TSTP Solution File: HEN004-3 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : HEN004-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n009.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 0.22s 0.45s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : HEN004-3 : TPTP v8.1.0. Released v1.0.0.
% 0.04/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.35 % Computer : n009.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Wed Aug 31 21:38:35 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.21/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.21/0.35 Usage: tptp [options] [-file:]file
% 0.21/0.35 -h, -? prints this message.
% 0.21/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.21/0.35 -m, -model generate model.
% 0.21/0.35 -p, -proof generate proof.
% 0.21/0.35 -c, -core generate unsat core of named formulas.
% 0.21/0.35 -st, -statistics display statistics.
% 0.21/0.35 -t:timeout set timeout (in second).
% 0.21/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.21/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.21/0.35 -<param>:<value> configuration parameter and value.
% 0.21/0.35 -o:<output-file> file to place output in.
% 0.22/0.45 % SZS status Unsatisfiable
% 0.22/0.45 % SZS output start Proof
% 0.22/0.45 tff(less_equal_type, type, (
% 0.22/0.45 less_equal: ( $i * $i ) > $o)).
% 0.22/0.45 tff(divide_type, type, (
% 0.22/0.45 divide: ( $i * $i ) > $i)).
% 0.22/0.45 tff(zero_type, type, (
% 0.22/0.45 zero: $i)).
% 0.22/0.45 tff(a_type, type, (
% 0.22/0.45 a: $i)).
% 0.22/0.45 tff(1,assumption,(~less_equal(divide(a, divide(a, zero)), zero)), introduced(assumption)).
% 0.22/0.45 tff(2,plain,
% 0.22/0.45 (^[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)))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(3,plain,
% 0.22/0.45 (![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))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[2])).
% 0.22/0.45 tff(4,plain,
% 0.22/0.45 (![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))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(5,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')).
% 0.22/0.45 tff(6,plain,
% 0.22/0.45 (![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.22/0.45 tff(7,plain,(
% 0.22/0.45 ![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))),
% 0.22/0.45 inference(skolemize,[status(sab)],[6])).
% 0.22/0.45 tff(8,plain,
% 0.22/0.45 (![Z: $i, Y: $i, X: $i] : less_equal(divide(divide(X, Z), divide(Y, Z)), divide(divide(X, Y), Z))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.22/0.45 tff(9,plain,
% 0.22/0.45 ((~![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, zero), divide(a, zero)), divide(divide(a, a), zero))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(10,plain,
% 0.22/0.45 (less_equal(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[9, 8])).
% 0.22/0.45 tff(11,plain,
% 0.22/0.45 (^[Y: $i, X: $i] : refl(((~less_equal(X, Y)) | (divide(X, Y) = zero)) <=> ((~less_equal(X, Y)) | (divide(X, Y) = zero)))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(12,plain,
% 0.22/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero)) <=> ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[11])).
% 0.22/0.45 tff(13,plain,
% 0.22/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero)) <=> ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(14,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')).
% 0.22/0.45 tff(15,plain,
% 0.22/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.22/0.45 tff(16,plain,(
% 0.22/0.45 ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.22/0.45 inference(skolemize,[status(sab)],[15])).
% 0.22/0.45 tff(17,plain,
% 0.22/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.22/0.45 tff(18,plain,
% 0.22/0.45 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero))) | (divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero))) | (divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)) = zero))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(19,plain,
% 0.22/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero))) | (divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)) = zero))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(20,plain,
% 0.22/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero))) | (divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)) = zero)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[19, 18])).
% 0.22/0.45 tff(21,plain,
% 0.22/0.45 (divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)) = zero),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[20, 17, 10])).
% 0.22/0.45 tff(22,plain,
% 0.22/0.45 (zero = divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero))),
% 0.22/0.45 inference(symmetry,[status(thm)],[21])).
% 0.22/0.45 tff(23,plain,
% 0.22/0.45 (less_equal(zero, divide(divide(a, zero), divide(a, zero))) <=> less_equal(divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)), divide(divide(a, zero), divide(a, zero)))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[22])).
% 0.22/0.45 tff(24,plain,
% 0.22/0.45 (less_equal(divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)), divide(divide(a, zero), divide(a, zero))) <=> less_equal(zero, divide(divide(a, zero), divide(a, zero)))),
% 0.22/0.45 inference(symmetry,[status(thm)],[23])).
% 0.22/0.45 tff(25,plain,
% 0.22/0.45 (^[Y: $i, X: $i] : refl(less_equal(divide(X, Y), X) <=> less_equal(divide(X, Y), X))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(26,plain,
% 0.22/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X) <=> ![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.22/0.45 inference(quant_intro,[status(thm)],[25])).
% 0.22/0.45 tff(27,plain,
% 0.22/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X) <=> ![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(28,axiom,(![Y: $i, X: $i] : less_equal(divide(X, Y), X)), file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax','quotient_smaller_than_numerator')).
% 0.22/0.45 tff(29,plain,
% 0.22/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.22/0.45 tff(30,plain,(
% 0.22/0.45 ![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.22/0.45 inference(skolemize,[status(sab)],[29])).
% 0.22/0.45 tff(31,plain,
% 0.22/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.22/0.45 tff(32,plain,
% 0.22/0.45 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)), divide(divide(a, zero), divide(a, zero)))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(33,plain,
% 0.22/0.45 (less_equal(divide(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)), divide(divide(a, zero), divide(a, zero)))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[32, 31])).
% 0.22/0.45 tff(34,plain,
% 0.22/0.45 (less_equal(zero, divide(divide(a, zero), divide(a, zero)))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[33, 24])).
% 0.22/0.45 tff(35,assumption,(~less_equal(zero, divide(divide(a, zero), divide(a, zero)))), introduced(assumption)).
% 0.22/0.45 tff(36,plain,
% 0.22/0.45 ($false),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[35, 34])).
% 0.22/0.45 tff(37,plain,(less_equal(zero, divide(divide(a, zero), divide(a, zero)))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.45 tff(38,assumption,(less_equal(zero, divide(divide(a, zero), divide(a, zero)))), introduced(assumption)).
% 0.22/0.45 tff(39,plain,
% 0.22/0.45 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(divide(a, a), zero), divide(a, a))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(40,plain,
% 0.22/0.45 (less_equal(divide(divide(a, a), zero), divide(a, a))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[39, 31])).
% 0.22/0.45 tff(41,plain,
% 0.22/0.45 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(divide(a, a), zero), divide(a, a))) | (divide(divide(divide(a, a), zero), divide(a, a)) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(divide(a, a), zero), divide(a, a))) | (divide(divide(divide(a, a), zero), divide(a, a)) = zero))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(42,plain,
% 0.22/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(divide(a, a), zero), divide(a, a))) | (divide(divide(divide(a, a), zero), divide(a, a)) = zero))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(43,plain,
% 0.22/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(divide(a, a), zero), divide(a, a))) | (divide(divide(divide(a, a), zero), divide(a, a)) = zero)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.22/0.45 tff(44,plain,
% 0.22/0.45 (divide(divide(divide(a, a), zero), divide(a, a)) = zero),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[43, 17, 40])).
% 0.22/0.45 tff(45,plain,
% 0.22/0.45 (zero = divide(divide(divide(a, a), zero), divide(a, a))),
% 0.22/0.45 inference(symmetry,[status(thm)],[44])).
% 0.22/0.45 tff(46,plain,
% 0.22/0.45 (less_equal(zero, divide(divide(a, a), zero)) <=> less_equal(divide(divide(divide(a, a), zero), divide(a, a)), divide(divide(a, a), zero))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[45])).
% 0.22/0.45 tff(47,plain,
% 0.22/0.45 (less_equal(divide(divide(divide(a, a), zero), divide(a, a)), divide(divide(a, a), zero)) <=> less_equal(zero, divide(divide(a, a), zero))),
% 0.22/0.45 inference(symmetry,[status(thm)],[46])).
% 0.22/0.45 tff(48,plain,
% 0.22/0.45 ((~less_equal(divide(divide(divide(a, a), zero), divide(a, a)), divide(divide(a, a), zero))) <=> (~less_equal(zero, divide(divide(a, a), zero)))),
% 0.22/0.45 inference(monotonicity,[status(thm)],[47])).
% 0.22/0.45 tff(49,assumption,(~less_equal(divide(divide(divide(a, a), zero), divide(a, a)), divide(divide(a, a), zero))), introduced(assumption)).
% 0.22/0.45 tff(50,plain,
% 0.22/0.45 (~less_equal(zero, divide(divide(a, a), zero))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.22/0.45 tff(51,plain,
% 0.22/0.45 (^[X: $i] : refl(less_equal(zero, X) <=> less_equal(zero, X))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(52,plain,
% 0.22/0.45 (![X: $i] : less_equal(zero, X) <=> ![X: $i] : less_equal(zero, X)),
% 0.22/0.45 inference(quant_intro,[status(thm)],[51])).
% 0.22/0.45 tff(53,plain,
% 0.22/0.45 (![X: $i] : less_equal(zero, X) <=> ![X: $i] : less_equal(zero, X)),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(54,axiom,(![X: $i] : less_equal(zero, X)), file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax','zero_is_smallest')).
% 0.22/0.45 tff(55,plain,
% 0.22/0.45 (![X: $i] : less_equal(zero, X)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.22/0.45 tff(56,plain,(
% 0.22/0.45 ![X: $i] : less_equal(zero, X)),
% 0.22/0.45 inference(skolemize,[status(sab)],[55])).
% 0.22/0.45 tff(57,plain,
% 0.22/0.45 (![X: $i] : less_equal(zero, X)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[56, 52])).
% 0.22/0.45 tff(58,plain,
% 0.22/0.45 ((~![X: $i] : less_equal(zero, X)) | less_equal(zero, divide(divide(a, a), zero))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(59,plain,
% 0.22/0.45 (less_equal(zero, divide(divide(a, a), zero))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.22/0.45 tff(60,plain,
% 0.22/0.45 ($false),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[59, 50])).
% 0.22/0.45 tff(61,plain,(less_equal(divide(divide(divide(a, a), zero), divide(a, a)), divide(divide(a, a), zero))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.45 tff(62,plain,
% 0.22/0.45 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(a, divide(a, zero)), a)),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(63,plain,
% 0.22/0.45 (less_equal(divide(a, divide(a, zero)), a)),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[62, 31])).
% 0.22/0.45 tff(64,plain,
% 0.22/0.45 (((~![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))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(65,plain,
% 0.22/0.45 ((~![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))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(66,plain,
% 0.22/0.45 ((~![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)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.22/0.45 tff(67,plain,
% 0.22/0.45 ((~less_equal(divide(a, divide(a, zero)), a)) | (divide(divide(a, divide(a, zero)), a) = zero)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[66, 17])).
% 0.22/0.46 tff(68,plain,
% 0.22/0.46 (divide(divide(a, divide(a, zero)), a) = zero),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[67, 63])).
% 0.22/0.46 tff(69,plain,
% 0.22/0.46 (zero = divide(divide(a, divide(a, zero)), a)),
% 0.22/0.46 inference(symmetry,[status(thm)],[68])).
% 0.22/0.46 tff(70,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(a, zero), a)),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(71,plain,
% 0.22/0.46 (less_equal(divide(a, zero), a)),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[70, 31])).
% 0.22/0.46 tff(72,plain,
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% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(73,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(a, zero), a)) | (divide(divide(a, zero), a) = zero))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(74,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(a, zero), a)) | (divide(divide(a, zero), a) = zero)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.22/0.46 tff(75,plain,
% 0.22/0.46 (divide(divide(a, zero), a) = zero),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[74, 17, 71])).
% 0.22/0.46 tff(76,plain,
% 0.22/0.46 (divide(divide(a, a), divide(divide(a, zero), a)) = divide(divide(a, a), zero)),
% 0.22/0.46 inference(monotonicity,[status(thm)],[75])).
% 0.22/0.46 tff(77,plain,
% 0.22/0.46 (divide(divide(a, a), zero) = divide(divide(a, a), divide(divide(a, zero), a))),
% 0.22/0.46 inference(symmetry,[status(thm)],[76])).
% 0.22/0.46 tff(78,plain,
% 0.22/0.46 (less_equal(divide(divide(a, a), zero), zero) <=> less_equal(divide(divide(a, a), divide(divide(a, zero), a)), divide(divide(a, divide(a, zero)), a))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[77, 69])).
% 0.22/0.46 tff(79,plain,
% 0.22/0.46 (less_equal(divide(divide(a, a), divide(divide(a, zero), a)), divide(divide(a, divide(a, zero)), a)) <=> less_equal(divide(divide(a, a), zero), zero)),
% 0.22/0.46 inference(symmetry,[status(thm)],[78])).
% 0.22/0.46 tff(80,plain,
% 0.22/0.46 ((~![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, a), divide(divide(a, zero), a)), divide(divide(a, divide(a, zero)), a))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(81,plain,
% 0.22/0.46 (less_equal(divide(divide(a, a), divide(divide(a, zero), a)), divide(divide(a, divide(a, zero)), a))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[80, 8])).
% 0.22/0.46 tff(82,plain,
% 0.22/0.46 (less_equal(divide(divide(a, a), zero), zero)),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[81, 79])).
% 0.22/0.46 tff(83,assumption,(less_equal(divide(divide(divide(a, a), zero), divide(a, a)), divide(divide(a, a), zero))), introduced(assumption)).
% 0.22/0.46 tff(84,plain,
% 0.22/0.46 (less_equal(zero, divide(divide(a, a), zero))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[83, 47])).
% 0.22/0.46 tff(85,assumption,(~(divide(divide(a, a), zero) = zero)), introduced(assumption)).
% 0.22/0.46 tff(86,plain,
% 0.22/0.46 (![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)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(87,plain,
% 0.22/0.46 (![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)))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(88,plain,
% 0.22/0.46 (^[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))))),
% 0.22/0.46 inference(bind,[status(th)],[])).
% 0.22/0.46 tff(89,plain,
% 0.22/0.46 (![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)))),
% 0.22/0.46 inference(quant_intro,[status(thm)],[88])).
% 0.22/0.46 tff(90,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')).
% 0.22/0.46 tff(91,plain,
% 0.22/0.46 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.22/0.46 tff(92,plain,
% 0.22/0.46 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[91, 87])).
% 0.22/0.46 tff(93,plain,(
% 0.22/0.46 ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.22/0.46 inference(skolemize,[status(sab)],[92])).
% 0.22/0.46 tff(94,plain,
% 0.22/0.46 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[93, 86])).
% 0.22/0.46 tff(95,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(divide(a, a), zero) = zero) | (~less_equal(divide(divide(a, a), zero), zero)) | (~less_equal(zero, divide(divide(a, a), zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, a), zero) = zero) | (~less_equal(divide(divide(a, a), zero), zero)) | (~less_equal(zero, divide(divide(a, a), zero))))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(96,plain,
% 0.22/0.46 (((~less_equal(divide(divide(a, a), zero), zero)) | (divide(divide(a, a), zero) = zero) | (~less_equal(zero, divide(divide(a, a), zero)))) <=> ((divide(divide(a, a), zero) = zero) | (~less_equal(divide(divide(a, a), zero), zero)) | (~less_equal(zero, divide(divide(a, a), zero))))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(97,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, a), zero), zero)) | (divide(divide(a, a), zero) = zero) | (~less_equal(zero, divide(divide(a, a), zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(divide(a, a), zero) = zero) | (~less_equal(divide(divide(a, a), zero), zero)) | (~less_equal(zero, divide(divide(a, a), zero)))))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[96])).
% 0.22/0.46 tff(98,plain,
% 0.22/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, a), zero), zero)) | (divide(divide(a, a), zero) = zero) | (~less_equal(zero, divide(divide(a, a), zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, a), zero) = zero) | (~less_equal(divide(divide(a, a), zero), zero)) | (~less_equal(zero, divide(divide(a, a), zero))))),
% 0.22/0.46 inference(transitivity,[status(thm)],[97, 95])).
% 0.22/0.46 tff(99,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, a), zero), zero)) | (divide(divide(a, a), zero) = zero) | (~less_equal(zero, divide(divide(a, a), zero))))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(100,plain,
% 0.22/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, a), zero) = zero) | (~less_equal(divide(divide(a, a), zero), zero)) | (~less_equal(zero, divide(divide(a, a), zero)))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.22/0.46 tff(101,plain,
% 0.22/0.46 ((~less_equal(divide(divide(a, a), zero), zero)) | (~less_equal(zero, divide(divide(a, a), zero)))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[100, 94, 85])).
% 0.22/0.46 tff(102,plain,
% 0.22/0.46 ($false),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[101, 84, 82])).
% 0.22/0.46 tff(103,plain,((~less_equal(divide(divide(divide(a, a), zero), divide(a, a)), divide(divide(a, a), zero))) | (divide(divide(a, a), zero) = zero)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.46 tff(104,plain,
% 0.22/0.46 (divide(divide(a, a), zero) = zero),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[103, 61])).
% 0.22/0.46 tff(105,plain,
% 0.22/0.46 (zero = divide(divide(a, a), zero)),
% 0.22/0.46 inference(symmetry,[status(thm)],[104])).
% 0.22/0.46 tff(106,plain,
% 0.22/0.46 (less_equal(divide(divide(a, zero), divide(a, zero)), zero) <=> less_equal(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[105])).
% 0.22/0.46 tff(107,plain,
% 0.22/0.46 (less_equal(divide(divide(a, zero), divide(a, zero)), divide(divide(a, a), zero)) <=> less_equal(divide(divide(a, zero), divide(a, zero)), zero)),
% 0.22/0.46 inference(symmetry,[status(thm)],[106])).
% 0.22/0.47 tff(108,plain,
% 0.22/0.47 (less_equal(divide(divide(a, zero), divide(a, zero)), zero)),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[10, 107])).
% 0.22/0.47 tff(109,assumption,(~(divide(divide(a, zero), divide(a, zero)) = zero)), introduced(assumption)).
% 0.22/0.47 tff(110,plain,
% 0.22/0.47 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(divide(a, zero), divide(a, zero)) = zero) | (~less_equal(zero, divide(divide(a, zero), divide(a, zero)))) | (~less_equal(divide(divide(a, zero), divide(a, zero)), zero)))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, zero), divide(a, zero)) = zero) | (~less_equal(zero, divide(divide(a, zero), divide(a, zero)))) | (~less_equal(divide(divide(a, zero), divide(a, zero)), zero)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(111,plain,
% 0.22/0.47 (((~less_equal(divide(divide(a, zero), divide(a, zero)), zero)) | (divide(divide(a, zero), divide(a, zero)) = zero) | (~less_equal(zero, divide(divide(a, zero), divide(a, zero))))) <=> ((divide(divide(a, zero), divide(a, zero)) = zero) | (~less_equal(zero, divide(divide(a, zero), divide(a, zero)))) | (~less_equal(divide(divide(a, zero), divide(a, zero)), zero)))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(112,plain,
% 0.22/0.47 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, zero), divide(a, zero)), zero)) | (divide(divide(a, zero), divide(a, zero)) = zero) | (~less_equal(zero, divide(divide(a, zero), divide(a, zero)))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(divide(a, zero), divide(a, zero)) = zero) | (~less_equal(zero, divide(divide(a, zero), divide(a, zero)))) | (~less_equal(divide(divide(a, zero), divide(a, zero)), zero))))),
% 0.22/0.47 inference(monotonicity,[status(thm)],[111])).
% 0.22/0.47 tff(113,plain,
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% 0.22/0.47 inference(transitivity,[status(thm)],[112, 110])).
% 0.22/0.47 tff(114,plain,
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% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
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% 0.22/0.47 inference(modus_ponens,[status(thm)],[114, 113])).
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% 0.22/0.47 ((~less_equal(zero, divide(divide(a, zero), divide(a, zero)))) | (~less_equal(divide(divide(a, zero), divide(a, zero)), zero))),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[115, 94, 109])).
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% 0.22/0.47 inference(unit_resolution,[status(thm)],[116, 108, 38])).
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% 0.22/0.47 (divide(divide(a, zero), divide(a, zero)) = zero),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[118, 37])).
% 0.22/0.47 tff(120,plain,
% 0.22/0.47 (divide(a, divide(divide(a, zero), divide(a, zero))) = divide(a, zero)),
% 0.22/0.47 inference(monotonicity,[status(thm)],[119])).
% 0.22/0.47 tff(121,plain,
% 0.22/0.47 (divide(divide(a, divide(divide(a, zero), divide(a, zero))), divide(a, zero)) = divide(divide(a, zero), divide(a, zero))),
% 0.22/0.47 inference(monotonicity,[status(thm)],[120])).
% 0.22/0.47 tff(122,plain,
% 0.22/0.47 (divide(divide(a, divide(divide(a, zero), divide(a, zero))), divide(a, zero)) = zero),
% 0.22/0.47 inference(transitivity,[status(thm)],[121, 119])).
% 0.22/0.47 tff(123,plain,
% 0.22/0.47 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(divide(a, zero), divide(a, zero)), divide(a, zero))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(124,plain,
% 0.22/0.47 (less_equal(divide(divide(a, zero), divide(a, zero)), divide(a, zero))),
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% 0.22/0.47 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(divide(a, zero), divide(a, zero)), divide(a, zero))) | (divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero)) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(divide(a, zero), divide(a, zero)), divide(a, zero))) | (divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero)) = zero))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(126,plain,
% 0.22/0.47 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(divide(a, zero), divide(a, zero)), divide(a, zero))) | (divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero)) = zero))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(127,plain,
% 0.22/0.47 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(divide(a, zero), divide(a, zero)), divide(a, zero))) | (divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero)) = zero)),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.22/0.47 tff(128,plain,
% 0.22/0.47 (divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero)) = zero),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[127, 17, 124])).
% 0.22/0.47 tff(129,plain,
% 0.22/0.47 (divide(divide(a, divide(a, zero)), divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero))) = divide(divide(a, divide(a, zero)), zero)),
% 0.22/0.47 inference(monotonicity,[status(thm)],[128])).
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% 0.22/0.47 (less_equal(divide(divide(a, divide(a, zero)), divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero))), divide(divide(a, divide(divide(a, zero), divide(a, zero))), divide(a, zero))) <=> less_equal(divide(divide(a, divide(a, zero)), zero), zero)),
% 0.22/0.47 inference(monotonicity,[status(thm)],[129, 122])).
% 0.22/0.47 tff(131,plain,
% 0.22/0.47 (less_equal(divide(divide(a, divide(a, zero)), zero), zero) <=> less_equal(divide(divide(a, divide(a, zero)), divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero))), divide(divide(a, divide(divide(a, zero), divide(a, zero))), divide(a, zero)))),
% 0.22/0.47 inference(symmetry,[status(thm)],[130])).
% 0.22/0.47 tff(132,plain,
% 0.22/0.47 ((~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) <=> (~less_equal(divide(divide(a, divide(a, zero)), divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero))), divide(divide(a, divide(divide(a, zero), divide(a, zero))), divide(a, zero))))),
% 0.22/0.47 inference(monotonicity,[status(thm)],[131])).
% 0.22/0.47 tff(133,assumption,(~less_equal(divide(divide(a, divide(a, zero)), zero), zero)), introduced(assumption)).
% 0.22/0.47 tff(134,plain,
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% 0.22/0.47 inference(modus_ponens,[status(thm)],[133, 132])).
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% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(136,plain,
% 0.22/0.47 (less_equal(divide(divide(a, divide(a, zero)), divide(divide(divide(a, zero), divide(a, zero)), divide(a, zero))), divide(divide(a, divide(divide(a, zero), divide(a, zero))), divide(a, zero)))),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[135, 8])).
% 0.22/0.47 tff(137,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[136, 134])).
% 0.22/0.47 tff(138,plain,(less_equal(divide(divide(a, divide(a, zero)), zero), zero)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.47 tff(139,assumption,(~less_equal(zero, divide(divide(a, divide(a, zero)), zero))), introduced(assumption)).
% 0.22/0.47 tff(140,plain,
% 0.22/0.47 ((~![X: $i] : less_equal(zero, X)) | less_equal(zero, divide(divide(a, divide(a, zero)), zero))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(141,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[140, 57, 139])).
% 0.22/0.47 tff(142,plain,(less_equal(zero, divide(divide(a, divide(a, zero)), zero))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.47 tff(143,assumption,(less_equal(zero, divide(divide(a, divide(a, zero)), zero))), introduced(assumption)).
% 0.22/0.47 tff(144,assumption,(less_equal(divide(divide(a, divide(a, zero)), zero), zero)), introduced(assumption)).
% 0.22/0.47 tff(145,assumption,(~(divide(divide(a, divide(a, zero)), zero) = zero)), introduced(assumption)).
% 0.22/0.47 tff(146,plain,
% 0.22/0.47 (((~![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))))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(147,plain,
% 0.22/0.47 (((~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))))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(148,plain,
% 0.22/0.47 (((~![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)))))),
% 0.22/0.47 inference(monotonicity,[status(thm)],[147])).
% 0.22/0.47 tff(149,plain,
% 0.22/0.47 (((~![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))))),
% 0.22/0.47 inference(transitivity,[status(thm)],[148, 146])).
% 0.22/0.47 tff(150,plain,
% 0.22/0.47 ((~![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))))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(151,plain,
% 0.22/0.47 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero)) | (~less_equal(zero, divide(divide(a, divide(a, zero)), zero)))),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[150, 149])).
% 0.22/0.47 tff(152,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[151, 94, 145, 144, 143])).
% 0.22/0.47 tff(153,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,[]))).
% 0.22/0.47 tff(154,plain,
% 0.22/0.47 ((divide(divide(a, divide(a, zero)), zero) = zero) | (~less_equal(divide(divide(a, divide(a, zero)), zero), zero))),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[153, 142])).
% 0.22/0.48 tff(155,plain,
% 0.22/0.48 (divide(divide(a, divide(a, zero)), zero) = zero),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[154, 138])).
% 0.22/0.48 tff(156,plain,
% 0.22/0.48 (^[Y: $i, X: $i] : refl(((~(divide(X, Y) = zero)) | less_equal(X, Y)) <=> ((~(divide(X, Y) = zero)) | less_equal(X, Y)))),
% 0.22/0.48 inference(bind,[status(th)],[])).
% 0.22/0.48 tff(157,plain,
% 0.22/0.48 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y)) <=> ![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.22/0.48 inference(quant_intro,[status(thm)],[156])).
% 0.22/0.48 tff(158,plain,
% 0.22/0.48 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y)) <=> ![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(159,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')).
% 0.22/0.48 tff(160,plain,
% 0.22/0.48 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[159, 158])).
% 0.22/0.48 tff(161,plain,(
% 0.22/0.48 ![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.22/0.48 inference(skolemize,[status(sab)],[160])).
% 0.22/0.48 tff(162,plain,
% 0.22/0.48 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[161, 157])).
% 0.22/0.48 tff(163,plain,
% 0.22/0.48 (((~![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))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(164,plain,
% 0.22/0.48 ((~![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))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(165,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(divide(a, divide(a, zero)), zero) = zero)) | less_equal(divide(a, divide(a, zero)), zero)),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[164, 163])).
% 0.22/0.48 tff(166,plain,
% 0.22/0.48 ($false),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[165, 162, 155, 1])).
% 0.22/0.48 tff(167,plain,(less_equal(divide(a, divide(a, zero)), zero)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.48 tff(168,assumption,(less_equal(divide(a, divide(a, zero)), zero)), introduced(assumption)).
% 0.22/0.48 tff(169,plain,
% 0.22/0.48 (less_equal(zero, divide(a, divide(a, zero))) <=> less_equal(divide(divide(a, divide(a, zero)), a), divide(a, divide(a, zero)))),
% 0.22/0.48 inference(monotonicity,[status(thm)],[69])).
% 0.22/0.48 tff(170,plain,
% 0.22/0.48 (less_equal(divide(divide(a, divide(a, zero)), a), divide(a, divide(a, zero))) <=> less_equal(zero, divide(a, divide(a, zero)))),
% 0.22/0.48 inference(symmetry,[status(thm)],[169])).
% 0.22/0.48 tff(171,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(divide(a, divide(a, zero)), a), divide(a, divide(a, zero)))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(172,plain,
% 0.22/0.48 (less_equal(divide(divide(a, divide(a, zero)), a), divide(a, divide(a, zero)))),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[171, 31])).
% 0.22/0.48 tff(173,plain,
% 0.22/0.48 (less_equal(zero, divide(a, divide(a, zero)))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[172, 170])).
% 0.22/0.48 tff(174,assumption,(~(divide(a, divide(a, zero)) = zero)), introduced(assumption)).
% 0.22/0.48 tff(175,plain,
% 0.22/0.48 (((~![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)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(176,plain,
% 0.22/0.48 (((~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)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(177,plain,
% 0.22/0.48 (((~![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))))),
% 0.22/0.48 inference(monotonicity,[status(thm)],[176])).
% 0.22/0.48 tff(178,plain,
% 0.22/0.48 (((~![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)))),
% 0.22/0.48 inference(transitivity,[status(thm)],[177, 175])).
% 0.22/0.48 tff(179,plain,
% 0.22/0.48 ((~![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)))))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(180,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, divide(a, zero)) = zero) | (~less_equal(zero, divide(a, divide(a, zero)))) | (~less_equal(divide(a, divide(a, zero)), zero))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[179, 178])).
% 0.22/0.48 tff(181,plain,
% 0.22/0.48 ($false),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[180, 94, 174, 173, 168])).
% 0.22/0.48 tff(182,plain,((~less_equal(divide(a, divide(a, zero)), zero)) | (divide(a, divide(a, zero)) = zero)), inference(lemma,lemma(discharge,[]))).
% 0.22/0.48 tff(183,plain,
% 0.22/0.48 (divide(a, divide(a, zero)) = zero),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[182, 167])).
% 0.22/0.48 tff(184,plain,
% 0.22/0.48 (((~![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)))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(185,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | ((~(divide(a, divide(a, zero)) = zero)) | less_equal(a, divide(a, zero)))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(186,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(a, divide(a, zero)) = zero)) | less_equal(a, divide(a, zero))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[185, 184])).
% 0.22/0.48 tff(187,plain,
% 0.22/0.48 ((~(divide(a, divide(a, zero)) = zero)) | less_equal(a, divide(a, zero))),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[186, 162])).
% 0.22/0.48 tff(188,plain,
% 0.22/0.48 (less_equal(a, divide(a, zero))),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[187, 183])).
% 0.22/0.48 tff(189,plain,
% 0.22/0.48 ((~(divide(a, zero) = a)) <=> (~(divide(a, zero) = a))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(190,axiom,(~(divide(a, zero) = a)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_x_divide_zero_is_x')).
% 0.22/0.48 tff(191,plain,
% 0.22/0.48 (~(divide(a, zero) = a)),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[190, 189])).
% 0.22/0.48 tff(192,plain,
% 0.22/0.48 (((~![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))))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(193,plain,
% 0.22/0.48 (((~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))))),
% 0.22/0.48 inference(rewrite,[status(thm)],[])).
% 0.22/0.48 tff(194,plain,
% 0.22/0.48 (((~![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)))))),
% 0.22/0.48 inference(monotonicity,[status(thm)],[193])).
% 0.22/0.48 tff(195,plain,
% 0.22/0.48 (((~![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))))),
% 0.22/0.48 inference(transitivity,[status(thm)],[194, 192])).
% 0.22/0.48 tff(196,plain,
% 0.22/0.48 ((~![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))))),
% 0.22/0.48 inference(quant_inst,[status(thm)],[])).
% 0.22/0.48 tff(197,plain,
% 0.22/0.48 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, zero) = a) | (~less_equal(divide(a, zero), a)) | (~less_equal(a, divide(a, zero)))),
% 0.22/0.48 inference(modus_ponens,[status(thm)],[196, 195])).
% 0.22/0.48 tff(198,plain,
% 0.22/0.48 ($false),
% 0.22/0.48 inference(unit_resolution,[status(thm)],[197, 94, 191, 71, 188])).
% 0.22/0.48 % SZS output end Proof
%------------------------------------------------------------------------------