TSTP Solution File: HEN005-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : HEN005-3 : 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 22:39:47 EDT 2022
% Result : Unsatisfiable 0.19s 0.44s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : HEN005-3 : 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 : n009.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 21:39:20 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.
% 0.19/0.44 % SZS status Unsatisfiable
% 0.19/0.44 % SZS output start Proof
% 0.19/0.44 tff(less_equal_type, type, (
% 0.19/0.44 less_equal: ( $i * $i ) > $o)).
% 0.19/0.44 tff(zero_type, type, (
% 0.19/0.44 zero: $i)).
% 0.19/0.44 tff(divide_type, type, (
% 0.19/0.44 divide: ( $i * $i ) > $i)).
% 0.19/0.44 tff(c_type, type, (
% 0.19/0.44 c: $i)).
% 0.19/0.44 tff(a_type, type, (
% 0.19/0.44 a: $i)).
% 0.19/0.44 tff(b_type, type, (
% 0.19/0.44 b: $i)).
% 0.19/0.44 tff(1,plain,
% 0.19/0.44 (less_equal(a, b) <=> less_equal(a, b)),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(2,axiom,(less_equal(a, b)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_LE_b')).
% 0.19/0.44 tff(3,plain,
% 0.19/0.44 (less_equal(a, b)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[2, 1])).
% 0.19/0.44 tff(4,plain,
% 0.19/0.44 (^[Y: $i, X: $i] : refl(((~less_equal(X, Y)) | (divide(X, Y) = zero)) <=> ((~less_equal(X, Y)) | (divide(X, Y) = zero)))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(5,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero)) <=> ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[4])).
% 0.19/0.44 tff(6,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero)) <=> ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(7,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')).
% 0.19/0.44 tff(8,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.19/0.44 tff(9,plain,(
% 0.19/0.44 ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.44 inference(skolemize,[status(sab)],[8])).
% 0.19/0.44 tff(10,plain,
% 0.19/0.44 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.19/0.44 tff(11,plain,
% 0.19/0.44 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(a, b)) | (divide(a, b) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(a, b)) | (divide(a, b) = zero))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(12,plain,
% 0.19/0.44 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(a, b)) | (divide(a, b) = zero))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(13,plain,
% 0.19/0.44 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(a, b)) | (divide(a, b) = zero)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[12, 11])).
% 0.19/0.44 tff(14,plain,
% 0.19/0.44 (divide(a, b) = zero),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[13, 10, 3])).
% 0.19/0.44 tff(15,plain,
% 0.19/0.44 (divide(divide(a, b), c) = divide(zero, c)),
% 0.19/0.44 inference(monotonicity,[status(thm)],[14])).
% 0.19/0.44 tff(16,plain,
% 0.19/0.44 (divide(zero, c) = divide(divide(a, b), c)),
% 0.19/0.44 inference(symmetry,[status(thm)],[15])).
% 0.19/0.44 tff(17,assumption,(~less_equal(zero, c)), introduced(assumption)).
% 0.19/0.44 tff(18,plain,
% 0.19/0.44 (^[X: $i] : refl(less_equal(zero, X) <=> less_equal(zero, X))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(19,plain,
% 0.19/0.44 (![X: $i] : less_equal(zero, X) <=> ![X: $i] : less_equal(zero, X)),
% 0.19/0.44 inference(quant_intro,[status(thm)],[18])).
% 0.19/0.44 tff(20,plain,
% 0.19/0.44 (![X: $i] : less_equal(zero, X) <=> ![X: $i] : less_equal(zero, X)),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(21,axiom,(![X: $i] : less_equal(zero, X)), file('/export/starexec/sandbox/benchmark/Axioms/HEN002-0.ax','zero_is_smallest')).
% 0.19/0.44 tff(22,plain,
% 0.19/0.44 (![X: $i] : less_equal(zero, X)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.19/0.44 tff(23,plain,(
% 0.19/0.44 ![X: $i] : less_equal(zero, X)),
% 0.19/0.44 inference(skolemize,[status(sab)],[22])).
% 0.19/0.44 tff(24,plain,
% 0.19/0.44 (![X: $i] : less_equal(zero, X)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[23, 19])).
% 0.19/0.44 tff(25,plain,
% 0.19/0.44 ((~![X: $i] : less_equal(zero, X)) | less_equal(zero, c)),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(26,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[25, 24, 17])).
% 0.19/0.44 tff(27,plain,(less_equal(zero, c)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.44 tff(28,plain,
% 0.19/0.44 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(zero, c)) | (divide(zero, c) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(zero, c)) | (divide(zero, c) = zero))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(29,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(zero, c)) | (divide(zero, c) = zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(30,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(zero, c)) | (divide(zero, c) = zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.19/0.45 tff(31,plain,
% 0.19/0.45 ((~less_equal(zero, c)) | (divide(zero, c) = zero)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[30, 10])).
% 0.19/0.45 tff(32,plain,
% 0.19/0.45 (divide(zero, c) = zero),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[31, 27])).
% 0.19/0.45 tff(33,plain,
% 0.19/0.45 (zero = divide(zero, c)),
% 0.19/0.45 inference(symmetry,[status(thm)],[32])).
% 0.19/0.45 tff(34,plain,
% 0.19/0.45 (zero = divide(divide(a, b), c)),
% 0.19/0.45 inference(transitivity,[status(thm)],[33, 16])).
% 0.19/0.45 tff(35,plain,
% 0.19/0.45 (less_equal(b, c) <=> less_equal(b, c)),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(36,axiom,(less_equal(b, c)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','b_LE_c')).
% 0.19/0.45 tff(37,plain,
% 0.19/0.45 (less_equal(b, c)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[36, 35])).
% 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(b, c)) | (divide(b, c) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(b, c)) | (divide(b, c) = zero))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(39,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(b, c)) | (divide(b, c) = zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(40,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(b, c)) | (divide(b, c) = zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.19/0.45 tff(41,plain,
% 0.19/0.45 (divide(b, c) = zero),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[40, 10, 37])).
% 0.19/0.45 tff(42,plain,
% 0.19/0.45 (divide(divide(a, c), divide(b, c)) = divide(divide(a, c), zero)),
% 0.19/0.45 inference(monotonicity,[status(thm)],[41])).
% 0.19/0.45 tff(43,plain,
% 0.19/0.45 (divide(divide(a, c), zero) = divide(divide(a, c), divide(b, c))),
% 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, c), zero), zero) <=> less_equal(divide(divide(a, c), divide(b, c)), divide(divide(a, b), c))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[43, 34])).
% 0.19/0.45 tff(45,plain,
% 0.19/0.45 (less_equal(divide(divide(a, c), divide(b, c)), divide(divide(a, b), c)) <=> less_equal(divide(divide(a, c), zero), zero)),
% 0.19/0.45 inference(symmetry,[status(thm)],[44])).
% 0.19/0.45 tff(46,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(47,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)],[46])).
% 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(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(49,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(50,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)],[49, 48])).
% 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(skolemize,[status(sab)],[50])).
% 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(modus_ponens,[status(thm)],[51, 47])).
% 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))) | less_equal(divide(divide(a, c), divide(b, c)), divide(divide(a, b), c))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(54,plain,
% 0.19/0.45 (less_equal(divide(divide(a, c), divide(b, c)), divide(divide(a, b), c))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[53, 52])).
% 0.19/0.45 tff(55,plain,
% 0.19/0.45 (less_equal(divide(divide(a, c), zero), zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[54, 45])).
% 0.19/0.45 tff(56,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(57,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)],[56])).
% 0.19/0.45 tff(58,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(59,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(60,plain,
% 0.19/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.19/0.45 tff(61,plain,(
% 0.19/0.45 ![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.19/0.45 inference(skolemize,[status(sab)],[60])).
% 0.19/0.45 tff(62,plain,
% 0.19/0.45 (![Y: $i, X: $i] : less_equal(divide(X, Y), X)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[61, 57])).
% 0.19/0.45 tff(63,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(divide(a, c), zero), divide(a, c))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(64,plain,
% 0.19/0.45 (less_equal(divide(divide(a, c), zero), divide(a, c))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[63, 62])).
% 0.19/0.45 tff(65,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(divide(a, c), zero), divide(a, c))) | (divide(divide(divide(a, c), zero), divide(a, c)) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(divide(a, c), zero), divide(a, c))) | (divide(divide(divide(a, c), zero), divide(a, c)) = zero))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(66,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(divide(a, c), zero), divide(a, c))) | (divide(divide(divide(a, c), zero), divide(a, c)) = zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(67,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(divide(a, c), zero), divide(a, c))) | (divide(divide(divide(a, c), zero), divide(a, c)) = zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.19/0.45 tff(68,plain,
% 0.19/0.45 (divide(divide(divide(a, c), zero), divide(a, c)) = zero),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[67, 10, 64])).
% 0.19/0.45 tff(69,plain,
% 0.19/0.45 (zero = divide(divide(divide(a, c), zero), divide(a, c))),
% 0.19/0.45 inference(symmetry,[status(thm)],[68])).
% 0.19/0.45 tff(70,plain,
% 0.19/0.45 (less_equal(zero, divide(divide(a, c), zero)) <=> less_equal(divide(divide(divide(a, c), zero), divide(a, c)), divide(divide(a, c), zero))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[69])).
% 0.19/0.45 tff(71,plain,
% 0.19/0.45 (less_equal(divide(divide(divide(a, c), zero), divide(a, c)), divide(divide(a, c), zero)) <=> less_equal(zero, divide(divide(a, c), zero))),
% 0.19/0.45 inference(symmetry,[status(thm)],[70])).
% 0.19/0.45 tff(72,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(divide(divide(a, c), zero), divide(a, c)), divide(divide(a, c), zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(73,plain,
% 0.19/0.45 (less_equal(divide(divide(divide(a, c), zero), divide(a, c)), divide(divide(a, c), zero))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[72, 62])).
% 0.19/0.45 tff(74,plain,
% 0.19/0.45 (less_equal(zero, divide(divide(a, c), zero))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[73, 71])).
% 0.19/0.45 tff(75,assumption,(~(divide(divide(a, c), zero) = zero)), introduced(assumption)).
% 0.19/0.45 tff(76,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(77,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(78,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(79,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)))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[78])).
% 0.19/0.45 tff(80,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')).
% 0.19/0.45 tff(81,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.19/0.45 tff(82,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[81, 77])).
% 0.19/0.45 tff(83,plain,(
% 0.19/0.45 ![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.45 inference(skolemize,[status(sab)],[82])).
% 0.19/0.45 tff(84,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[83, 76])).
% 0.19/0.45 tff(85,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero))) | (~less_equal(divide(divide(a, c), zero), zero)))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero))) | (~less_equal(divide(divide(a, c), zero), zero)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(86,plain,
% 0.19/0.45 (((~less_equal(divide(divide(a, c), zero), zero)) | (divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero)))) <=> ((divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero))) | (~less_equal(divide(divide(a, c), zero), zero)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(87,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, c), zero), zero)) | (divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero))) | (~less_equal(divide(divide(a, c), zero), zero))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[86])).
% 0.19/0.45 tff(88,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, c), zero), zero)) | (divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero))) | (~less_equal(divide(divide(a, c), zero), zero)))),
% 0.19/0.45 inference(transitivity,[status(thm)],[87, 85])).
% 0.19/0.45 tff(89,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(divide(a, c), zero), zero)) | (divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero))))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(90,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(divide(a, c), zero) = zero) | (~less_equal(zero, divide(divide(a, c), zero))) | (~less_equal(divide(divide(a, c), zero), zero))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.19/0.45 tff(91,plain,
% 0.19/0.45 ($false),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[90, 84, 75, 74, 55])).
% 0.19/0.45 tff(92,plain,(divide(divide(a, c), zero) = zero), inference(lemma,lemma(discharge,[]))).
% 0.19/0.45 tff(93,plain,
% 0.19/0.45 (^[Y: $i, X: $i] : refl(((~(divide(X, Y) = zero)) | less_equal(X, Y)) <=> ((~(divide(X, Y) = zero)) | less_equal(X, Y)))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(94,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y)) <=> ![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[93])).
% 0.19/0.45 tff(95,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y)) <=> ![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(96,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')).
% 0.19/0.45 tff(97,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.19/0.45 tff(98,plain,(
% 0.19/0.45 ![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.45 inference(skolemize,[status(sab)],[97])).
% 0.19/0.45 tff(99,plain,
% 0.19/0.45 (![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[98, 94])).
% 0.19/0.45 tff(100,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | ((~(divide(divide(a, c), zero) = zero)) | less_equal(divide(a, c), zero))) <=> ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(divide(a, c), zero) = zero)) | less_equal(divide(a, c), zero))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(101,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | ((~(divide(divide(a, c), zero) = zero)) | less_equal(divide(a, c), zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(102,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(divide(a, c), zero) = zero)) | less_equal(divide(a, c), zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.19/0.45 tff(103,plain,
% 0.19/0.45 ((~(divide(divide(a, c), zero) = zero)) | less_equal(divide(a, c), zero)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[102, 99])).
% 0.19/0.45 tff(104,plain,
% 0.19/0.45 (less_equal(divide(a, c), zero)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[103, 92])).
% 0.19/0.45 tff(105,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(a, c), a)),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(106,plain,
% 0.19/0.45 (less_equal(divide(a, c), a)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[105, 62])).
% 0.19/0.45 tff(107,plain,
% 0.19/0.45 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(a, c), a)) | (divide(divide(a, c), a) = zero))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(a, c), a)) | (divide(divide(a, c), a) = zero))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(108,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | ((~less_equal(divide(a, c), a)) | (divide(divide(a, c), a) = zero))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(109,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (divide(X, Y) = zero))) | (~less_equal(divide(a, c), a)) | (divide(divide(a, c), a) = zero)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.19/0.45 tff(110,plain,
% 0.19/0.45 (divide(divide(a, c), a) = zero),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[109, 10, 106])).
% 0.19/0.45 tff(111,plain,
% 0.19/0.45 (zero = divide(divide(a, c), a)),
% 0.19/0.45 inference(symmetry,[status(thm)],[110])).
% 0.19/0.45 tff(112,plain,
% 0.19/0.45 (less_equal(zero, divide(a, c)) <=> less_equal(divide(divide(a, c), a), divide(a, c))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[111])).
% 0.19/0.45 tff(113,plain,
% 0.19/0.45 (less_equal(divide(divide(a, c), a), divide(a, c)) <=> less_equal(zero, divide(a, c))),
% 0.19/0.45 inference(symmetry,[status(thm)],[112])).
% 0.19/0.45 tff(114,plain,
% 0.19/0.45 ((~![Y: $i, X: $i] : less_equal(divide(X, Y), X)) | less_equal(divide(divide(a, c), a), divide(a, c))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(115,plain,
% 0.19/0.45 (less_equal(divide(divide(a, c), a), divide(a, c))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[114, 62])).
% 0.19/0.45 tff(116,plain,
% 0.19/0.45 (less_equal(zero, divide(a, c))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[115, 113])).
% 0.19/0.46 tff(117,plain,
% 0.19/0.46 ((~less_equal(a, c)) <=> (~less_equal(a, c))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(118,axiom,(~less_equal(a, c)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_a_LE_c')).
% 0.19/0.46 tff(119,plain,
% 0.19/0.46 (~less_equal(a, c)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.19/0.46 tff(120,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | ((~(divide(a, c) = zero)) | less_equal(a, c))) <=> ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(a, c) = zero)) | less_equal(a, c))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(121,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | ((~(divide(a, c) = zero)) | less_equal(a, c))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(122,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~(divide(X, Y) = zero)) | less_equal(X, Y))) | (~(divide(a, c) = zero)) | less_equal(a, c)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[121, 120])).
% 0.19/0.46 tff(123,plain,
% 0.19/0.46 (~(divide(a, c) = zero)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[122, 99, 119])).
% 0.19/0.46 tff(124,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(a, c) = zero) | (~less_equal(zero, divide(a, c))) | (~less_equal(divide(a, c), zero)))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, c) = zero) | (~less_equal(zero, divide(a, c))) | (~less_equal(divide(a, c), zero)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(125,plain,
% 0.19/0.46 (((~less_equal(divide(a, c), zero)) | (divide(a, c) = zero) | (~less_equal(zero, divide(a, c)))) <=> ((divide(a, c) = zero) | (~less_equal(zero, divide(a, c))) | (~less_equal(divide(a, c), zero)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(126,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(a, c), zero)) | (divide(a, c) = zero) | (~less_equal(zero, divide(a, c))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((divide(a, c) = zero) | (~less_equal(zero, divide(a, c))) | (~less_equal(divide(a, c), zero))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[125])).
% 0.19/0.46 tff(127,plain,
% 0.19/0.46 (((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(a, c), zero)) | (divide(a, c) = zero) | (~less_equal(zero, divide(a, c))))) <=> ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, c) = zero) | (~less_equal(zero, divide(a, c))) | (~less_equal(divide(a, c), zero)))),
% 0.19/0.46 inference(transitivity,[status(thm)],[126, 124])).
% 0.19/0.46 tff(128,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | ((~less_equal(divide(a, c), zero)) | (divide(a, c) = zero) | (~less_equal(zero, divide(a, c))))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(129,plain,
% 0.19/0.46 ((~![Y: $i, X: $i] : ((~less_equal(X, Y)) | (X = Y) | (~less_equal(Y, X)))) | (divide(a, c) = zero) | (~less_equal(zero, divide(a, c))) | (~less_equal(divide(a, c), zero))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[128, 127])).
% 0.19/0.46 tff(130,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[129, 84, 123, 116, 104])).
% 0.19/0.46 % SZS output end Proof
%------------------------------------------------------------------------------