TSTP Solution File: ANA003-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : ANA003-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 6 16:23:20 EDT 2022
% Result : Unsatisfiable 0.76s 0.74s
% Output : Proof 0.76s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.11 % Problem : ANA003-3 : TPTP v8.1.0. Released v1.0.0.
% 0.09/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.33 % Computer : n006.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Mon Aug 29 17:45:55 EDT 2022
% 0.13/0.33 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 0.76/0.74 % SZS status Unsatisfiable
% 0.76/0.74 % SZS output start Proof
% 0.76/0.74 tff(less_or_equal_type, type, (
% 0.76/0.74 less_or_equal: ( $i * $i ) > $o)).
% 0.76/0.74 tff(n0_type, type, (
% 0.76/0.74 n0: $i)).
% 0.76/0.74 tff(half_type, type, (
% 0.76/0.74 half: $i > $i)).
% 0.76/0.74 tff(epsilon_0_type, type, (
% 0.76/0.74 epsilon_0: $i)).
% 0.76/0.74 tff(absolute_type, type, (
% 0.76/0.74 absolute: $i > $i)).
% 0.76/0.74 tff(add_type, type, (
% 0.76/0.74 add: ( $i * $i ) > $i)).
% 0.76/0.74 tff(negate_type, type, (
% 0.76/0.74 negate: $i > $i)).
% 0.76/0.74 tff(g_type, type, (
% 0.76/0.74 g: $i > $i)).
% 0.76/0.74 tff(a_real_number_type, type, (
% 0.76/0.74 a_real_number: $i)).
% 0.76/0.74 tff(xs_type, type, (
% 0.76/0.74 xs: $i > $i)).
% 0.76/0.74 tff(delta_1_type, type, (
% 0.76/0.74 delta_1: $i > $i)).
% 0.76/0.74 tff(f_type, type, (
% 0.76/0.74 f: $i > $i)).
% 0.76/0.74 tff(delta_2_type, type, (
% 0.76/0.74 delta_2: $i > $i)).
% 0.76/0.74 tff(1,assumption,(~less_or_equal(half(epsilon_0), n0)), introduced(assumption)).
% 0.76/0.74 tff(2,plain,
% 0.76/0.74 (^[Epsilon: $i] : refl((less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0))) <=> (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0))))),
% 0.76/0.74 inference(bind,[status(th)],[])).
% 0.76/0.74 tff(3,plain,
% 0.76/0.74 (![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0))) <=> ![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))),
% 0.76/0.74 inference(quant_intro,[status(thm)],[2])).
% 0.76/0.74 tff(4,plain,
% 0.76/0.74 (![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0))) <=> ![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))),
% 0.76/0.74 inference(rewrite,[status(thm)],[])).
% 0.76/0.74 tff(5,axiom,(![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause_1')).
% 0.76/0.74 tff(6,plain,
% 0.76/0.74 (![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))),
% 0.76/0.74 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.76/0.74 tff(7,plain,(
% 0.76/0.74 ![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))),
% 0.76/0.74 inference(skolemize,[status(sab)],[6])).
% 0.76/0.74 tff(8,plain,
% 0.76/0.74 (![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))),
% 0.76/0.74 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.76/0.74 tff(9,plain,
% 0.76/0.74 (((~![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))) | (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(delta_1(half(epsilon_0)), n0)))) <=> ((~![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(delta_1(half(epsilon_0)), n0)))),
% 0.76/0.74 inference(rewrite,[status(thm)],[])).
% 0.76/0.74 tff(10,plain,
% 0.76/0.74 ((~![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))) | (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(delta_1(half(epsilon_0)), n0)))),
% 0.76/0.74 inference(quant_inst,[status(thm)],[])).
% 0.76/0.74 tff(11,plain,
% 0.76/0.74 ((~![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_1(Epsilon), n0)))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(delta_1(half(epsilon_0)), n0))),
% 0.76/0.74 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.76/0.74 tff(12,plain,
% 0.76/0.74 (~less_or_equal(delta_1(half(epsilon_0)), n0)),
% 0.76/0.74 inference(unit_resolution,[status(thm)],[11, 8, 1])).
% 0.76/0.74 tff(13,plain,
% 0.76/0.74 (^[Delta: $i] : refl((less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta)) <=> (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta)))),
% 0.76/0.74 inference(bind,[status(th)],[])).
% 0.76/0.74 tff(14,plain,
% 0.76/0.74 (![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta)) <=> ![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))),
% 0.76/0.74 inference(quant_intro,[status(thm)],[13])).
% 0.76/0.74 tff(15,plain,
% 0.76/0.74 (![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta)) <=> ![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))),
% 0.76/0.74 inference(rewrite,[status(thm)],[])).
% 0.76/0.74 tff(16,axiom,(![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause_6')).
% 0.76/0.75 tff(17,plain,
% 0.76/0.75 (![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.76/0.75 tff(18,plain,(
% 0.76/0.75 ![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))),
% 0.76/0.75 inference(skolemize,[status(sab)],[17])).
% 0.76/0.75 tff(19,plain,
% 0.76/0.75 (![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.76/0.75 tff(20,plain,
% 0.76/0.75 (((~![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))) | (less_or_equal(delta_1(half(epsilon_0)), n0) | less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0))))) <=> ((~![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))) | less_or_equal(delta_1(half(epsilon_0)), n0) | less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0))))),
% 0.76/0.75 inference(rewrite,[status(thm)],[])).
% 0.76/0.75 tff(21,plain,
% 0.76/0.75 ((~![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))) | (less_or_equal(delta_1(half(epsilon_0)), n0) | less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0))))),
% 0.76/0.75 inference(quant_inst,[status(thm)],[])).
% 0.76/0.75 tff(22,plain,
% 0.76/0.75 ((~![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))) | less_or_equal(delta_1(half(epsilon_0)), n0) | less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.76/0.75 tff(23,plain,
% 0.76/0.75 (less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))),
% 0.76/0.75 inference(unit_resolution,[status(thm)],[22, 19, 12])).
% 0.76/0.75 tff(24,plain,
% 0.76/0.75 (^[Epsilon: $i, Z: $i] : refl((less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) <=> (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))))),
% 0.76/0.75 inference(bind,[status(th)],[])).
% 0.76/0.75 tff(25,plain,
% 0.76/0.75 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) <=> ![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))),
% 0.76/0.75 inference(quant_intro,[status(thm)],[24])).
% 0.76/0.75 tff(26,plain,
% 0.76/0.75 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) <=> ![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))),
% 0.76/0.75 inference(rewrite,[status(thm)],[])).
% 0.76/0.75 tff(27,plain,
% 0.76/0.75 (^[Epsilon: $i, Z: $i] : trans(monotonicity(rewrite((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) <=> (less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))), (((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon)) <=> ((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon)))), rewrite(((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon)) <=> (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))), (((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon)) <=> (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))))),
% 0.76/0.75 inference(bind,[status(th)],[])).
% 0.76/0.75 tff(28,plain,
% 0.76/0.75 (![Epsilon: $i, Z: $i] : ((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon)) <=> ![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))),
% 0.76/0.75 inference(quant_intro,[status(thm)],[27])).
% 0.76/0.75 tff(29,axiom,(![Epsilon: $i, Z: $i] : ((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon)))) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause_3')).
% 0.76/0.75 tff(30,plain,
% 0.76/0.75 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.76/0.75 tff(31,plain,
% 0.76/0.75 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.76/0.75 tff(32,plain,(
% 0.76/0.75 ![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))),
% 0.76/0.75 inference(skolemize,[status(sab)],[31])).
% 0.76/0.75 tff(33,plain,
% 0.76/0.75 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[32, 25])).
% 0.76/0.75 tff(34,plain,
% 0.76/0.75 (((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)))) <=> ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)))),
% 0.76/0.75 inference(rewrite,[status(thm)],[])).
% 0.76/0.75 tff(35,plain,
% 0.76/0.75 ((less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0))))) <=> (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)))),
% 0.76/0.75 inference(rewrite,[status(thm)],[])).
% 0.76/0.75 tff(36,plain,
% 0.76/0.75 (((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))))) <=> ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))))),
% 0.76/0.75 inference(monotonicity,[status(thm)],[35])).
% 0.76/0.75 tff(37,plain,
% 0.76/0.75 (((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))))) <=> ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)))),
% 0.76/0.75 inference(transitivity,[status(thm)],[36, 34])).
% 0.76/0.75 tff(38,plain,
% 0.76/0.75 ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))))),
% 0.76/0.75 inference(quant_inst,[status(thm)],[])).
% 0.76/0.75 tff(39,plain,
% 0.76/0.75 ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.76/0.75 tff(40,plain,
% 0.76/0.75 (less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))),
% 0.76/0.75 inference(unit_resolution,[status(thm)],[39, 33, 1, 23])).
% 0.76/0.75 tff(41,plain,
% 0.76/0.75 (^[Delta: $i] : refl((less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0))) <=> (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0))))),
% 0.76/0.75 inference(bind,[status(th)],[])).
% 0.76/0.75 tff(42,plain,
% 0.76/0.75 (![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0))) <=> ![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))),
% 0.76/0.75 inference(quant_intro,[status(thm)],[41])).
% 0.76/0.75 tff(43,plain,
% 0.76/0.75 (![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0))) <=> ![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))),
% 0.76/0.75 inference(rewrite,[status(thm)],[])).
% 0.76/0.75 tff(44,axiom,(![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause_7_2')).
% 0.76/0.75 tff(45,plain,
% 0.76/0.75 (![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.76/0.75 tff(46,plain,(
% 0.76/0.75 ![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))),
% 0.76/0.75 inference(skolemize,[status(sab)],[45])).
% 0.76/0.75 tff(47,plain,
% 0.76/0.75 (![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.76/0.75 tff(48,plain,
% 0.76/0.75 (((~![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))) | (less_or_equal(delta_1(half(epsilon_0)), n0) | (~less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0)))) <=> ((~![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))) | less_or_equal(delta_1(half(epsilon_0)), n0) | (~less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0)))),
% 0.76/0.75 inference(rewrite,[status(thm)],[])).
% 0.76/0.75 tff(49,plain,
% 0.76/0.75 ((~![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))) | (less_or_equal(delta_1(half(epsilon_0)), n0) | (~less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0)))),
% 0.76/0.75 inference(quant_inst,[status(thm)],[])).
% 0.76/0.75 tff(50,plain,
% 0.76/0.75 ((~![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))) | less_or_equal(delta_1(half(epsilon_0)), n0) | (~less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.76/0.75 tff(51,plain,
% 0.76/0.75 (~less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0)),
% 0.76/0.75 inference(unit_resolution,[status(thm)],[50, 47, 12])).
% 0.76/0.75 tff(52,plain,
% 0.76/0.75 (^[Z: $i, Y: $i, X: $i] : refl((less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z)))) <=> (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z)))))),
% 0.76/0.75 inference(bind,[status(th)],[])).
% 0.76/0.75 tff(53,plain,
% 0.76/0.75 (![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z)))) <=> ![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))),
% 0.76/0.75 inference(quant_intro,[status(thm)],[52])).
% 0.76/0.75 tff(54,plain,
% 0.76/0.75 (![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z)))) <=> ![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))),
% 0.76/0.75 inference(rewrite,[status(thm)],[])).
% 0.76/0.75 tff(55,plain,
% 0.76/0.75 (^[Z: $i, Y: $i, X: $i] : rewrite((((~less_or_equal(X, half(Z))) | (~less_or_equal(Y, half(Z)))) | less_or_equal(add(X, Y), Z)) <=> (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z)))))),
% 0.76/0.75 inference(bind,[status(th)],[])).
% 0.76/0.75 tff(56,plain,
% 0.76/0.75 (![Z: $i, Y: $i, X: $i] : (((~less_or_equal(X, half(Z))) | (~less_or_equal(Y, half(Z)))) | less_or_equal(add(X, Y), Z)) <=> ![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))),
% 0.76/0.75 inference(quant_intro,[status(thm)],[55])).
% 0.76/0.75 tff(57,axiom,(![Z: $i, Y: $i, X: $i] : (((~less_or_equal(X, half(Z))) | (~less_or_equal(Y, half(Z)))) | less_or_equal(add(X, Y), Z))), file('/export/starexec/sandbox2/benchmark/Axioms/ANA002-0.ax','less_or_equal_sum_of_halves')).
% 0.76/0.75 tff(58,plain,
% 0.76/0.75 (![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.76/0.75 tff(59,plain,
% 0.76/0.75 (![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[58, 54])).
% 0.76/0.75 tff(60,plain,(
% 0.76/0.75 ![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))),
% 0.76/0.75 inference(skolemize,[status(sab)],[59])).
% 0.76/0.75 tff(61,plain,
% 0.76/0.75 (![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[60, 53])).
% 0.76/0.75 tff(62,plain,
% 0.76/0.75 (((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))))) <=> ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))))),
% 0.76/0.75 inference(rewrite,[status(thm)],[])).
% 0.76/0.75 tff(63,plain,
% 0.76/0.75 ((less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)))) <=> (less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))))),
% 0.76/0.75 inference(rewrite,[status(thm)],[])).
% 0.76/0.75 tff(64,plain,
% 0.76/0.75 (((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))))) <=> ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)))))),
% 0.76/0.75 inference(monotonicity,[status(thm)],[63])).
% 0.76/0.75 tff(65,plain,
% 0.76/0.75 (((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))))) <=> ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))))),
% 0.76/0.75 inference(transitivity,[status(thm)],[64, 62])).
% 0.76/0.75 tff(66,plain,
% 0.76/0.75 ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))))),
% 0.76/0.75 inference(quant_inst,[status(thm)],[])).
% 0.76/0.75 tff(67,plain,
% 0.76/0.75 ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | less_or_equal(add(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_1(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)))),
% 0.76/0.75 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.76/0.75 tff(68,plain,
% 0.76/0.75 (~less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))),
% 0.76/0.75 inference(unit_resolution,[status(thm)],[67, 61, 51, 40])).
% 0.76/0.75 tff(69,plain,
% 0.76/0.75 (^[Epsilon: $i] : refl((less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0))) <=> (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0))))),
% 0.76/0.75 inference(bind,[status(th)],[])).
% 0.76/0.75 tff(70,plain,
% 0.76/0.75 (![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0))) <=> ![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))),
% 0.76/0.76 inference(quant_intro,[status(thm)],[69])).
% 0.76/0.76 tff(71,plain,
% 0.76/0.76 (![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0))) <=> ![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(72,axiom,(![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause_2')).
% 0.76/0.76 tff(73,plain,
% 0.76/0.76 (![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.76/0.76 tff(74,plain,(
% 0.76/0.76 ![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))),
% 0.76/0.76 inference(skolemize,[status(sab)],[73])).
% 0.76/0.76 tff(75,plain,
% 0.76/0.76 (![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[74, 70])).
% 0.76/0.76 tff(76,plain,
% 0.76/0.76 (((~![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))) | (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(delta_2(half(epsilon_0)), n0)))) <=> ((~![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(delta_2(half(epsilon_0)), n0)))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(77,plain,
% 0.76/0.76 ((~![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))) | (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(delta_2(half(epsilon_0)), n0)))),
% 0.76/0.76 inference(quant_inst,[status(thm)],[])).
% 0.76/0.76 tff(78,plain,
% 0.76/0.76 ((~![Epsilon: $i] : (less_or_equal(Epsilon, n0) | (~less_or_equal(delta_2(Epsilon), n0)))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(delta_2(half(epsilon_0)), n0))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.76/0.76 tff(79,plain,
% 0.76/0.76 (~less_or_equal(delta_2(half(epsilon_0)), n0)),
% 0.76/0.76 inference(unit_resolution,[status(thm)],[78, 75, 1])).
% 0.76/0.76 tff(80,plain,
% 0.76/0.76 (((~![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))) | (less_or_equal(delta_2(half(epsilon_0)), n0) | less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))))) <=> ((~![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))) | less_or_equal(delta_2(half(epsilon_0)), n0) | less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(81,plain,
% 0.76/0.76 ((~![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))) | (less_or_equal(delta_2(half(epsilon_0)), n0) | less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))))),
% 0.76/0.76 inference(quant_inst,[status(thm)],[])).
% 0.76/0.76 tff(82,plain,
% 0.76/0.76 ((~![Delta: $i] : (less_or_equal(Delta, n0) | less_or_equal(absolute(add(xs(Delta), negate(a_real_number))), Delta))) | less_or_equal(delta_2(half(epsilon_0)), n0) | less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.76/0.76 tff(83,plain,
% 0.76/0.76 (less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))),
% 0.76/0.76 inference(unit_resolution,[status(thm)],[82, 19, 79])).
% 0.76/0.76 tff(84,plain,
% 0.76/0.76 (^[Epsilon: $i, Z: $i] : refl((less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) <=> (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))))),
% 0.76/0.76 inference(bind,[status(th)],[])).
% 0.76/0.76 tff(85,plain,
% 0.76/0.76 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) <=> ![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))),
% 0.76/0.76 inference(quant_intro,[status(thm)],[84])).
% 0.76/0.76 tff(86,plain,
% 0.76/0.76 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) <=> ![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(87,plain,
% 0.76/0.76 (^[Epsilon: $i, Z: $i] : trans(monotonicity(rewrite((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) <=> (less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))), (((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon)) <=> ((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon)))), rewrite(((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon)) <=> (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))), (((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon)) <=> (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))))),
% 0.76/0.76 inference(bind,[status(th)],[])).
% 0.76/0.76 tff(88,plain,
% 0.76/0.76 (![Epsilon: $i, Z: $i] : ((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon)) <=> ![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))),
% 0.76/0.76 inference(quant_intro,[status(thm)],[87])).
% 0.76/0.76 tff(89,axiom,(![Epsilon: $i, Z: $i] : ((less_or_equal(Epsilon, n0) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon)))) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause_4')).
% 0.76/0.76 tff(90,plain,
% 0.76/0.76 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.76/0.76 tff(91,plain,
% 0.76/0.76 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[90, 86])).
% 0.76/0.76 tff(92,plain,(
% 0.76/0.76 ![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))),
% 0.76/0.76 inference(skolemize,[status(sab)],[91])).
% 0.76/0.76 tff(93,plain,
% 0.76/0.76 (![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[92, 85])).
% 0.76/0.76 tff(94,plain,
% 0.76/0.76 (((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)))) <=> ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(95,plain,
% 0.76/0.76 ((less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))))) <=> (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(96,plain,
% 0.76/0.76 (((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))))) <=> ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))))),
% 0.76/0.76 inference(monotonicity,[status(thm)],[95])).
% 0.76/0.76 tff(97,plain,
% 0.76/0.76 (((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))))) <=> ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)))),
% 0.76/0.76 inference(transitivity,[status(thm)],[96, 94])).
% 0.76/0.76 tff(98,plain,
% 0.76/0.76 ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))))),
% 0.76/0.76 inference(quant_inst,[status(thm)],[])).
% 0.76/0.76 tff(99,plain,
% 0.76/0.76 ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.76/0.76 tff(100,plain,
% 0.76/0.76 (less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))),
% 0.76/0.76 inference(unit_resolution,[status(thm)],[99, 93, 1, 83])).
% 0.76/0.76 tff(101,plain,
% 0.76/0.76 (((~![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))) | (less_or_equal(delta_2(half(epsilon_0)), n0) | (~less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0)))) <=> ((~![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))) | less_or_equal(delta_2(half(epsilon_0)), n0) | (~less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0)))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(102,plain,
% 0.76/0.76 ((~![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))) | (less_or_equal(delta_2(half(epsilon_0)), n0) | (~less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0)))),
% 0.76/0.76 inference(quant_inst,[status(thm)],[])).
% 0.76/0.76 tff(103,plain,
% 0.76/0.76 ((~![Delta: $i] : (less_or_equal(Delta, n0) | (~less_or_equal(add(absolute(add(f(xs(Delta)), negate(f(a_real_number)))), absolute(add(g(xs(Delta)), negate(g(a_real_number))))), epsilon_0)))) | less_or_equal(delta_2(half(epsilon_0)), n0) | (~less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.76/0.76 tff(104,plain,
% 0.76/0.76 (~less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0)),
% 0.76/0.76 inference(unit_resolution,[status(thm)],[103, 47, 79])).
% 0.76/0.76 tff(105,plain,
% 0.76/0.76 (((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))))) <=> ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(106,plain,
% 0.76/0.76 ((less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)))) <=> (less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(107,plain,
% 0.76/0.76 (((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))))) <=> ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)))))),
% 0.76/0.76 inference(monotonicity,[status(thm)],[106])).
% 0.76/0.76 tff(108,plain,
% 0.76/0.76 (((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))))) <=> ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))))),
% 0.76/0.76 inference(transitivity,[status(thm)],[107, 105])).
% 0.76/0.76 tff(109,plain,
% 0.76/0.76 ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | (less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))))),
% 0.76/0.76 inference(quant_inst,[status(thm)],[])).
% 0.76/0.76 tff(110,plain,
% 0.76/0.76 ((~![Z: $i, Y: $i, X: $i] : (less_or_equal(add(X, Y), Z) | (~less_or_equal(Y, half(Z))) | (~less_or_equal(X, half(Z))))) | less_or_equal(add(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number))))), epsilon_0) | (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))) | (~less_or_equal(absolute(add(g(xs(delta_2(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[109, 108])).
% 0.76/0.76 tff(111,plain,
% 0.76/0.76 (~less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0))),
% 0.76/0.76 inference(unit_resolution,[status(thm)],[110, 61, 104, 100])).
% 0.76/0.76 tff(112,plain,
% 0.76/0.76 (((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))))) <=> ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(113,plain,
% 0.76/0.76 ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))))),
% 0.76/0.76 inference(quant_inst,[status(thm)],[])).
% 0.76/0.76 tff(114,plain,
% 0.76/0.76 ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(f(Z), negate(f(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_1(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(f(xs(delta_2(half(epsilon_0)))), negate(f(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0))))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.76/0.76 tff(115,plain,
% 0.76/0.76 (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))),
% 0.76/0.76 inference(unit_resolution,[status(thm)],[114, 33, 1, 111])).
% 0.76/0.76 tff(116,plain,
% 0.76/0.76 (^[Y: $i, X: $i] : refl((less_or_equal(X, Y) | less_or_equal(Y, X)) <=> (less_or_equal(X, Y) | less_or_equal(Y, X)))),
% 0.76/0.76 inference(bind,[status(th)],[])).
% 0.76/0.76 tff(117,plain,
% 0.76/0.76 (![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X)) <=> ![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))),
% 0.76/0.76 inference(quant_intro,[status(thm)],[116])).
% 0.76/0.76 tff(118,plain,
% 0.76/0.76 (![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X)) <=> ![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(119,axiom,(![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))), file('/export/starexec/sandbox2/benchmark/Axioms/ANA002-0.ax','commutativity_of_less_or_equal')).
% 0.76/0.76 tff(120,plain,
% 0.76/0.76 (![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.76/0.76 tff(121,plain,(
% 0.76/0.76 ![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))),
% 0.76/0.76 inference(skolemize,[status(sab)],[120])).
% 0.76/0.76 tff(122,plain,
% 0.76/0.76 (![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[121, 117])).
% 0.76/0.76 tff(123,plain,
% 0.76/0.76 (((~![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))) | (less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number)))) | less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0))))) <=> ((~![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))) | less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number)))) | less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0))))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(124,plain,
% 0.76/0.76 ((~![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))) | (less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number)))) | less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0))))),
% 0.76/0.76 inference(quant_inst,[status(thm)],[])).
% 0.76/0.76 tff(125,plain,
% 0.76/0.76 ((~![Y: $i, X: $i] : (less_or_equal(X, Y) | less_or_equal(Y, X))) | less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number)))) | less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.76/0.76 tff(126,plain,
% 0.76/0.76 (less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))),
% 0.76/0.76 inference(unit_resolution,[status(thm)],[125, 122, 115])).
% 0.76/0.76 tff(127,plain,
% 0.76/0.76 (![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(128,plain,
% 0.76/0.76 (![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))),
% 0.76/0.76 inference(rewrite,[status(thm)],[])).
% 0.76/0.76 tff(129,plain,
% 0.76/0.76 (^[Z: $i, Y: $i, X: $i] : rewrite((((~less_or_equal(X, Y)) | (~less_or_equal(Y, Z))) | less_or_equal(X, Z)) <=> ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z))))),
% 0.76/0.76 inference(bind,[status(th)],[])).
% 0.76/0.76 tff(130,plain,
% 0.76/0.76 (![Z: $i, Y: $i, X: $i] : (((~less_or_equal(X, Y)) | (~less_or_equal(Y, Z))) | less_or_equal(X, Z)) <=> ![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))),
% 0.76/0.76 inference(quant_intro,[status(thm)],[129])).
% 0.76/0.76 tff(131,axiom,(![Z: $i, Y: $i, X: $i] : (((~less_or_equal(X, Y)) | (~less_or_equal(Y, Z))) | less_or_equal(X, Z))), file('/export/starexec/sandbox2/benchmark/Axioms/ANA002-0.ax','transitivity_of_less_or_equal')).
% 0.76/0.76 tff(132,plain,
% 0.76/0.76 (![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.76/0.76 tff(133,plain,
% 0.76/0.76 (![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[132, 128])).
% 0.76/0.76 tff(134,plain,(
% 0.76/0.76 ![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))),
% 0.76/0.76 inference(skolemize,[status(sab)],[133])).
% 0.76/0.76 tff(135,plain,
% 0.76/0.76 (![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))),
% 0.76/0.76 inference(modus_ponens,[status(thm)],[134, 127])).
% 0.76/0.76 tff(136,plain,
% 0.76/0.76 (((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | ((~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | (~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | (~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0))))),
% 0.76/0.77 inference(rewrite,[status(thm)],[])).
% 0.76/0.77 tff(137,plain,
% 0.76/0.77 (((~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0))) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))))) <=> ((~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | (~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0))))),
% 0.76/0.77 inference(rewrite,[status(thm)],[])).
% 0.76/0.77 tff(138,plain,
% 0.76/0.77 (((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | ((~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0))) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | ((~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | (~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0)))))),
% 0.76/0.77 inference(monotonicity,[status(thm)],[137])).
% 0.76/0.77 tff(139,plain,
% 0.76/0.77 (((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | ((~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0))) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | (~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0))))),
% 0.76/0.77 inference(transitivity,[status(thm)],[138, 136])).
% 0.76/0.77 tff(140,plain,
% 0.76/0.77 ((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | ((~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0))) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))))),
% 0.76/0.77 inference(quant_inst,[status(thm)],[])).
% 0.76/0.77 tff(141,plain,
% 0.76/0.77 ((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | (~less_or_equal(absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))) | (~less_or_equal(delta_1(half(epsilon_0)), absolute(add(xs(delta_2(half(epsilon_0))), negate(a_real_number))))) | less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0)))),
% 0.76/0.77 inference(modus_ponens,[status(thm)],[140, 139])).
% 0.76/0.77 tff(142,plain,
% 0.76/0.77 (less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0)))),
% 0.76/0.77 inference(unit_resolution,[status(thm)],[141, 135, 83, 126])).
% 0.76/0.77 tff(143,plain,
% 0.76/0.77 (((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | ((~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))) | (~less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))) | (~less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0)))))),
% 0.76/0.77 inference(rewrite,[status(thm)],[])).
% 0.76/0.77 tff(144,plain,
% 0.76/0.77 ((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | ((~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))) | (~less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0)))))),
% 0.76/0.77 inference(quant_inst,[status(thm)],[])).
% 0.76/0.77 tff(145,plain,
% 0.76/0.77 ((~![Z: $i, Y: $i, X: $i] : ((~less_or_equal(X, Y)) | less_or_equal(X, Z) | (~less_or_equal(Y, Z)))) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_1(half(epsilon_0)))) | less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))) | (~less_or_equal(delta_1(half(epsilon_0)), delta_2(half(epsilon_0))))),
% 0.76/0.77 inference(modus_ponens,[status(thm)],[144, 143])).
% 0.76/0.77 tff(146,plain,
% 0.76/0.77 (less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))),
% 0.76/0.77 inference(unit_resolution,[status(thm)],[145, 135, 142, 23])).
% 0.76/0.77 tff(147,plain,
% 0.76/0.77 (((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))))) <=> ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))))),
% 0.76/0.77 inference(rewrite,[status(thm)],[])).
% 0.76/0.77 tff(148,plain,
% 0.76/0.77 ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | (less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0)))))),
% 0.76/0.77 inference(quant_inst,[status(thm)],[])).
% 0.76/0.77 tff(149,plain,
% 0.76/0.77 ((~![Epsilon: $i, Z: $i] : (less_or_equal(Epsilon, n0) | less_or_equal(absolute(add(g(Z), negate(g(a_real_number)))), Epsilon) | (~less_or_equal(absolute(add(Z, negate(a_real_number))), delta_2(Epsilon))))) | less_or_equal(half(epsilon_0), n0) | less_or_equal(absolute(add(g(xs(delta_1(half(epsilon_0)))), negate(g(a_real_number)))), half(epsilon_0)) | (~less_or_equal(absolute(add(xs(delta_1(half(epsilon_0))), negate(a_real_number))), delta_2(half(epsilon_0))))),
% 0.76/0.77 inference(modus_ponens,[status(thm)],[148, 147])).
% 0.76/0.77 tff(150,plain,
% 0.76/0.77 ($false),
% 0.76/0.77 inference(unit_resolution,[status(thm)],[149, 93, 1, 146, 68])).
% 0.76/0.77 tff(151,plain,(less_or_equal(half(epsilon_0), n0)), inference(lemma,lemma(discharge,[]))).
% 0.76/0.77 tff(152,plain,
% 0.76/0.77 ((~less_or_equal(epsilon_0, n0)) <=> (~less_or_equal(epsilon_0, n0))),
% 0.76/0.77 inference(rewrite,[status(thm)],[])).
% 0.76/0.77 tff(153,axiom,(~less_or_equal(epsilon_0, n0)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','clause_5')).
% 0.76/0.77 tff(154,plain,
% 0.76/0.77 (~less_or_equal(epsilon_0, n0)),
% 0.76/0.77 inference(modus_ponens,[status(thm)],[153, 152])).
% 0.76/0.77 tff(155,plain,
% 0.76/0.77 (^[X: $i] : refl((less_or_equal(X, n0) | (~less_or_equal(half(X), n0))) <=> (less_or_equal(X, n0) | (~less_or_equal(half(X), n0))))),
% 0.76/0.77 inference(bind,[status(th)],[])).
% 0.76/0.77 tff(156,plain,
% 0.76/0.77 (![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0))) <=> ![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))),
% 0.76/0.77 inference(quant_intro,[status(thm)],[155])).
% 0.76/0.77 tff(157,plain,
% 0.76/0.77 (![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0))) <=> ![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))),
% 0.76/0.77 inference(rewrite,[status(thm)],[])).
% 0.76/0.77 tff(158,axiom,(![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))), file('/export/starexec/sandbox2/benchmark/Axioms/ANA002-0.ax','zero_and_half')).
% 0.76/0.77 tff(159,plain,
% 0.76/0.77 (![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))),
% 0.76/0.77 inference(modus_ponens,[status(thm)],[158, 157])).
% 0.76/0.77 tff(160,plain,(
% 0.76/0.77 ![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))),
% 0.76/0.77 inference(skolemize,[status(sab)],[159])).
% 0.76/0.77 tff(161,plain,
% 0.76/0.77 (![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))),
% 0.76/0.77 inference(modus_ponens,[status(thm)],[160, 156])).
% 0.76/0.77 tff(162,plain,
% 0.76/0.77 (((~![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))) | (less_or_equal(epsilon_0, n0) | (~less_or_equal(half(epsilon_0), n0)))) <=> ((~![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))) | less_or_equal(epsilon_0, n0) | (~less_or_equal(half(epsilon_0), n0)))),
% 0.76/0.77 inference(rewrite,[status(thm)],[])).
% 0.76/0.77 tff(163,plain,
% 0.76/0.77 ((~![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))) | (less_or_equal(epsilon_0, n0) | (~less_or_equal(half(epsilon_0), n0)))),
% 0.76/0.77 inference(quant_inst,[status(thm)],[])).
% 0.76/0.77 tff(164,plain,
% 0.76/0.77 ((~![X: $i] : (less_or_equal(X, n0) | (~less_or_equal(half(X), n0)))) | less_or_equal(epsilon_0, n0) | (~less_or_equal(half(epsilon_0), n0))),
% 0.76/0.77 inference(modus_ponens,[status(thm)],[163, 162])).
% 0.76/0.77 tff(165,plain,
% 0.76/0.77 ($false),
% 0.76/0.77 inference(unit_resolution,[status(thm)],[164, 161, 154, 151])).
% 0.76/0.77 % SZS output end Proof
%------------------------------------------------------------------------------