference(transitivity,[status(thm)],[101, 99])).
% 74.75/48.43 tff(103,plain,
% 74.75/48.43 ((~![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(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))))),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(104,plain,
% 74.75/48.43 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(X, V3, V4) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)) | (~product(V1, V2, V4)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[103, 102])).
% 74.75/48.43 tff(105,plain,
% 74.75/48.43 ($false),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[104, 98, 88, 79, 66, 57])).
% 74.75/48.43 tff(106,plain,(sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x))), inference(lemma,lemma(discharge,[]))).
% 74.75/48.43 tff(107,plain,
% 74.75/48.43 (((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x))) | sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x))) | sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x)))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(108,plain,
% 74.75/48.43 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | ((~sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x))) | sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x)))),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(109,plain,
% 74.75/48.43 ((~![Z: $i, Y: $i, X: $i] : ((~sum(X, Y, Z)) | sum(Y, X, Z))) | (~sum(additive_identity, multiply(additive_identity, x), multiply(additive_identity, x))) | sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[108, 107])).
% 74.75/48.43 tff(110,plain,
% 74.75/48.43 (sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x))),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[109, 75, 106])).
% 74.75/48.43 tff(111,plain,
% 74.75/48.43 (^[Z: $i, Y: $i, X: $i] : refl(((~product(X, Y, Z)) | product(Y, X, Z)) <=> ((~product(X, Y, Z)) | product(Y, X, Z)))),
% 74.75/48.43 inference(bind,[status(th)],[])).
% 74.75/48.43 tff(112,plain,
% 74.75/48.43 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 74.75/48.43 inference(quant_intro,[status(thm)],[111])).
% 74.75/48.43 tff(113,plain,
% 74.75/48.43 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(114,axiom,(![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','commutativity_of_multiplication')).
% 74.75/48.43 tff(115,plain,
% 74.75/48.43 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[114, 113])).
% 74.75/48.43 tff(116,plain,(
% 74.75/48.43 ![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 74.75/48.43 inference(skolemize,[status(sab)],[115])).
% 74.75/48.43 tff(117,plain,
% 74.75/48.43 (![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[116, 112])).
% 74.75/48.43 tff(118,plain,
% 74.75/48.43 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(additive_identity, x, multiply(additive_identity, x))) | product(x, additive_identity, multiply(additive_identity, x)))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(additive_identity, x, multiply(additive_identity, x))) | product(x, additive_identity, multiply(additive_identity, x)))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(119,plain,
% 74.75/48.43 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(additive_identity, x, multiply(additive_identity, x))) | product(x, additive_identity, multiply(additive_identity, x)))),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(120,plain,
% 74.75/48.43 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(additive_identity, x, multiply(additive_identity, x))) | product(x, additive_identity, multiply(additive_identity, x))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[119, 118])).
% 74.75/48.43 tff(121,plain,
% 74.75/48.43 (product(x, additive_identity, multiply(additive_identity, x))),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[120, 117, 66])).
% 74.75/48.43 tff(122,plain,
% 74.75/48.43 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(multiplicative_identity, x, add(multiplicative_identity, x))),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(123,plain,
% 74.75/48.43 (sum(multiplicative_identity, x, add(multiplicative_identity, x))),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[122, 16])).
% 74.75/48.43 tff(124,plain,
% 74.75/48.43 (((~![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)))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(125,plain,
% 74.75/48.43 ((~![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)))),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(126,plain,
% 74.75/48.43 ((~![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))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[125, 124])).
% 74.75/48.43 tff(127,plain,
% 74.75/48.43 (sum(x, multiplicative_identity, add(multiplicative_identity, x))),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[126, 75, 123])).
% 74.75/48.43 tff(128,plain,
% 74.75/48.43 (^[X: $i] : refl(product(multiplicative_identity, X, X) <=> product(multiplicative_identity, X, X))),
% 74.75/48.43 inference(bind,[status(th)],[])).
% 74.75/48.43 tff(129,plain,
% 74.75/48.43 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 74.75/48.43 inference(quant_intro,[status(thm)],[128])).
% 74.75/48.43 tff(130,plain,
% 74.75/48.43 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(131,axiom,(![X: $i] : product(multiplicative_identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_identity1')).
% 74.75/48.43 tff(132,plain,
% 74.75/48.43 (![X: $i] : product(multiplicative_identity, X, X)),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[131, 130])).
% 74.75/48.43 tff(133,plain,(
% 74.75/48.43 ![X: $i] : product(multiplicative_identity, X, X)),
% 74.75/48.43 inference(skolemize,[status(sab)],[132])).
% 74.75/48.43 tff(134,plain,
% 74.75/48.43 (![X: $i] : product(multiplicative_identity, X, X)),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[133, 129])).
% 74.75/48.43 tff(135,plain,
% 74.75/48.43 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, additive_identity, additive_identity)),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(136,plain,
% 74.75/48.43 (product(multiplicative_identity, additive_identity, additive_identity)),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[135, 134])).
% 74.75/48.43 tff(137,plain,
% 74.75/48.43 (^[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))))),
% 74.75/48.43 inference(bind,[status(th)],[])).
% 74.75/48.43 tff(138,plain,
% 74.75/48.43 (![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)))),
% 74.75/48.43 inference(quant_intro,[status(thm)],[137])).
% 74.75/48.43 tff(139,plain,
% 74.75/48.43 (![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)))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(140,plain,
% 74.75/48.43 (^[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)))))),
% 74.75/48.43 inference(bind,[status(th)],[])).
% 74.75/48.43 tff(141,plain,
% 74.75/48.43 (![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)))),
% 74.75/48.43 inference(quant_intro,[status(thm)],[140])).
% 74.75/48.43 tff(142,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')).
% 74.75/48.43 tff(143,plain,
% 74.75/48.43 (![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)))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[142, 141])).
% 74.75/48.43 tff(144,plain,
% 74.75/48.43 (![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)))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[143, 139])).
% 74.75/48.43 tff(145,plain,(
% 74.75/48.43 ![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)))),
% 74.75/48.43 inference(skolemize,[status(sab)],[144])).
% 74.75/48.43 tff(146,plain,
% 74.75/48.43 (![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)))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[145, 138])).
% 74.75/48.43 tff(147,plain,
% 74.75/48.43 (((~![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(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_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)))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(148,plain,
% 74.75/48.43 (((~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_identity, x)))) <=> ((~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(149,plain,
% 74.75/48.43 (((~![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(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_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)))) | ((~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x)))))),
% 74.75/48.43 inference(monotonicity,[status(thm)],[148])).
% 74.75/48.43 tff(150,plain,
% 74.75/48.43 (((~![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(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_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)))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.43 inference(transitivity,[status(thm)],[149, 147])).
% 74.75/48.43 tff(151,plain,
% 74.75/48.43 ((~![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(x, multiplicative_identity, add(multiplicative_identity, x))) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_identity, x))))),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(152,plain,
% 74.75/48.43 ((~![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(multiplicative_identity, additive_identity, additive_identity)) | (~product(x, additive_identity, multiply(additive_identity, x))) | (~sum(x, multiplicative_identity, add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, x), additive_identity, multiply(additive_identity, x)))),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[151, 150])).
% 74.75/48.43 tff(153,plain,
% 74.75/48.43 (product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x))),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[152, 146, 136, 127, 121, 110])).
% 74.75/48.43 tff(154,plain,
% 74.75/48.43 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(155,plain,
% 74.75/48.43 (sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[154, 86])).
% 74.75/48.43 tff(156,plain,
% 74.75/48.43 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(157,plain,
% 74.75/48.43 (product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[156, 64])).
% 74.75/48.43 tff(158,plain,
% 74.75/48.43 ((~product(x, additive_identity, additive_identity)) <=> (~product(x, additive_identity, additive_identity))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(159,axiom,(~product(x, additive_identity, additive_identity)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_equations')).
% 74.75/48.43 tff(160,plain,
% 74.75/48.43 (~product(x, additive_identity, additive_identity)),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[159, 158])).
% 74.75/48.43 tff(161,plain,
% 74.75/48.43 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(additive_identity, x, additive_identity)) | product(x, additive_identity, additive_identity))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(additive_identity, x, additive_identity)) | product(x, additive_identity, additive_identity))),
% 74.75/48.43 inference(rewrite,[status(thm)],[])).
% 74.75/48.43 tff(162,plain,
% 74.75/48.43 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(additive_identity, x, additive_identity)) | product(x, additive_identity, additive_identity))),
% 74.75/48.43 inference(quant_inst,[status(thm)],[])).
% 74.75/48.43 tff(163,plain,
% 74.75/48.43 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(additive_identity, x, additive_identity)) | product(x, additive_identity, additive_identity)),
% 74.75/48.43 inference(modus_ponens,[status(thm)],[162, 161])).
% 74.75/48.43 tff(164,plain,
% 74.75/48.43 (~product(additive_identity, x, additive_identity)),
% 74.75/48.43 inference(unit_resolution,[status(thm)],[163, 117, 160])).
% 74.75/48.43 tff(165,plain,
% 74.75/48.43 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))))),
% 74.75/48.43 inference(bind,[status(th)],[])).
% 74.75/48.43 tff(166,plain,
% 74.75/48.43 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 74.75/48.44 inference(quant_intro,[status(thm)],[165])).
% 74.75/48.44 tff(167,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(168,plain,
% 74.75/48.44 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) | product(X, V3, V4)))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))))),
% 74.75/48.44 inference(bind,[status(th)],[])).
% 74.75/48.44 tff(169,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 74.75/48.44 inference(quant_intro,[status(thm)],[168])).
% 74.75/48.44 tff(170,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~sum(V1, V2, V4))) | product(X, V3, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity2')).
% 74.75/48.44 tff(171,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[170, 169])).
% 74.75/48.44 tff(172,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[171, 167])).
% 74.75/48.44 tff(173,plain,(
% 74.75/48.44 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 74.75/48.44 inference(skolemize,[status(sab)],[172])).
% 74.75/48.44 tff(174,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[173, 166])).
% 74.75/48.44 tff(175,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(additive_identity, x, additive_identity) | (~sum(x, additive_identity, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | product(additive_identity, x, additive_identity) | (~sum(x, additive_identity, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity)))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(176,plain,
% 74.75/48.44 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(additive_identity, x, additive_identity) | (~sum(x, additive_identity, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity)))),
% 74.75/48.44 inference(quant_inst,[status(thm)],[])).
% 74.75/48.44 tff(177,plain,
% 74.75/48.44 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | product(additive_identity, x, additive_identity) | (~sum(x, additive_identity, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[176, 175])).
% 74.75/48.44 tff(178,plain,
% 74.75/48.44 (~sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity)),
% 74.75/48.44 inference(unit_resolution,[status(thm)],[177, 174, 164, 157, 68, 66])).
% 74.75/48.44 tff(179,plain,
% 74.75/48.44 (^[X: $i] : refl(product(X, multiplicative_identity, X) <=> product(X, multiplicative_identity, X))),
% 74.75/48.44 inference(bind,[status(th)],[])).
% 74.75/48.44 tff(180,plain,
% 74.75/48.44 (![X: $i] : product(X, multiplicative_identity, X) <=> ![X: $i] : product(X, multiplicative_identity, X)),
% 74.75/48.44 inference(quant_intro,[status(thm)],[179])).
% 74.75/48.44 tff(181,plain,
% 74.75/48.44 (![X: $i] : product(X, multiplicative_identity, X) <=> ![X: $i] : product(X, multiplicative_identity, X)),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(182,axiom,(![X: $i] : product(X, multiplicative_identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_identity2')).
% 74.75/48.44 tff(183,plain,
% 74.75/48.44 (![X: $i] : product(X, multiplicative_identity, X)),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[182, 181])).
% 74.75/48.44 tff(184,plain,(
% 74.75/48.44 ![X: $i] : product(X, multiplicative_identity, X)),
% 74.75/48.44 inference(skolemize,[status(sab)],[183])).
% 74.75/48.44 tff(185,plain,
% 74.75/48.44 (![X: $i] : product(X, multiplicative_identity, X)),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[184, 180])).
% 74.75/48.44 tff(186,plain,
% 74.75/48.44 ((~![X: $i] : product(X, multiplicative_identity, X)) | product(additive_identity, multiplicative_identity, additive_identity)),
% 74.75/48.44 inference(quant_inst,[status(thm)],[])).
% 74.75/48.44 tff(187,plain,
% 74.75/48.44 (product(additive_identity, multiplicative_identity, additive_identity)),
% 74.75/48.44 inference(unit_resolution,[status(thm)],[186, 185])).
% 74.75/48.44 tff(188,plain,
% 74.75/48.44 ((~![X: $i] : sum(X, additive_identity, X)) | sum(multiplicative_identity, additive_identity, multiplicative_identity)),
% 74.75/48.44 inference(quant_inst,[status(thm)],[])).
% 74.75/48.44 tff(189,plain,
% 74.75/48.44 (sum(multiplicative_identity, additive_identity, multiplicative_identity)),
% 74.75/48.44 inference(unit_resolution,[status(thm)],[188, 7])).
% 74.75/48.44 tff(190,plain,
% 74.75/48.44 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))))),
% 74.75/48.44 inference(bind,[status(th)],[])).
% 74.75/48.44 tff(191,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 74.75/48.44 inference(quant_intro,[status(thm)],[190])).
% 74.75/48.44 tff(192,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(193,plain,
% 74.75/48.44 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) <=> (((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~product(X, V3, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | (~product(X, V3, V4))) <=> ((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) <=> ((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> (((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | sum(V1, V2, V4)))), rewrite((((~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))), ((((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))))),
% 74.75/48.44 inference(bind,[status(th)],[])).
% 74.75/48.44 tff(194,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 74.75/48.44 inference(quant_intro,[status(thm)],[193])).
% 74.75/48.44 tff(195,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(X, Y, V1)) | (~product(X, Z, V2))) | (~sum(Y, Z, V3))) | (~product(X, V3, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity1')).
% 74.75/48.44 tff(196,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[195, 194])).
% 74.75/48.44 tff(197,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[196, 192])).
% 74.75/48.44 tff(198,plain,(
% 74.75/48.44 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 74.75/48.44 inference(skolemize,[status(sab)],[197])).
% 74.75/48.44 tff(199,plain,
% 74.75/48.44 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[198, 191])).
% 74.75/48.44 tff(200,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x))))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(201,plain,
% 74.75/48.44 ((sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x)))) <=> ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x))))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(202,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x)))))),
% 74.75/48.44 inference(monotonicity,[status(thm)],[201])).
% 74.75/48.44 tff(203,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x))))),
% 74.75/48.44 inference(transitivity,[status(thm)],[202, 200])).
% 74.75/48.44 tff(204,plain,
% 74.75/48.44 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x))))),
% 74.75/48.44 inference(quant_inst,[status(thm)],[])).
% 74.75/48.44 tff(205,plain,
% 74.75/48.44 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(multiply(additive_identity, x), multiply(additive_identity, additive_identity), additive_identity) | (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x)))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[204, 203])).
% 74.75/48.44 tff(206,plain,
% 74.75/48.44 (~product(additive_identity, multiplicative_identity, multiply(additive_identity, x))),
% 74.75/48.44 inference(unit_resolution,[status(thm)],[205, 199, 157, 189, 187, 178])).
% 74.75/48.44 tff(207,plain,
% 74.75/48.44 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, multiplicative_identity, multiplicative_identity)),
% 74.75/48.44 inference(quant_inst,[status(thm)],[])).
% 74.75/48.44 tff(208,plain,
% 74.75/48.44 (sum(additive_identity, multiplicative_identity, multiplicative_identity)),
% 74.75/48.44 inference(unit_resolution,[status(thm)],[207, 86])).
% 74.75/48.44 tff(209,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(210,plain,
% 74.75/48.44 ((product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)))) <=> ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(211,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)))))),
% 74.75/48.44 inference(monotonicity,[status(thm)],[210])).
% 74.75/48.44 tff(212,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.44 inference(transitivity,[status(thm)],[211, 209])).
% 74.75/48.44 tff(213,plain,
% 74.75/48.44 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.44 inference(quant_inst,[status(thm)],[])).
% 74.75/48.44 tff(214,plain,
% 74.75/48.44 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiplicative_identity, multiplicative_identity)) | (~product(additive_identity, multiplicative_identity, additive_identity)) | product(additive_identity, multiplicative_identity, multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[213, 212])).
% 74.75/48.44 tff(215,plain,
% 74.75/48.44 (~sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x))),
% 74.75/48.44 inference(unit_resolution,[status(thm)],[214, 174, 157, 208, 187, 206])).
% 74.75/48.44 tff(216,plain,
% 74.75/48.44 (^[X: $i] : refl(product(inverse(X), X, additive_identity) <=> product(inverse(X), X, additive_identity))),
% 74.75/48.44 inference(bind,[status(th)],[])).
% 74.75/48.44 tff(217,plain,
% 74.75/48.44 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 74.75/48.44 inference(quant_intro,[status(thm)],[216])).
% 74.75/48.44 tff(218,plain,
% 74.75/48.44 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(219,axiom,(![X: $i] : product(inverse(X), X, additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse1')).
% 74.75/48.44 tff(220,plain,
% 74.75/48.44 (![X: $i] : product(inverse(X), X, additive_identity)),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[219, 218])).
% 74.75/48.44 tff(221,plain,(
% 74.75/48.44 ![X: $i] : product(inverse(X), X, additive_identity)),
% 74.75/48.44 inference(skolemize,[status(sab)],[220])).
% 74.75/48.44 tff(222,plain,
% 74.75/48.44 (![X: $i] : product(inverse(X), X, additive_identity)),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[221, 217])).
% 74.75/48.44 tff(223,plain,
% 74.75/48.44 ((~![X: $i] : product(inverse(X), X, additive_identity)) | product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)),
% 74.75/48.44 inference(quant_inst,[status(thm)],[])).
% 74.75/48.44 tff(224,plain,
% 74.75/48.44 (product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)),
% 74.75/48.44 inference(unit_resolution,[status(thm)],[223, 222])).
% 74.75/48.44 tff(225,plain,
% 74.75/48.44 (^[X: $i] : refl(sum(X, inverse(X), multiplicative_identity) <=> sum(X, inverse(X), multiplicative_identity))),
% 74.75/48.44 inference(bind,[status(th)],[])).
% 74.75/48.44 tff(226,plain,
% 74.75/48.44 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 74.75/48.44 inference(quant_intro,[status(thm)],[225])).
% 74.75/48.44 tff(227,plain,
% 74.75/48.44 (![X: $i] : sum(X, inverse(X), multiplicative_identity) <=> ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(228,axiom,(![X: $i] : sum(X, inverse(X), multiplicative_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','additive_inverse2')).
% 74.75/48.44 tff(229,plain,
% 74.75/48.44 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[228, 227])).
% 74.75/48.44 tff(230,plain,(
% 74.75/48.44 ![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 74.75/48.44 inference(skolemize,[status(sab)],[229])).
% 74.75/48.44 tff(231,plain,
% 74.75/48.44 (![X: $i] : sum(X, inverse(X), multiplicative_identity)),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[230, 226])).
% 74.75/48.44 tff(232,plain,
% 74.75/48.44 ((~![X: $i] : sum(X, inverse(X), multiplicative_identity)) | sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)),
% 74.75/48.44 inference(quant_inst,[status(thm)],[])).
% 74.75/48.44 tff(233,plain,
% 74.75/48.44 (sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)),
% 74.75/48.44 inference(unit_resolution,[status(thm)],[232, 231])).
% 74.75/48.44 tff(234,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(235,plain,
% 74.75/48.44 ((sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity)))) <=> ((~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(236,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity)))))),
% 74.75/48.44 inference(monotonicity,[status(thm)],[235])).
% 74.75/48.44 tff(237,plain,
% 74.75/48.44 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))))),
% 74.75/48.44 inference(transitivity,[status(thm)],[236, 234])).
% 74.75/48.44 tff(238,plain,
% 74.75/48.44 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))))),
% 74.75/48.44 inference(quant_inst,[status(thm)],[])).
% 74.75/48.44 tff(239,plain,
% 74.75/48.44 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~product(X, V3, V4)) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity)))),
% 74.75/48.44 inference(modus_ponens,[status(thm)],[238, 237])).
% 74.75/48.44 tff(240,plain,
% 74.75/48.44 (sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))),
% 74.75/48.44 inference(unit_resolution,[status(thm)],[239, 199, 157, 88])).
% 74.75/48.44 tff(241,plain,
% 74.75/48.44 (((~![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)))) | ((~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_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)))) | (~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x))))),
% 74.75/48.44 inference(rewrite,[status(thm)],[])).
% 74.75/48.44 tff(242,plain,
% 74.75/48.44 ((sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)))) <=> ((~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x))))),
% 74.75/48.45 inference(rewrite,[status(thm)],[])).
% 74.75/48.45 tff(243,plain,
% 74.75/48.45 (((~![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(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_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)))) | ((~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)))))),
% 74.75/48.45 inference(monotonicity,[status(thm)],[242])).
% 74.75/48.45 tff(244,plain,
% 74.75/48.45 (((~![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(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_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)))) | (~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x))))),
% 74.75/48.45 inference(transitivity,[status(thm)],[243, 241])).
% 74.75/48.45 tff(245,plain,
% 74.75/48.45 ((~![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(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x))))),
% 74.75/48.45 inference(quant_inst,[status(thm)],[])).
% 74.75/48.45 tff(246,plain,
% 74.75/48.45 ((~![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)))) | (~product(inverse(multiply(additive_identity, additive_identity)), multiply(additive_identity, additive_identity), additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), inverse(multiply(additive_identity, additive_identity)), multiplicative_identity)) | sum(multiply(additive_identity, additive_identity), additive_identity, multiply(additive_identity, x)) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)))),
% 74.75/48.45 inference(modus_ponens,[status(thm)],[245, 244])).
% 74.75/48.45 tff(247,plain,
% 74.75/48.45 (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x))),
% 74.75/48.45 inference(unit_resolution,[status(thm)],[246, 98, 240, 233, 224, 215])).
% 74.75/48.45 tff(248,plain,
% 74.75/48.45 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))),
% 74.75/48.45 inference(quant_inst,[status(thm)],[])).
% 74.75/48.45 tff(249,plain,
% 74.75/48.45 (product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))),
% 74.75/48.45 inference(unit_resolution,[status(thm)],[248, 134])).
% 74.75/48.45 tff(250,plain,
% 74.75/48.45 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))))),
% 74.75/48.45 inference(rewrite,[status(thm)],[])).
% 74.75/48.45 tff(251,plain,
% 74.75/48.45 ((product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)))) <=> ((~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))))),
% 74.75/48.45 inference(rewrite,[status(thm)],[])).
% 74.75/48.45 tff(252,plain,
% 74.75/48.45 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)))))),
% 74.75/48.45 inference(monotonicity,[status(thm)],[251])).
% 74.75/48.45 tff(253,plain,
% 74.75/48.45 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))))),
% 74.75/48.45 inference(transitivity,[status(thm)],[252, 250])).
% 74.75/48.45 tff(254,plain,
% 74.75/48.45 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))))),
% 74.75/48.45 inference(quant_inst,[status(thm)],[])).
% 74.75/48.45 tff(255,plain,
% 74.75/48.45 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | product(multiplicative_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)))),
% 74.75/48.45 inference(modus_ponens,[status(thm)],[254, 253])).
% 74.75/48.45 tff(256,plain,
% 74.75/48.45 (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))),
% 74.75/48.45 inference(unit_resolution,[status(thm)],[255, 174, 249, 240, 247])).
% 74.75/48.45 tff(257,plain,
% 74.75/48.45 (^[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))))),
% 74.75/48.45 inference(bind,[status(th)],[])).
% 74.75/48.45 tff(258,plain,
% 74.75/48.45 (![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)))),
% 74.75/48.45 inference(quant_intro,[status(thm)],[257])).
% 74.75/48.45 tff(259,plain,
% 74.75/48.45 (![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)))),
% 74.75/48.45 inference(rewrite,[status(thm)],[])).
% 74.75/48.45 tff(260,plain,
% 74.75/48.45 (^[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)))))),
% 74.75/48.45 inference(bind,[status(th)],[])).
% 74.75/48.45 tff(261,plain,
% 74.75/48.45 (![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)))),
% 74.75/48.45 inference(quant_intro,[status(thm)],[260])).
% 74.75/48.45 tff(262,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity3')).
% 74.75/48.45 tff(263,plain,
% 74.75/48.45 (![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)))),
% 74.75/48.45 inference(modus_ponens,[status(thm)],[262, 261])).
% 74.75/48.45 tff(264,plain,
% 74.75/48.45 (![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)))),
% 74.75/48.45 inference(modus_ponens,[status(thm)],[263, 259])).
% 74.75/48.45 tff(265,plain,(
% 74.75/48.45 ![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)))),
% 74.75/48.45 inference(skolemize,[status(sab)],[264])).
% 74.75/48.45 tff(266,plain,
% 74.75/48.45 (![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)))),
% 74.75/48.45 inference(modus_ponens,[status(thm)],[265, 258])).
% 74.75/48.45 tff(267,plain,
% 74.75/48.45 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)))),
% 74.75/48.45 inference(rewrite,[status(thm)],[])).
% 74.75/48.45 tff(268,plain,
% 74.75/48.45 ((sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity)))) <=> ((~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)))),
% 74.75/48.45 inference(rewrite,[status(thm)],[])).
% 74.75/48.45 tff(269,plain,
% 74.75/48.45 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | ((~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))))),
% 74.75/48.45 inference(monotonicity,[status(thm)],[268])).
% 74.75/48.45 tff(270,plain,
% 74.75/48.45 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)))),
% 74.75/48.45 inference(transitivity,[status(thm)],[269, 267])).
% 74.75/48.45 tff(271,plain,
% 74.75/48.45 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))))),
% 74.75/48.45 inference(quant_inst,[status(thm)],[])).
% 74.75/48.45 tff(272,plain,
% 74.75/48.45 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, x, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity), multiply(additive_identity, x))),
% 74.75/48.45 inference(modus_ponens,[status(thm)],[271, 270])).
% 74.75/48.45 tff(273,plain,
% 74.75/48.45 (~product(additive_identity, x, multiply(additive_identity, additive_identity))),
% 74.75/48.45 inference(unit_resolution,[status(thm)],[272, 266, 88, 66, 256])).
% 74.75/48.45 tff(274,plain,
% 74.75/48.45 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))))),
% 74.75/48.45 inference(rewrite,[status(thm)],[])).
% 74.75/48.45 tff(275,plain,
% 74.75/48.45 ((product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)))) <=> ((~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))))),
% 74.75/48.45 inference(rewrite,[status(thm)],[])).
% 74.75/48.45 tff(276,plain,
% 74.75/48.45 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | ((~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)))))),
% 74.75/48.45 inference(monotonicity,[status(thm)],[275])).
% 74.75/48.45 tff(277,plain,
% 74.75/48.45 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))))),
% 74.75/48.45 inference(transitivity,[status(thm)],[276, 274])).
% 74.75/48.45 tff(278,plain,
% 74.75/48.45 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~product(additive_identity, x, multiply(additive_identity, x))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))))),
% 74.75/48.45 inference(quant_inst,[status(thm)],[])).
% 74.75/48.45 tff(279,plain,
% 74.75/48.45 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(X, V3, V4) | (~sum(Y, Z, V3)) | (~product(X, Z, V2)) | (~product(X, Y, V1)) | (~sum(V1, V2, V4)))) | (~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, x, multiply(additive_identity, x))) | product(additive_identity, x, multiply(additive_identity, additive_identity)) | (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)))),
% 74.75/48.45 inference(modus_ponens,[status(thm)],[278, 277])).
% 74.75/48.45 tff(280,plain,
% 74.75/48.45 (~sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))),
% 74.75/48.45 inference(unit_resolution,[status(thm)],[279, 174, 157, 79, 66, 273])).
% 74.75/48.45 tff(281,plain,
% 74.75/48.45 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl(((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))),
% 74.75/48.45 inference(bind,[status(th)],[])).
% 74.75/48.45 tff(282,plain,
% 74.75/48.45 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.45 inference(quant_intro,[status(thm)],[281])).
% 74.75/48.45 tff(283,plain,
% 74.75/48.45 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.45 inference(rewrite,[status(thm)],[])).
% 74.75/48.45 tff(284,plain,
% 74.75/48.45 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) <=> ((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) <=> (((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~product(V1, V2, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4)) <=> (((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | sum(V3, X, V4)))), rewrite((((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | sum(V3, X, V4)) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4)) <=> ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))))),
% 74.75/48.46 inference(bind,[status(th)],[])).
% 74.75/48.46 tff(285,plain,
% 74.75/48.46 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.46 inference(quant_intro,[status(thm)],[284])).
% 74.75/48.46 tff(286,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~product(V1, V2, V4))) | sum(V3, X, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity8')).
% 74.75/48.46 tff(287,plain,
% 74.75/48.46 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[286, 285])).
% 74.75/48.46 tff(288,plain,
% 74.75/48.46 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[287, 283])).
% 74.75/48.46 tff(289,plain,(
% 74.75/48.46 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.46 inference(skolemize,[status(sab)],[288])).
% 74.75/48.46 tff(290,plain,
% 74.75/48.46 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[289, 282])).
% 74.75/48.46 tff(291,plain,
% 74.75/48.46 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(292,plain,
% 74.75/48.46 (((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity))) <=> ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(293,plain,
% 74.75/48.46 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))))),
% 74.75/48.46 inference(monotonicity,[status(thm)],[292])).
% 74.75/48.46 tff(294,plain,
% 74.75/48.46 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)))),
% 74.75/48.46 inference(transitivity,[status(thm)],[293, 291])).
% 74.75/48.46 tff(295,plain,
% 74.75/48.46 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity)) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)))),
% 74.75/48.46 inference(quant_inst,[status(thm)],[])).
% 74.75/48.46 tff(296,plain,
% 74.75/48.46 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, x), additive_identity)) | sum(multiply(additive_identity, additive_identity), multiply(additive_identity, x), multiply(additive_identity, additive_identity))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[295, 294])).
% 74.75/48.46 tff(297,plain,
% 74.75/48.46 (~sum(additive_identity, multiply(additive_identity, x), additive_identity)),
% 74.75/48.46 inference(unit_resolution,[status(thm)],[296, 290, 157, 280])).
% 74.75/48.46 tff(298,plain,
% 74.75/48.46 (((~![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(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, multiply(additive_identity, x), additive_identity) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_identity)) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_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(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, multiply(additive_identity, x), additive_identity) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_identity)) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(299,plain,
% 74.75/48.46 ((sum(additive_identity, multiply(additive_identity, x), additive_identity) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_identity))) <=> ((~sum(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, multiply(additive_identity, x), additive_identity) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_identity)) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(300,plain,
% 74.75/48.46 (((~![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(additive_identity, multiply(additive_identity, x), additive_identity) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_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(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, multiply(additive_identity, x), additive_identity) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_identity)) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)))))),
% 74.75/48.46 inference(monotonicity,[status(thm)],[299])).
% 74.75/48.46 tff(301,plain,
% 74.75/48.46 (((~![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(additive_identity, multiply(additive_identity, x), additive_identity) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_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(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, multiply(additive_identity, x), additive_identity) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_identity)) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x))))),
% 74.75/48.46 inference(transitivity,[status(thm)],[300, 298])).
% 74.75/48.46 tff(302,plain,
% 74.75/48.46 ((~![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(additive_identity, multiply(additive_identity, x), additive_identity) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x))) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_identity)))),
% 74.75/48.46 inference(quant_inst,[status(thm)],[])).
% 74.75/48.46 tff(303,plain,
% 74.75/48.46 ((~![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(additive_identity, additive_identity, additive_identity)) | sum(additive_identity, multiply(additive_identity, x), additive_identity) | (~sum(additive_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))) | (~product(add(multiplicative_identity, x), additive_identity, additive_identity)) | (~product(add(multiplicative_identity, x), additive_identity, multiply(additive_identity, x)))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[302, 301])).
% 74.75/48.46 tff(304,plain,
% 74.75/48.46 (~product(add(multiplicative_identity, x), additive_identity, additive_identity)),
% 74.75/48.46 inference(unit_resolution,[status(thm)],[303, 98, 88, 297, 155, 153])).
% 74.75/48.46 tff(305,plain,
% 74.75/48.46 (^[X: $i] : refl(product(X, inverse(X), additive_identity) <=> product(X, inverse(X), additive_identity))),
% 74.75/48.46 inference(bind,[status(th)],[])).
% 74.75/48.46 tff(306,plain,
% 74.75/48.46 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 74.75/48.46 inference(quant_intro,[status(thm)],[305])).
% 74.75/48.46 tff(307,plain,
% 74.75/48.46 (![X: $i] : product(X, inverse(X), additive_identity) <=> ![X: $i] : product(X, inverse(X), additive_identity)),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(308,axiom,(![X: $i] : product(X, inverse(X), additive_identity)), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse2')).
% 74.75/48.46 tff(309,plain,
% 74.75/48.46 (![X: $i] : product(X, inverse(X), additive_identity)),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[308, 307])).
% 74.75/48.46 tff(310,plain,(
% 74.75/48.46 ![X: $i] : product(X, inverse(X), additive_identity)),
% 74.75/48.46 inference(skolemize,[status(sab)],[309])).
% 74.75/48.46 tff(311,plain,
% 74.75/48.46 (![X: $i] : product(X, inverse(X), additive_identity)),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[310, 306])).
% 74.75/48.46 tff(312,plain,
% 74.75/48.46 ((~![X: $i] : product(X, inverse(X), additive_identity)) | product(additive_identity, inverse(additive_identity), additive_identity)),
% 74.75/48.46 inference(quant_inst,[status(thm)],[])).
% 74.75/48.46 tff(313,plain,
% 74.75/48.46 (product(additive_identity, inverse(additive_identity), additive_identity)),
% 74.75/48.46 inference(unit_resolution,[status(thm)],[312, 311])).
% 74.75/48.46 tff(314,plain,
% 74.75/48.46 (((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(additive_identity, inverse(additive_identity), additive_identity)) | product(inverse(additive_identity), additive_identity, additive_identity))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(additive_identity, inverse(additive_identity), additive_identity)) | product(inverse(additive_identity), additive_identity, additive_identity))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(315,plain,
% 74.75/48.46 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | ((~product(additive_identity, inverse(additive_identity), additive_identity)) | product(inverse(additive_identity), additive_identity, additive_identity))),
% 74.75/48.46 inference(quant_inst,[status(thm)],[])).
% 74.75/48.46 tff(316,plain,
% 74.75/48.46 ((~![Z: $i, Y: $i, X: $i] : ((~product(X, Y, Z)) | product(Y, X, Z))) | (~product(additive_identity, inverse(additive_identity), additive_identity)) | product(inverse(additive_identity), additive_identity, additive_identity)),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[315, 314])).
% 74.75/48.46 tff(317,plain,
% 74.75/48.46 (product(inverse(additive_identity), additive_identity, additive_identity)),
% 74.75/48.46 inference(unit_resolution,[status(thm)],[316, 117, 313])).
% 74.75/48.46 tff(318,plain,
% 74.75/48.46 (^[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))))),
% 74.75/48.46 inference(bind,[status(th)],[])).
% 74.75/48.46 tff(319,plain,
% 74.75/48.46 (![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)))),
% 74.75/48.46 inference(quant_intro,[status(thm)],[318])).
% 74.75/48.46 tff(320,plain,
% 74.75/48.46 (![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)))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(321,plain,
% 74.75/48.46 (^[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)))))),
% 74.75/48.46 inference(bind,[status(th)],[])).
% 74.75/48.46 tff(322,plain,
% 74.75/48.46 (![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)))),
% 74.75/48.46 inference(quant_intro,[status(thm)],[321])).
% 74.75/48.46 tff(323,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')).
% 74.75/48.46 tff(324,plain,
% 74.75/48.46 (![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)))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[323, 322])).
% 74.75/48.46 tff(325,plain,
% 74.75/48.46 (![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)))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[324, 320])).
% 74.75/48.46 tff(326,plain,(
% 74.75/48.46 ![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)))),
% 74.75/48.46 inference(skolemize,[status(sab)],[325])).
% 74.75/48.46 tff(327,plain,
% 74.75/48.46 (![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)))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[326, 319])).
% 74.75/48.46 tff(328,plain,
% 74.75/48.46 (((~![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(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, additive_identity))) <=> ((~![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(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, additive_identity))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(329,plain,
% 74.75/48.46 ((product(add(multiplicative_identity, x), additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)))) <=> ((~sum(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, additive_identity))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(330,plain,
% 74.75/48.46 (((~![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), additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_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(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, additive_identity)))),
% 74.75/48.46 inference(monotonicity,[status(thm)],[329])).
% 74.75/48.46 tff(331,plain,
% 74.75/48.46 (((~![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), additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_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(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, additive_identity))),
% 74.75/48.46 inference(transitivity,[status(thm)],[330, 328])).
% 74.75/48.46 tff(332,plain,
% 74.75/48.46 ((~![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), additive_identity, additive_identity) | (~sum(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x))))),
% 74.75/48.46 inference(quant_inst,[status(thm)],[])).
% 74.75/48.46 tff(333,plain,
% 74.75/48.46 ((~![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(additive_identity, additive_identity, additive_identity)) | (~product(inverse(additive_identity), additive_identity, additive_identity)) | (~sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x))) | product(add(multiplicative_identity, x), additive_identity, additive_identity)),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[332, 331])).
% 74.75/48.46 tff(334,plain,
% 74.75/48.46 (~sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x))),
% 74.75/48.46 inference(unit_resolution,[status(thm)],[333, 327, 88, 317, 304])).
% 74.75/48.46 tff(335,plain,
% 74.75/48.46 (inverse(additive_identity) = add(inverse(additive_identity), additive_identity)),
% 74.75/48.46 inference(symmetry,[status(thm)],[35])).
% 74.75/48.46 tff(336,plain,
% 74.75/48.46 (inverse(additive_identity) = multiplicative_identity),
% 74.75/48.46 inference(transitivity,[status(thm)],[335, 51])).
% 74.75/48.46 tff(337,plain,
% 74.75/48.46 (product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x)) <=> product(multiplicative_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))),
% 74.75/48.46 inference(monotonicity,[status(thm)],[336])).
% 74.75/48.46 tff(338,plain,
% 74.75/48.46 (product(multiplicative_identity, add(multiplicative_identity, x), add(multiplicative_identity, x)) <=> product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))),
% 74.75/48.46 inference(symmetry,[status(thm)],[337])).
% 74.75/48.46 tff(339,plain,
% 74.75/48.46 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))),
% 74.75/48.46 inference(quant_inst,[status(thm)],[])).
% 74.75/48.46 tff(340,plain,
% 74.75/48.46 (product(multiplicative_identity, add(multiplicative_identity, x), add(multiplicative_identity, x))),
% 74.75/48.46 inference(unit_resolution,[status(thm)],[339, 134])).
% 74.75/48.46 tff(341,plain,
% 74.75/48.46 (product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[340, 338])).
% 74.75/48.46 tff(342,plain,
% 74.75/48.46 (sum(x, multiplicative_identity, add(multiplicative_identity, x)) <=> sum(x, inverse(additive_identity), add(multiplicative_identity, x))),
% 74.75/48.46 inference(monotonicity,[status(thm)],[53])).
% 74.75/48.46 tff(343,plain,
% 74.75/48.46 (sum(x, inverse(additive_identity), add(multiplicative_identity, x))),
% 74.75/48.46 inference(modus_ponens,[status(thm)],[127, 342])).
% 74.75/48.46 tff(344,plain,
% 74.75/48.46 ((~![X: $i] : product(inverse(X), X, additive_identity)) | product(inverse(x), x, additive_identity)),
% 74.75/48.46 inference(quant_inst,[status(thm)],[])).
% 74.75/48.46 tff(345,plain,
% 74.75/48.46 (product(inverse(x), x, additive_identity)),
% 74.75/48.46 inference(unit_resolution,[status(thm)],[344, 222])).
% 74.75/48.46 tff(346,plain,
% 74.75/48.46 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(x), x, additive_identity)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(inverse(x), x, additive_identity)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(347,plain,
% 74.75/48.46 (((~product(inverse(x), x, additive_identity)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity)))) <=> ((~product(inverse(x), x, additive_identity)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))))),
% 74.75/48.46 inference(rewrite,[status(thm)],[])).
% 74.75/48.46 tff(348,plain,
% 74.75/48.46 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(x), x, additive_identity)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(x), x, additive_identity)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x)))))),
% 74.75/48.46 inference(monotonicity,[status(thm)],[347])).
% 74.75/48.46 tff(349,plain,
% 74.75/48.46 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(x), x, additive_identity)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(inverse(x), x, additive_identity)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))))),
% 74.75/48.47 inference(transitivity,[status(thm)],[348, 346])).
% 74.75/48.47 tff(350,plain,
% 74.75/48.47 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~product(inverse(x), x, additive_identity)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))))),
% 74.75/48.47 inference(quant_inst,[status(thm)],[])).
% 74.75/48.47 tff(351,plain,
% 74.75/48.47 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : ((~product(Y, Z, V3)) | (~product(V1, V2, V4)) | sum(V3, X, V4) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~product(inverse(x), x, additive_identity)) | (~sum(x, inverse(additive_identity), add(multiplicative_identity, x))) | (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))) | sum(additive_identity, inverse(additive_identity), add(multiplicative_identity, x)) | (~product(inverse(additive_identity), add(multiplicative_identity, x), add(multiplicative_identity, x)))),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[350, 349])).
% 74.75/48.47 tff(352,plain,
% 74.75/48.47 (~sum(inverse(x), inverse(additive_identity), inverse(additive_identity))),
% 74.75/48.47 inference(unit_resolution,[status(thm)],[351, 290, 345, 343, 341, 334])).
% 74.75/48.47 tff(353,plain,
% 74.75/48.47 (product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity)) <=> product(multiplicative_identity, multiplicative_identity, multiplicative_identity)),
% 74.75/48.47 inference(monotonicity,[status(thm)],[336, 336, 336])).
% 74.75/48.47 tff(354,plain,
% 74.75/48.47 (product(multiplicative_identity, multiplicative_identity, multiplicative_identity) <=> product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))),
% 74.75/48.47 inference(symmetry,[status(thm)],[353])).
% 74.75/48.47 tff(355,plain,
% 74.75/48.47 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, multiplicative_identity, multiplicative_identity)),
% 74.75/48.47 inference(quant_inst,[status(thm)],[])).
% 74.75/48.47 tff(356,plain,
% 74.75/48.47 (product(multiplicative_identity, multiplicative_identity, multiplicative_identity)),
% 74.75/48.47 inference(unit_resolution,[status(thm)],[355, 134])).
% 74.75/48.47 tff(357,plain,
% 74.75/48.47 (product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[356, 354])).
% 74.75/48.47 tff(358,plain,
% 74.75/48.47 ((~![Y: $i, X: $i] : sum(X, Y, add(X, Y))) | sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))),
% 74.75/48.47 inference(quant_inst,[status(thm)],[])).
% 74.75/48.47 tff(359,plain,
% 74.75/48.47 (sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))),
% 74.75/48.47 inference(unit_resolution,[status(thm)],[358, 16])).
% 74.75/48.47 tff(360,plain,
% 74.75/48.47 (((~![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(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_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(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity))))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(361,plain,
% 74.75/48.47 ((sum(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)))) <=> (sum(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity))))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(362,plain,
% 74.75/48.47 (((~![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(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_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(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)))))),
% 74.75/48.47 inference(monotonicity,[status(thm)],[361])).
% 74.75/48.47 tff(363,plain,
% 74.75/48.47 (((~![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(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_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(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity))))),
% 74.75/48.47 inference(transitivity,[status(thm)],[362, 360])).
% 74.75/48.47 tff(364,plain,
% 74.75/48.47 ((~![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(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity))))),
% 74.75/48.47 inference(quant_inst,[status(thm)],[])).
% 74.75/48.47 tff(365,plain,
% 74.75/48.47 ((~![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(inverse(x), inverse(additive_identity), inverse(additive_identity)) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(inverse(additive_identity), inverse(additive_identity), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)))),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[364, 363])).
% 74.75/48.47 tff(366,plain,
% 74.75/48.47 (~product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity))),
% 74.75/48.47 inference(unit_resolution,[status(thm)],[365, 98, 359, 357, 352])).
% 74.75/48.47 tff(367,assumption,(~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)), introduced(assumption)).
% 74.75/48.47 tff(368,plain,
% 74.75/48.47 (((~![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)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(additive_identity, multiply(additive_identity, additive_identity), 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)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(369,plain,
% 74.75/48.47 ((sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, additive_identity))) <=> ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(370,plain,
% 74.75/48.47 (((~![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, multiply(additive_identity, additive_identity), additive_identity) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, 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)))) | ((~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)))),
% 74.75/48.47 inference(monotonicity,[status(thm)],[369])).
% 74.75/48.47 tff(371,plain,
% 74.75/48.47 (((~![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, multiply(additive_identity, additive_identity), additive_identity) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, 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)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity))),
% 74.75/48.47 inference(transitivity,[status(thm)],[370, 368])).
% 74.75/48.47 tff(372,plain,
% 74.75/48.47 ((~![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, multiply(additive_identity, additive_identity), additive_identity) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, additive_identity)))),
% 74.75/48.47 inference(quant_inst,[status(thm)],[])).
% 74.75/48.47 tff(373,plain,
% 74.75/48.47 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(additive_identity, additive_identity, multiply(additive_identity, additive_identity))) | (~product(multiplicative_identity, additive_identity, additive_identity)) | (~sum(multiplicative_identity, additive_identity, multiplicative_identity)) | sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[372, 371])).
% 74.75/48.47 tff(374,plain,
% 74.75/48.47 ($false),
% 74.75/48.47 inference(unit_resolution,[status(thm)],[373, 266, 157, 189, 136, 367])).
% 74.75/48.47 tff(375,plain,(sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)), inference(lemma,lemma(discharge,[]))).
% 74.75/48.47 tff(376,plain,
% 74.75/48.47 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))),
% 74.75/48.47 inference(quant_inst,[status(thm)],[])).
% 74.75/48.47 tff(377,plain,
% 74.75/48.47 (sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))),
% 74.75/48.47 inference(unit_resolution,[status(thm)],[376, 86])).
% 74.75/48.47 tff(378,plain,
% 74.75/48.47 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)) | (additive_identity = multiply(additive_identity, additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)) | (additive_identity = multiply(additive_identity, additive_identity)))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(379,plain,
% 74.75/48.47 (((additive_identity = multiply(additive_identity, additive_identity)) | (~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity))) <=> ((~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)) | (additive_identity = multiply(additive_identity, additive_identity)))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(380,plain,
% 74.75/48.47 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((additive_identity = multiply(additive_identity, additive_identity)) | (~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)) | (additive_identity = multiply(additive_identity, additive_identity))))),
% 74.75/48.47 inference(monotonicity,[status(thm)],[379])).
% 74.75/48.47 tff(381,plain,
% 74.75/48.47 (((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((additive_identity = multiply(additive_identity, additive_identity)) | (~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)))) <=> ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)) | (additive_identity = multiply(additive_identity, additive_identity)))),
% 74.75/48.47 inference(transitivity,[status(thm)],[380, 378])).
% 74.75/48.47 tff(382,plain,
% 74.75/48.47 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | ((additive_identity = multiply(additive_identity, additive_identity)) | (~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)))),
% 74.75/48.47 inference(quant_inst,[status(thm)],[])).
% 74.75/48.47 tff(383,plain,
% 74.75/48.47 ((~![V: $i, Y: $i, U: $i, X: $i] : ((U = V) | (~sum(X, Y, V)) | (~sum(X, Y, U)))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), multiply(additive_identity, additive_identity))) | (~sum(additive_identity, multiply(additive_identity, additive_identity), additive_identity)) | (additive_identity = multiply(additive_identity, additive_identity))),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[382, 381])).
% 74.75/48.47 tff(384,plain,
% 74.75/48.47 (additive_identity = multiply(additive_identity, additive_identity)),
% 74.75/48.47 inference(unit_resolution,[status(thm)],[383, 28, 377, 375])).
% 74.75/48.47 tff(385,plain,
% 74.75/48.47 (multiply(additive_identity, additive_identity) = additive_identity),
% 74.75/48.47 inference(symmetry,[status(thm)],[384])).
% 74.75/48.47 tff(386,plain,
% 74.75/48.47 (inverse(multiply(additive_identity, additive_identity)) = inverse(additive_identity)),
% 74.75/48.47 inference(monotonicity,[status(thm)],[385])).
% 74.75/48.47 tff(387,plain,
% 74.75/48.47 (inverse(additive_identity) = inverse(multiply(additive_identity, additive_identity))),
% 74.75/48.47 inference(symmetry,[status(thm)],[386])).
% 74.75/48.47 tff(388,plain,
% 74.75/48.47 (sum(inverse(x), x, inverse(additive_identity)) <=> sum(inverse(x), x, inverse(multiply(additive_identity, additive_identity)))),
% 74.75/48.47 inference(monotonicity,[status(thm)],[387])).
% 74.75/48.47 tff(389,plain,
% 74.75/48.47 (sum(inverse(x), x, inverse(multiply(additive_identity, additive_identity))) <=> sum(inverse(x), x, inverse(additive_identity))),
% 74.75/48.47 inference(symmetry,[status(thm)],[388])).
% 74.75/48.47 tff(390,plain,
% 74.75/48.47 (multiplicative_identity = inverse(multiply(additive_identity, additive_identity))),
% 74.75/48.47 inference(transitivity,[status(thm)],[52, 35, 387])).
% 74.75/48.47 tff(391,plain,
% 74.75/48.47 (sum(inverse(x), x, multiplicative_identity) <=> sum(inverse(x), x, inverse(multiply(additive_identity, additive_identity)))),
% 74.75/48.47 inference(monotonicity,[status(thm)],[390])).
% 74.75/48.47 tff(392,plain,
% 74.75/48.47 (sum(inverse(x), x, multiplicative_identity) <=> sum(inverse(x), x, inverse(additive_identity))),
% 74.75/48.47 inference(transitivity,[status(thm)],[391, 389])).
% 74.75/48.47 tff(393,plain,
% 74.75/48.47 ((~![X: $i] : sum(inverse(X), X, multiplicative_identity)) | sum(inverse(x), x, multiplicative_identity)),
% 74.75/48.47 inference(quant_inst,[status(thm)],[])).
% 74.75/48.47 tff(394,plain,
% 74.75/48.47 (sum(inverse(x), x, multiplicative_identity)),
% 74.75/48.47 inference(unit_resolution,[status(thm)],[393, 42])).
% 74.75/48.47 tff(395,plain,
% 74.75/48.47 (sum(inverse(x), x, inverse(additive_identity))),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[394, 392])).
% 74.75/48.47 tff(396,assumption,(~product(x, x, x)), introduced(assumption)).
% 74.75/48.47 tff(397,plain,
% 74.75/48.47 (product(additive_identity, additive_identity, multiply(additive_identity, additive_identity)) <=> product(additive_identity, additive_identity, additive_identity)),
% 74.75/48.47 inference(monotonicity,[status(thm)],[385])).
% 74.75/48.47 tff(398,plain,
% 74.75/48.47 (product(additive_identity, additive_identity, additive_identity)),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[157, 397])).
% 74.75/48.47 tff(399,plain,
% 74.75/48.47 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))),
% 74.75/48.47 inference(bind,[status(th)],[])).
% 74.75/48.47 tff(400,plain,
% 74.75/48.47 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.47 inference(quant_intro,[status(thm)],[399])).
% 74.75/48.47 tff(401,plain,
% 74.75/48.47 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(402,plain,
% 74.75/48.47 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) <=> ((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) <=> (((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~sum(V3, X, V4))))), rewrite((((~product(Y, Z, V3)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | (~sum(V3, X, V4))) <=> ((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) <=> ((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4)) <=> (((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | product(V1, V2, V4)))), rewrite((((~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))), ((((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4)) <=> (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))))),
% 74.75/48.47 inference(bind,[status(th)],[])).
% 74.75/48.47 tff(403,plain,
% 74.75/48.47 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.47 inference(quant_intro,[status(thm)],[402])).
% 74.75/48.47 tff(404,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~sum(Y, X, V1)) | (~sum(Z, X, V2))) | (~product(Y, Z, V3))) | (~sum(V3, X, V4))) | product(V1, V2, V4))), file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax','distributivity7')).
% 74.75/48.47 tff(405,plain,
% 74.75/48.47 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[404, 403])).
% 74.75/48.47 tff(406,plain,
% 74.75/48.47 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[405, 401])).
% 74.75/48.47 tff(407,plain,(
% 74.75/48.47 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.47 inference(skolemize,[status(sab)],[406])).
% 74.75/48.47 tff(408,plain,
% 74.75/48.47 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[407, 400])).
% 74.75/48.47 tff(409,plain,
% 74.75/48.47 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, additive_identity)) | product(x, x, x))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, additive_identity)) | product(x, x, x))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(410,plain,
% 74.75/48.47 ((product(x, x, x) | (~product(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x))) <=> ((~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, additive_identity)) | product(x, x, x))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(411,plain,
% 74.75/48.47 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(x, x, x) | (~product(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | ((~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, additive_identity)) | product(x, x, x)))),
% 74.75/48.47 inference(monotonicity,[status(thm)],[410])).
% 74.75/48.47 tff(412,plain,
% 74.75/48.47 (((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(x, x, x) | (~product(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)))) <=> ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, additive_identity)) | product(x, x, x))),
% 74.75/48.47 inference(transitivity,[status(thm)],[411, 409])).
% 74.75/48.47 tff(413,plain,
% 74.75/48.47 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (product(x, x, x) | (~product(additive_identity, additive_identity, additive_identity)) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)))),
% 74.75/48.47 inference(quant_inst,[status(thm)],[])).
% 74.75/48.47 tff(414,plain,
% 74.75/48.47 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~product(Y, Z, V3)) | (~sum(V3, X, V4)) | (~sum(Z, X, V2)) | (~sum(Y, X, V1)))) | (~sum(additive_identity, x, x)) | (~product(additive_identity, additive_identity, additive_identity)) | product(x, x, x)),
% 74.75/48.47 inference(modus_ponens,[status(thm)],[413, 412])).
% 74.75/48.47 tff(415,plain,
% 74.75/48.47 ($false),
% 74.75/48.47 inference(unit_resolution,[status(thm)],[414, 408, 79, 398, 396])).
% 74.75/48.47 tff(416,plain,(product(x, x, x)), inference(lemma,lemma(discharge,[]))).
% 74.75/48.47 tff(417,plain,
% 74.75/48.47 (((~![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(x, x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)))) <=> ((~![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(x, x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(418,plain,
% 74.75/48.47 ((product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(x, x, x)) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity))))) <=> ((~product(x, x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)))),
% 74.75/48.47 inference(rewrite,[status(thm)],[])).
% 74.75/48.47 tff(419,plain,
% 74.75/48.47 (((~![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(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(x, x, x)) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))))) <=> ((~![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(x, x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity))))),
% 74.75/48.47 inference(monotonicity,[status(thm)],[418])).
% 74.75/48.47 tff(420,plain,
% 74.75/48.47 (((~![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(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(x, x, x)) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))))) <=> ((~![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(x, x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)))),
% 74.75/48.48 inference(transitivity,[status(thm)],[419, 417])).
% 74.75/48.48 tff(421,plain,
% 74.75/48.48 ((~![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(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(x, x, x)) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))))),
% 74.75/48.48 inference(quant_inst,[status(thm)],[])).
% 74.75/48.48 tff(422,plain,
% 74.75/48.48 ((~![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(x, x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))) | product(add(inverse(x), inverse(additive_identity)), add(inverse(x), inverse(additive_identity)), inverse(additive_identity))),
% 74.75/48.48 inference(modus_ponens,[status(thm)],[421, 420])).
% 74.75/48.48 tff(423,plain,
% 74.75/48.48 (~sum(inverse(x), x, add(inverse(x), inverse(additive_identity)))),
% 74.75/48.48 inference(unit_resolution,[status(thm)],[422, 327, 416, 395, 366])).
% 74.75/48.48 tff(424,plain,
% 74.75/48.48 (product(inverse(additive_identity), x, x) <=> product(inverse(multiply(additive_identity, additive_identity)), x, x)),
% 74.75/48.48 inference(monotonicity,[status(thm)],[387])).
% 74.75/48.48 tff(425,plain,
% 74.75/48.48 (product(inverse(multiply(additive_identity, additive_identity)), x, x) <=> product(inverse(additive_identity), x, x)),
% 74.75/48.48 inference(symmetry,[status(thm)],[424])).
% 74.75/48.48 tff(426,plain,
% 74.75/48.48 (product(multiplicative_identity, x, x) <=> product(inverse(multiply(additive_identity, additive_identity)), x, x)),
% 74.75/48.48 inference(monotonicity,[status(thm)],[390])).
% 74.75/48.48 tff(427,plain,
% 74.75/48.48 (product(multiplicative_identity, x, x) <=> product(inverse(additive_identity), x, x)),
% 74.75/48.48 inference(transitivity,[status(thm)],[426, 425])).
% 74.75/48.48 tff(428,plain,
% 74.75/48.48 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, x, x)),
% 74.75/48.48 inference(quant_inst,[status(thm)],[])).
% 74.75/48.48 tff(429,plain,
% 74.75/48.48 (product(multiplicative_identity, x, x)),
% 74.75/48.48 inference(unit_resolution,[status(thm)],[428, 134])).
% 74.75/48.48 tff(430,plain,
% 74.75/48.48 (product(inverse(additive_identity), x, x)),
% 74.75/48.48 inference(modus_ponens,[status(thm)],[429, 427])).
% 74.75/48.48 tff(431,plain,
% 74.75/48.48 (((~![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(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | sum(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_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(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | sum(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))))),
% 74.75/48.48 inference(rewrite,[status(thm)],[])).
% 74.75/48.48 tff(432,plain,
% 74.75/48.48 ((sum(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity))))) <=> ((~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | sum(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))))),
% 74.75/48.48 inference(rewrite,[status(thm)],[])).
% 74.75/48.48 tff(433,plain,
% 74.75/48.48 (((~![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(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_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(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | sum(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity))))))),
% 74.75/48.48 inference(monotonicity,[status(thm)],[432])).
% 74.75/48.48 tff(434,plain,
% 74.75/48.48 (((~![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(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_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(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | sum(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))))),
% 74.75/48.48 inference(transitivity,[status(thm)],[433, 431])).
% 74.75/48.48 tff(435,plain,
% 74.75/48.48 ((~![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(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | (~sum(inverse(x), x, inverse(additive_identity))) | (~sum(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))))),
% 74.75/48.48 inference(quant_inst,[status(thm)],[])).
% 74.75/48.48 tff(436,plain,
% 74.75/48.48 ((~![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(inverse(x), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))) | (~sum(inverse(x), x, inverse(additive_identity))) | (~product(inverse(additive_identity), x, x)) | sum(inverse(x), x, add(inverse(x), inverse(additive_identity))) | (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity))))),
% 74.84/48.52 inference(modus_ponens,[status(thm)],[435, 434])).
% 74.84/48.52 tff(437,plain,
% 74.84/48.52 (~product(add(inverse(x), inverse(additive_identity)), inverse(additive_identity), add(inverse(x), inverse(additive_identity)))),
% 74.84/48.52 inference(unit_resolution,[status(thm)],[436, 98, 359, 430, 395, 423])).
% 74.84/48.52 tff(438,plain,
% 74.84/48.52 (~product(add(inverse(x), inverse(additive_identity)), multiplicative_identity, add(inverse(x), inverse(additive_identity)))),
% 74.84/48.52 inference(modus_ponens,[status(thm)],[437, 56])).
% 74.84/48.52 tff(439,plain,
% 74.84/48.52 ((~![X: $i] : product(X, multiplicative_identity, X)) | product(add(inverse(x), inverse(additive_identity)), multiplicative_identity, add(inverse(x), inverse(additive_identity)))),
% 74.84/48.52 inference(quant_inst,[status(thm)],[])).
% 74.84/48.52 tff(440,plain,
% 74.84/48.52 (product(add(inverse(x), inverse(additive_identity)), multiplicative_identity, add(inverse(x), inverse(additive_identity)))),
% 74.84/48.52 inference(unit_resolution,[status(thm)],[439, 185])).
% 74.84/48.52 tff(441,plain,
% 74.84/48.52 ($false),
% 74.84/48.52 inference(unit_resolution,[status(thm)],[440, 438])).
% 74.84/48.52 % SZS output end Proof
%------------------------------------------------------------------------------