TSTP Solution File: BOO003-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : BOO003-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n026.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 17:18:34 EDT 2022
% Result : Unsatisfiable 0.86s 0.82s
% Output : Proof 0.86s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.11 % Problem : BOO003-1 : TPTP v8.1.0. Released v1.0.0.
% 0.09/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n026.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 30 02:40:35 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.33 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.33 Usage: tptp [options] [-file:]file
% 0.12/0.33 -h, -? prints this message.
% 0.12/0.33 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.33 -m, -model generate model.
% 0.12/0.33 -p, -proof generate proof.
% 0.12/0.33 -c, -core generate unsat core of named formulas.
% 0.12/0.33 -st, -statistics display statistics.
% 0.12/0.33 -t:timeout set timeout (in second).
% 0.12/0.33 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.33 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.33 -<param>:<value> configuration parameter and value.
% 0.12/0.33 -o:<output-file> file to place output in.
% 0.86/0.82 % SZS status Unsatisfiable
% 0.86/0.82 % SZS output start Proof
% 0.86/0.82 tff(sum_type, type, (
% 0.86/0.82 sum: ( $i * $i * $i ) > $o)).
% 0.86/0.82 tff(x_type, type, (
% 0.86/0.82 x: $i)).
% 0.86/0.82 tff(multiply_type, type, (
% 0.86/0.82 multiply: ( $i * $i ) > $i)).
% 0.86/0.82 tff(additive_identity_type, type, (
% 0.86/0.82 additive_identity: $i)).
% 0.86/0.82 tff(product_type, type, (
% 0.86/0.82 product: ( $i * $i * $i ) > $o)).
% 0.86/0.82 tff(inverse_type, type, (
% 0.86/0.82 inverse: $i > $i)).
% 0.86/0.82 tff(multiplicative_identity_type, type, (
% 0.86/0.82 multiplicative_identity: $i)).
% 0.86/0.82 tff(1,assumption,(~sum(additive_identity, multiply(x, x), x)), introduced(assumption)).
% 0.86/0.82 tff(2,plain,
% 0.86/0.82 (^[X: $i] : refl(product(inverse(X), X, additive_identity) <=> product(inverse(X), X, additive_identity))),
% 0.86/0.82 inference(bind,[status(th)],[])).
% 0.86/0.82 tff(3,plain,
% 0.86/0.82 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 0.86/0.82 inference(quant_intro,[status(thm)],[2])).
% 0.86/0.82 tff(4,plain,
% 0.86/0.82 (![X: $i] : product(inverse(X), X, additive_identity) <=> ![X: $i] : product(inverse(X), X, additive_identity)),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(5,axiom,(![X: $i] : product(inverse(X), X, additive_identity)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','multiplicative_inverse1')).
% 0.86/0.82 tff(6,plain,
% 0.86/0.82 (![X: $i] : product(inverse(X), X, additive_identity)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.86/0.82 tff(7,plain,(
% 0.86/0.82 ![X: $i] : product(inverse(X), X, additive_identity)),
% 0.86/0.82 inference(skolemize,[status(sab)],[6])).
% 0.86/0.82 tff(8,plain,
% 0.86/0.82 (![X: $i] : product(inverse(X), X, additive_identity)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.86/0.82 tff(9,plain,
% 0.86/0.82 ((~![X: $i] : product(inverse(X), X, additive_identity)) | product(inverse(x), x, additive_identity)),
% 0.86/0.82 inference(quant_inst,[status(thm)],[])).
% 0.86/0.82 tff(10,plain,
% 0.86/0.82 (product(inverse(x), x, additive_identity)),
% 0.86/0.82 inference(unit_resolution,[status(thm)],[9, 8])).
% 0.86/0.82 tff(11,plain,
% 0.86/0.82 (^[X: $i] : refl(sum(inverse(X), X, multiplicative_identity) <=> sum(inverse(X), X, multiplicative_identity))),
% 0.86/0.82 inference(bind,[status(th)],[])).
% 0.86/0.82 tff(12,plain,
% 0.86/0.82 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/0.82 inference(quant_intro,[status(thm)],[11])).
% 0.86/0.82 tff(13,plain,
% 0.86/0.82 (![X: $i] : sum(inverse(X), X, multiplicative_identity) <=> ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(14,axiom,(![X: $i] : sum(inverse(X), X, multiplicative_identity)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','additive_inverse1')).
% 0.86/0.82 tff(15,plain,
% 0.86/0.82 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.86/0.82 tff(16,plain,(
% 0.86/0.82 ![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/0.82 inference(skolemize,[status(sab)],[15])).
% 0.86/0.82 tff(17,plain,
% 0.86/0.82 (![X: $i] : sum(inverse(X), X, multiplicative_identity)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.86/0.82 tff(18,plain,
% 0.86/0.82 ((~![X: $i] : sum(inverse(X), X, multiplicative_identity)) | sum(inverse(x), x, multiplicative_identity)),
% 0.86/0.82 inference(quant_inst,[status(thm)],[])).
% 0.86/0.82 tff(19,plain,
% 0.86/0.82 (sum(inverse(x), x, multiplicative_identity)),
% 0.86/0.82 inference(unit_resolution,[status(thm)],[18, 17])).
% 0.86/0.82 tff(20,plain,
% 0.86/0.82 (^[X: $i] : refl(product(multiplicative_identity, X, X) <=> product(multiplicative_identity, X, X))),
% 0.86/0.82 inference(bind,[status(th)],[])).
% 0.86/0.82 tff(21,plain,
% 0.86/0.82 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 0.86/0.82 inference(quant_intro,[status(thm)],[20])).
% 0.86/0.82 tff(22,plain,
% 0.86/0.82 (![X: $i] : product(multiplicative_identity, X, X) <=> ![X: $i] : product(multiplicative_identity, X, X)),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(23,axiom,(![X: $i] : product(multiplicative_identity, X, X)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','multiplicative_identity1')).
% 0.86/0.82 tff(24,plain,
% 0.86/0.82 (![X: $i] : product(multiplicative_identity, X, X)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.86/0.82 tff(25,plain,(
% 0.86/0.82 ![X: $i] : product(multiplicative_identity, X, X)),
% 0.86/0.82 inference(skolemize,[status(sab)],[24])).
% 0.86/0.82 tff(26,plain,
% 0.86/0.82 (![X: $i] : product(multiplicative_identity, X, X)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.86/0.82 tff(27,plain,
% 0.86/0.82 ((~![X: $i] : product(multiplicative_identity, X, X)) | product(multiplicative_identity, x, x)),
% 0.86/0.82 inference(quant_inst,[status(thm)],[])).
% 0.86/0.82 tff(28,plain,
% 0.86/0.82 (product(multiplicative_identity, x, x)),
% 0.86/0.82 inference(unit_resolution,[status(thm)],[27, 26])).
% 0.86/0.82 tff(29,plain,
% 0.86/0.82 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 0.86/0.82 inference(bind,[status(th)],[])).
% 0.86/0.82 tff(30,plain,
% 0.86/0.82 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/0.82 inference(quant_intro,[status(thm)],[29])).
% 0.86/0.82 tff(31,plain,
% 0.86/0.82 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(32,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','closure_of_multiplication')).
% 0.86/0.82 tff(33,plain,
% 0.86/0.82 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.86/0.82 tff(34,plain,(
% 0.86/0.82 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/0.82 inference(skolemize,[status(sab)],[33])).
% 0.86/0.82 tff(35,plain,
% 0.86/0.82 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[34, 30])).
% 0.86/0.82 tff(36,plain,
% 0.86/0.82 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(x, x, multiply(x, x))),
% 0.86/0.82 inference(quant_inst,[status(thm)],[])).
% 0.86/0.82 tff(37,plain,
% 0.86/0.82 (product(x, x, multiply(x, x))),
% 0.86/0.82 inference(unit_resolution,[status(thm)],[36, 35])).
% 0.86/0.82 tff(38,plain,
% 0.86/0.82 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : refl((sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))))),
% 0.86/0.82 inference(bind,[status(th)],[])).
% 0.86/0.82 tff(39,plain,
% 0.86/0.82 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.86/0.82 inference(quant_intro,[status(thm)],[38])).
% 0.86/0.82 tff(40,plain,
% 0.86/0.82 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(41,plain,
% 0.86/0.82 (^[Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) <=> ((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) <=> (((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~product(V3, X, V4))))), rewrite((((~sum(Y, Z, V3)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | (~product(V3, X, V4))) <=> ((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) <=> ((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4)) <=> (((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | sum(V1, V2, V4)))), rewrite((((~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))), ((((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4)) <=> (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))))),
% 0.86/0.82 inference(bind,[status(th)],[])).
% 0.86/0.82 tff(42,plain,
% 0.86/0.82 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4)) <=> ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.86/0.82 inference(quant_intro,[status(thm)],[41])).
% 0.86/0.82 tff(43,axiom,(![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (((((~product(Y, X, V1)) | (~product(Z, X, V2))) | (~sum(Y, Z, V3))) | (~product(V3, X, V4))) | sum(V1, V2, V4))), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','distributivity3')).
% 0.86/0.82 tff(44,plain,
% 0.86/0.82 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.86/0.82 tff(45,plain,
% 0.86/0.82 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[44, 40])).
% 0.86/0.82 tff(46,plain,(
% 0.86/0.82 ![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.86/0.82 inference(skolemize,[status(sab)],[45])).
% 0.86/0.82 tff(47,plain,
% 0.86/0.82 (![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[46, 39])).
% 0.86/0.82 tff(48,plain,
% 0.86/0.82 (((~![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(inverse(x), x, additive_identity)) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), 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)))) | (~product(inverse(x), x, additive_identity)) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), x))),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(49,plain,
% 0.86/0.82 ((sum(additive_identity, multiply(x, x), x) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity))) <=> ((~product(inverse(x), x, additive_identity)) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), x))),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(50,plain,
% 0.86/0.82 (((~![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(x, x), x) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, 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(inverse(x), x, additive_identity)) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), x)))),
% 0.86/0.82 inference(monotonicity,[status(thm)],[49])).
% 0.86/0.82 tff(51,plain,
% 0.86/0.82 (((~![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(x, x), x) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, 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(inverse(x), x, additive_identity)) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), x))),
% 0.86/0.82 inference(transitivity,[status(thm)],[50, 48])).
% 0.86/0.82 tff(52,plain,
% 0.86/0.82 ((~![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(x, x), x) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(multiplicative_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~product(inverse(x), x, additive_identity)))),
% 0.86/0.82 inference(quant_inst,[status(thm)],[])).
% 0.86/0.82 tff(53,plain,
% 0.86/0.82 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (sum(V1, V2, V4) | (~sum(Y, Z, V3)) | (~product(V3, X, V4)) | (~product(Z, X, V2)) | (~product(Y, X, V1)))) | (~product(inverse(x), x, additive_identity)) | (~product(multiplicative_identity, x, x)) | (~sum(inverse(x), x, multiplicative_identity)) | (~product(x, x, multiply(x, x))) | sum(additive_identity, multiply(x, x), x)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.86/0.82 tff(54,plain,
% 0.86/0.82 ($false),
% 0.86/0.82 inference(unit_resolution,[status(thm)],[53, 47, 37, 28, 19, 10, 1])).
% 0.86/0.82 tff(55,plain,(sum(additive_identity, multiply(x, x), x)), inference(lemma,lemma(discharge,[]))).
% 0.86/0.82 tff(56,plain,
% 0.86/0.82 (^[X: $i] : refl(sum(additive_identity, X, X) <=> sum(additive_identity, X, X))),
% 0.86/0.82 inference(bind,[status(th)],[])).
% 0.86/0.82 tff(57,plain,
% 0.86/0.82 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.86/0.82 inference(quant_intro,[status(thm)],[56])).
% 0.86/0.82 tff(58,plain,
% 0.86/0.82 (![X: $i] : sum(additive_identity, X, X) <=> ![X: $i] : sum(additive_identity, X, X)),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(59,axiom,(![X: $i] : sum(additive_identity, X, X)), file('/export/starexec/sandbox/benchmark/Axioms/BOO002-0.ax','additive_identity1')).
% 0.86/0.82 tff(60,plain,
% 0.86/0.82 (![X: $i] : sum(additive_identity, X, X)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.86/0.82 tff(61,plain,(
% 0.86/0.82 ![X: $i] : sum(additive_identity, X, X)),
% 0.86/0.82 inference(skolemize,[status(sab)],[60])).
% 0.86/0.82 tff(62,plain,
% 0.86/0.82 (![X: $i] : sum(additive_identity, X, X)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[61, 57])).
% 0.86/0.82 tff(63,plain,
% 0.86/0.82 ((~![X: $i] : sum(additive_identity, X, X)) | sum(additive_identity, x, x)),
% 0.86/0.82 inference(quant_inst,[status(thm)],[])).
% 0.86/0.82 tff(64,plain,
% 0.86/0.82 (sum(additive_identity, x, x)),
% 0.86/0.82 inference(unit_resolution,[status(thm)],[63, 62])).
% 0.86/0.82 tff(65,plain,
% 0.86/0.82 ((~product(x, x, x)) <=> (~product(x, x, x))),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(66,axiom,(~product(x, x, x)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_both_equalities')).
% 0.86/0.82 tff(67,plain,
% 0.86/0.82 (~product(x, x, x)),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.86/0.82 tff(68,plain,
% 0.86/0.82 (^[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))))),
% 0.86/0.82 inference(bind,[status(th)],[])).
% 0.86/0.82 tff(69,plain,
% 0.86/0.82 (![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)))),
% 0.86/0.82 inference(quant_intro,[status(thm)],[68])).
% 0.86/0.82 tff(70,plain,
% 0.86/0.82 (![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)))),
% 0.86/0.82 inference(rewrite,[status(thm)],[])).
% 0.86/0.82 tff(71,plain,
% 0.86/0.82 (^[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)))))),
% 0.86/0.82 inference(bind,[status(th)],[])).
% 0.86/0.82 tff(72,plain,
% 0.86/0.82 (![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)))),
% 0.86/0.82 inference(quant_intro,[status(thm)],[71])).
% 0.86/0.82 tff(73,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/sandbox/benchmark/Axioms/BOO002-0.ax','distributivity5')).
% 0.86/0.82 tff(74,plain,
% 0.86/0.82 (![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)))),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.86/0.82 tff(75,plain,
% 0.86/0.82 (![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)))),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[74, 70])).
% 0.86/0.82 tff(76,plain,(
% 0.86/0.82 ![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)))),
% 0.86/0.82 inference(skolemize,[status(sab)],[75])).
% 0.86/0.82 tff(77,plain,
% 0.86/0.82 (![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)))),
% 0.86/0.82 inference(modus_ponens,[status(thm)],[76, 69])).
% 0.86/0.82 tff(78,plain,
% 0.86/0.82 (((~![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(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, multiply(x, x), 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)))) | product(x, x, x) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, multiply(x, x), x)))),
% 0.86/0.83 inference(rewrite,[status(thm)],[])).
% 0.86/0.83 tff(79,plain,
% 0.86/0.83 ((product(x, x, x) | (~sum(additive_identity, multiply(x, x), x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x))) <=> (product(x, x, x) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, multiply(x, x), x)))),
% 0.86/0.83 inference(rewrite,[status(thm)],[])).
% 0.86/0.83 tff(80,plain,
% 0.86/0.83 (((~![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(additive_identity, multiply(x, x), x)) | (~product(x, x, multiply(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) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | (product(x, x, x) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, multiply(x, x), x))))),
% 0.86/0.83 inference(monotonicity,[status(thm)],[79])).
% 0.86/0.83 tff(81,plain,
% 0.86/0.83 (((~![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(additive_identity, multiply(x, x), x)) | (~product(x, x, multiply(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) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | product(x, x, x) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, multiply(x, x), x)))),
% 0.86/0.83 inference(transitivity,[status(thm)],[80, 78])).
% 0.86/0.83 tff(82,plain,
% 0.86/0.83 ((~![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(additive_identity, multiply(x, x), x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, x, x)) | (~sum(additive_identity, x, x)))),
% 0.86/0.83 inference(quant_inst,[status(thm)],[])).
% 0.86/0.83 tff(83,plain,
% 0.86/0.83 ((~![Z: $i, Y: $i, V1: $i, X: $i, V2: $i, V3: $i, V4: $i] : (product(V1, V2, V4) | (~sum(X, V3, V4)) | (~product(Y, Z, V3)) | (~sum(X, Z, V2)) | (~sum(X, Y, V1)))) | product(x, x, x) | (~sum(additive_identity, x, x)) | (~product(x, x, multiply(x, x))) | (~sum(additive_identity, multiply(x, x), x))),
% 0.86/0.83 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.86/0.83 tff(84,plain,
% 0.86/0.83 ($false),
% 0.86/0.83 inference(unit_resolution,[status(thm)],[83, 77, 67, 37, 64, 55])).
% 0.86/0.83 % SZS output end Proof
%------------------------------------------------------------------------------