TSTP Solution File: HEN004-4 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : HEN004-4 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n025.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 : Fri Sep 16 22:39:46 EDT 2022
% Result : Unsatisfiable 0.19s 0.45s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : HEN004-4 : TPTP v8.1.0. Released v1.0.0.
% 0.07/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n025.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 : Wed Aug 31 21:50:31 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.19/0.45 % SZS status Unsatisfiable
% 0.19/0.45 % SZS output start Proof
% 0.19/0.45 tff(less_equal_type, type, (
% 0.19/0.45 less_equal: ( $i * $i ) > $o)).
% 0.19/0.45 tff(divide_type, type, (
% 0.19/0.45 divide: ( $i * $i ) > $i)).
% 0.19/0.45 tff(zero_type, type, (
% 0.19/0.45 zero: $i)).
% 0.19/0.45 tff(a_type, type, (
% 0.19/0.45 a: $i)).
% 0.19/0.45 tff(1,plain,
% 0.19/0.45 (^[X: $i] : refl((divide(X, X) = zero) <=> (divide(X, X) = zero))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(2,plain,
% 0.19/0.45 (![X: $i] : (divide(X, X) = zero) <=> ![X: $i] : (divide(X, X) = zero)),
% 0.19/0.45 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.45 tff(3,plain,
% 0.19/0.45 (![X: $i] : (divide(X, X) = zero) <=> ![X: $i] : (divide(X, X) = zero)),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(4,axiom,(![X: $i] : (divide(X, X) = zero)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','x_divide_x_is_zero')).
% 0.19/0.45 tff(5,plain,
% 0.19/0.45 (![X: $i] : (divide(X, X) = zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.45 tff(6,plain,(
% 0.19/0.45 ![X: $i] : (divide(X, X) = zero)),
% 0.19/0.45 inference(skolemize,[status(sab)],[5])).
% 0.19/0.45 tff(7,plain,
% 0.19/0.45 (![X: $i] : (divide(X, X) = zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.45 tff(8,plain,
% 0.19/0.45 ((~![X: $i] : (divide(X, X) = zero)) | (divide(divide(a, zero), divide(a, zero)) = zero)),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(9,plain,
% 0.19/0.45 (divide(divide(a, zero), divide(a, zero)) = zero),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.45 tff(10,plain,
% 0.19/0.45 (^[Y: $i, X: $i] : refl(less_equal(divide(X, Y), X) <=> less_equal(divide(X, Y), X))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(11,plain,
% 0.19/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X) <=> ![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.19/0.45 inference(quant_intro,[status(thm)],[10])).
% 0.19/0.45 tff(12,plain,
% 0.19/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X) <=> ![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(13,axiom,(![Y: $i, X: $i] : less_equal(divide(X, Y), X)), file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax','quotient_smaller_than_numerator')).
% 0.19/0.45 tff(14,plain,
% 0.19/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.19/0.45 tff(15,plain,(
% 0.19/0.45 ![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.19/0.45 inference(skolemize,[status(sab)],[14])).
% 0.19/0.45 tff(16,plain,
% 0.19/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.19/0.45 tff(17,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(a, zero), a)),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(18,plain,
% 0.19/0.45 (less_equal(divide(a, zero), a)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.19/0.45 tff(19,plain,
% 0.19/0.45 (^[Y: $i, X: $i] : refl(((~less_equal(X, Y)) | (divide(X, Y) = zero)) <=> ((~less_equal(X, Y)) | (divide(X, Y) = zero)))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(20,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero)) <=> ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[19])).
% 0.19/0.45 tff(21,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero)) <=> ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(22,axiom,(![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))), file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax','quotient_less_equal1')).
% 0.19/0.45 tff(23,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.19/0.45 tff(24,plain,(
% 0.19/0.45 ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.45 inference(skolemize,[status(sab)],[23])).
% 0.19/0.45 tff(25,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[24, 20])).
% 0.19/0.45 tff(26,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(a, zero), a)) | (divide(divide(a, zero), a) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(a, zero), a)) | (divide(divide(a, zero), a) = zero))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(27,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(a, zero), a)) | (divide(divide(a, zero), a) = zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(28,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(a, zero), a)) | (divide(divide(a, zero), a) = zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.19/0.45 tff(29,plain,
% 0.19/0.45 (divide(divide(a, zero), a) = zero),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[28, 25, 18])).
% 0.19/0.45 tff(30,plain,
% 0.19/0.45 (zero = divide(divide(a, zero), a)),
% 0.19/0.45 inference(symmetry,[status(thm)],[29])).
% 0.19/0.45 tff(31,plain,
% 0.19/0.45 (less_equal(zero, divide(a, zero)) <=> less_equal(divide(divide(a, zero), a), divide(a, zero))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[30])).
% 0.19/0.45 tff(32,plain,
% 0.19/0.45 (less_equal(divide(divide(a, zero), a), divide(a, zero)) <=> less_equal(zero, divide(a, zero))),
% 0.19/0.45 inference(symmetry,[status(thm)],[31])).
% 0.19/0.45 tff(33,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(divide(a, zero), a), divide(a, zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(34,plain,
% 0.19/0.45 (less_equal(divide(divide(a, zero), a), divide(a, zero))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[33, 16])).
% 0.19/0.45 tff(35,plain,
% 0.19/0.45 (less_equal(zero, divide(a, zero))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[34, 32])).
% 0.19/0.45 tff(36,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(zero, divide(a, zero))) | (divide(zero, divide(a, zero)) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(zero, divide(a, zero))) | (divide(zero, divide(a, zero)) = zero))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(37,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(zero, divide(a, zero))) | (divide(zero, divide(a, zero)) = zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(38,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(zero, divide(a, zero))) | (divide(zero, divide(a, zero)) = zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.19/0.45 tff(39,plain,
% 0.19/0.45 ((~less_equal(zero, divide(a, zero))) | (divide(zero, divide(a, zero)) = zero)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[38, 25])).
% 0.19/0.45 tff(40,plain,
% 0.19/0.45 (divide(zero, divide(a, zero)) = zero),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[39, 35])).
% 0.19/0.45 tff(41,plain,
% 0.19/0.45 (divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))) = divide(divide(a, divide(a, zero)), zero)),
% 0.19/0.45 inference(monotonicity,[status(thm)],[40])).
% 0.19/0.45 tff(42,plain,
% 0.19/0.45 (less_equal(divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))), divide(divide(a, zero), divide(a, zero))) <=> less_equal(divide(divide(a, divide(a, zero)), zero), zero)),
% 0.19/0.45 inference(monotonicity,[status(thm)],[41, 9])).
% 0.19/0.45 tff(43,plain,
% 0.19/0.45 (less_equal(divide(divide(a, divide(a, zero)), zero), zero) <=> less_equal(divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))), divide(divide(a, zero), divide(a, zero)))),
% 0.19/0.45 inference(symmetry,[status(thm)],[42])).
% 0.19/0.45 tff(44,plain,
% 0.19/0.45 ((~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) <=> (~less_equal(divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))), divide(divide(a, zero), divide(a, zero))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[43])).
% 0.19/0.45 tff(45,assumption,(~less_equal(divide(divide(a, divide(a, zero)), zero), zero)), introduced(assumption)).
% 0.19/0.45 tff(46,plain,
% 0.19/0.45 (~less_equal(divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))), divide(divide(a, zero), divide(a, zero)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.19/0.45 tff(47,plain,
% 0.19/0.45 (^[Z: $i, Y: $i, X: $i] : refl(less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z)) <=> less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z)))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(48,plain,
% 0.19/0.45 (![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z)) <=> ![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[47])).
% 0.19/0.45 tff(49,plain,
% 0.19/0.45 (![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z)) <=> ![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(50,axiom,(![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))), file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax','quotient_property')).
% 0.19/0.45 tff(51,plain,
% 0.19/0.45 (![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.19/0.45 tff(52,plain,(
% 0.19/0.45 ![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))),
% 0.19/0.45 inference(skolemize,[status(sab)],[51])).
% 0.19/0.45 tff(53,plain,
% 0.19/0.45 (![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.19/0.45 tff(54,plain,
% 0.19/0.45 ((~![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))) | less_equal(divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))), divide(divide(a, zero), divide(a, zero)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(55,plain,
% 0.19/0.45 (less_equal(divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))), divide(divide(a, zero), divide(a, zero)))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[54, 53])).
% 0.19/0.45 tff(56,plain,
% 0.19/0.45 ($false),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[55, 46])).
% 0.19/0.45 tff(57,plain,(less_equal(divide(divide(a, divide(a, zero)), zero), zero)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.45 tff(58,assumption,(~less_equal(zero, divide(divide(a, divide(a, zero)), zero))), introduced(assumption)).
% 0.19/0.45 tff(59,plain,
% 0.19/0.45 (^[X: $i] : refl(less_equal(zero, X) <=> less_equal(zero, X))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(60,plain,
% 0.19/0.45 (![X: $i] : less_equal(zero, X) <=> ![X: $i] : less_equal(zero, X)),
% 0.19/0.45 inference(quant_intro,[status(thm)],[59])).
% 0.19/0.45 tff(61,plain,
% 0.19/0.45 (![X: $i] : less_equal(zero, X) <=> ![X: $i] : less_equal(zero, X)),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(62,axiom,(![X: $i] : less_equal(zero, X)), file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax','zero_is_smallest')).
% 0.19/0.45 tff(63,plain,
% 0.19/0.45 (![X: $i] : less_equal(zero, X)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.19/0.45 tff(64,plain,(
% 0.19/0.45 ![X: $i] : less_equal(zero, X)),
% 0.19/0.45 inference(skolemize,[status(sab)],[63])).
% 0.19/0.45 tff(65,plain,
% 0.19/0.45 (![X: $i] : less_equal(zero, X)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[64, 60])).
% 0.19/0.45 tff(66,plain,
% 0.19/0.45 ((~![X: $i] : less_equal(zero, X)) | less_equal(zero, divide(divide(a, divide(a, zero)), zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(67,plain,
% 0.19/0.45 ($false),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[66, 65, 58])).
% 0.19/0.45 tff(68,plain,(less_equal(zero, divide(divide(a, divide(a, zero)), zero))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.45 tff(69,assumption,(less_equal(zero, divide(divide(a, divide(a, zero)), zero))), introduced(assumption)).
% 0.19/0.45 tff(70,assumption,(less_equal(divide(divide(a, divide(a, zero)), zero), zero)), introduced(assumption)).
% 0.19/0.45 tff(71,assumption,(~(divide(divide(a, divide(a, zero)), zero) = zero)), introduced(assumption)).
% 0.19/0.45 tff(72,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X))) <=> ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(73,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X))) <=> ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(74,plain,
% 0.19/0.45 (^[Y: $i, X: $i] : rewrite((((~less_equal(X, Y)) | (~less_equal(Y, X))) | (X = Y)) <=> ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X))))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(75,plain,
% 0.19/0.45 (![Y: $i, X: $i] : (((~less_equal(X, Y)) | (~less_equal(Y, X))) | (X = Y)) <=> ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[74])).
% 0.19/0.46 tff(76,axiom,(![Y: $i, X: $i] : (((~less_equal(X, Y)) | (~less_equal(Y, X))) | (X = Y))), file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax','less_equal_and_equal')).
% 0.19/0.46 tff(77,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.19/0.46 tff(78,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[77, 73])).
% 0.19/0.46 tff(79,plain,(
% 0.19/0.46 ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.46 inference(skolemize,[status(sab)],[78])).
% 0.19/0.46 tff(80,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[79, 72])).
% 0.19/0.46 tff(81,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero))))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(82,plain,
% 0.19/0.46 (((~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero)))) <=> ((divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero))))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(83,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero)))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[82])).
% 0.19/0.46 tff(84,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero))))),
% 0.19/0.46 inference(transitivity,[status(thm)],[83, 81])).
% 0.19/0.46 tff(85,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero))))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(86,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.19/0.46 tff(87,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[86, 80, 71, 70, 69])).
% 0.19/0.46 tff(88,plain,((~less_equal(zero, divide(divide(a, divide(a, zero)), zero))) | (divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.46 tff(89,plain,
% 0.19/0.46 ((divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[88, 68])).
% 0.19/0.46 tff(90,plain,
% 0.19/0.46 (divide(divide(a, divide(a, zero)), zero) = zero),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[89, 57])).
% 0.19/0.46 tff(91,plain,
% 0.19/0.46 (^[Y: $i, X: $i] : refl(((~(divide(X, Y) = zero)) | less_equal(X, Y)) <=> ((~(divide(X, Y) = zero)) | less_equal(X, Y)))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(92,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y)) <=> ![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[91])).
% 0.19/0.46 tff(93,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y)) <=> ![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(94,axiom,(![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax','quotient_less_equal2')).
% 0.19/0.46 tff(95,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.19/0.46 tff(96,plain,(
% 0.19/0.46 ![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.46 inference(skolemize,[status(sab)],[95])).
% 0.19/0.46 tff(97,plain,
% 0.19/0.46 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[96, 92])).
% 0.19/0.46 tff(98,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | ((~(divide(divide(a, divide(a, zero)), zero) = zero)) | less_equal(divide(a, divide(a, zero)), zero))) <=> ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(divide(a, divide(a, zero)), zero) = zero)) | less_equal(divide(a, divide(a, zero)), zero))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(99,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | ((~(divide(divide(a, divide(a, zero)), zero) = zero)) | less_equal(divide(a, divide(a, zero)), zero))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(100,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(divide(a, divide(a, zero)), zero) = zero)) | less_equal(divide(a, divide(a, zero)), zero)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.19/0.46 tff(101,plain,
% 0.19/0.46 (less_equal(divide(a, divide(a, zero)), zero)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[100, 97, 90])).
% 0.19/0.46 tff(102,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(a, divide(a, zero)), a)),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(103,plain,
% 0.19/0.46 (less_equal(divide(a, divide(a, zero)), a)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[102, 16])).
% 0.19/0.46 tff(104,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(a, divide(a, zero)), a)) | (divide(divide(a, divide(a, zero)), a) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(a, divide(a, zero)), a)) | (divide(divide(a, divide(a, zero)), a) = zero))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(105,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(a, divide(a, zero)), a)) | (divide(divide(a, divide(a, zero)), a) = zero))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(106,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(a, divide(a, zero)), a)) | (divide(divide(a, divide(a, zero)), a) = zero)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.19/0.46 tff(107,plain,
% 0.19/0.46 ((~less_equal(divide(a, divide(a, zero)), a)) | (divide(divide(a, divide(a, zero)), a) = zero)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[106, 25])).
% 0.19/0.46 tff(108,plain,
% 0.19/0.46 (divide(divide(a, divide(a, zero)), a) = zero),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[107, 103])).
% 0.19/0.46 tff(109,plain,
% 0.19/0.46 (zero = divide(divide(a, divide(a, zero)), a)),
% 0.19/0.46 inference(symmetry,[status(thm)],[108])).
% 0.19/0.46 tff(110,plain,
% 0.19/0.46 (less_equal(zero, divide(a, divide(a, zero))) <=> less_equal(divide(divide(a, divide(a, zero)), a), divide(a, divide(a, zero)))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[109])).
% 0.19/0.46 tff(111,plain,
% 0.19/0.46 (less_equal(divide(divide(a, divide(a, zero)), a), divide(a, divide(a, zero))) <=> less_equal(zero, divide(a, divide(a, zero)))),
% 0.19/0.46 inference(symmetry,[status(thm)],[110])).
% 0.19/0.46 tff(112,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(divide(a, divide(a, zero)), a), divide(a, divide(a, zero)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(113,plain,
% 0.19/0.46 (less_equal(divide(divide(a, divide(a, zero)), a), divide(a, divide(a, zero)))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[112, 16])).
% 0.19/0.46 tff(114,plain,
% 0.19/0.46 (less_equal(zero, divide(a, divide(a, zero)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[113, 111])).
% 0.19/0.46 tff(115,assumption,(~(divide(a, divide(a, zero)) = zero)), introduced(assumption)).
% 0.19/0.46 tff(116,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))) | (~less_equal(divide(a, divide(a, zero)), zero)))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))) | (~less_equal(divide(a, divide(a, zero)), zero)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(117,plain,
% 0.19/0.46 (((~less_equal(divide(a, divide(a, zero)), zero)) | (divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero))))) <=> ((divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))) | (~less_equal(divide(a, divide(a, zero)), zero)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(118,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(a, divide(a, zero)), zero)) | (divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))) | (~less_equal(divide(a, divide(a, zero)), zero))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[117])).
% 0.19/0.46 tff(119,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(a, divide(a, zero)), zero)) | (divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))) | (~less_equal(divide(a, divide(a, zero)), zero)))),
% 0.19/0.46 inference(transitivity,[status(thm)],[118, 116])).
% 0.19/0.46 tff(120,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(a, divide(a, zero)), zero)) | (divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(121,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))) | (~less_equal(divide(a, divide(a, zero)), zero))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.19/0.46 tff(122,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[121, 80, 115, 114, 101])).
% 0.19/0.46 tff(123,plain,(divide(a, divide(a, zero)) = zero), inference(lemma,lemma(discharge,[]))).
% 0.19/0.46 tff(124,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | ((~(divide(a, divide(a, zero)) = zero)) | less_equal(a, divide(a, zero)))) <=> ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(a, divide(a, zero)) = zero)) | less_equal(a, divide(a, zero)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(125,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | ((~(divide(a, divide(a, zero)) = zero)) | less_equal(a, divide(a, zero)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(126,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(a, divide(a, zero)) = zero)) | less_equal(a, divide(a, zero))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.19/0.46 tff(127,plain,
% 0.19/0.46 ((~(divide(a, divide(a, zero)) = zero)) | less_equal(a, divide(a, zero))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[126, 97])).
% 0.19/0.46 tff(128,plain,
% 0.19/0.46 (less_equal(a, divide(a, zero))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[127, 123])).
% 0.19/0.46 tff(129,plain,
% 0.19/0.46 ((~(divide(a, zero) = a)) <=> (~(divide(a, zero) = a))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(130,axiom,(~(divide(a, zero) = a)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_x_divide_zero_is_x')).
% 0.19/0.46 tff(131,plain,
% 0.19/0.46 (~(divide(a, zero) = a)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[130, 129])).
% 0.19/0.46 tff(132,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(a, zero) = a) | (~less_equal(divide(a, zero), a)) | (~less_equal(a, divide(a, zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, zero) = a) | (~less_equal(divide(a, zero), a)) | (~less_equal(a, divide(a, zero))))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(133,plain,
% 0.19/0.46 (((~less_equal(divide(a, zero), a)) | (divide(a, zero) = a) | (~less_equal(a, divide(a, zero)))) <=> ((divide(a, zero) = a) | (~less_equal(divide(a, zero), a)) | (~less_equal(a, divide(a, zero))))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(134,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(a, zero), a)) | (divide(a, zero) = a) | (~less_equal(a, divide(a, zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(a, zero) = a) | (~less_equal(divide(a, zero), a)) | (~less_equal(a, divide(a, zero)))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[133])).
% 0.19/0.46 tff(135,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(a, zero), a)) | (divide(a, zero) = a) | (~less_equal(a, divide(a, zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, zero) = a) | (~less_equal(divide(a, zero), a)) | (~less_equal(a, divide(a, zero))))),
% 0.19/0.46 inference(transitivity,[status(thm)],[134, 132])).
% 0.19/0.46 tff(136,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(a, zero), a)) | (divide(a, zero) = a) | (~less_equal(a, divide(a, zero))))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(137,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, zero) = a) | (~less_equal(divide(a, zero), a)) | (~less_equal(a, divide(a, zero)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[136, 135])).
% 0.19/0.46 tff(138,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[137, 80, 131, 18, 128])).
% 0.19/0.46 % SZS output end Proof
%------------------------------------------------------------------------------