TSTP Solution File: SET005-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET005-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n001.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 20 05:04:40 EDT 2022
% Result : Unsatisfiable 0.19s 0.39s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET005-1 : TPTP v8.1.0. Released v1.0.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n001.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sat Sep 3 01:21:54 EDT 2022
% 0.13/0.34 % 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.
% 0.19/0.39 % SZS status Unsatisfiable
% 0.19/0.39 % SZS output start Proof
% 0.19/0.39 tff(member_type, type, (
% 0.19/0.39 member: ( $i * $i ) > $o)).
% 0.19/0.39 tff(bIc_type, type, (
% 0.19/0.39 bIc: $i)).
% 0.19/0.39 tff(h_type, type, (
% 0.19/0.39 h: ( $i * $i * $i ) > $i)).
% 0.19/0.39 tff(aIbIc_type, type, (
% 0.19/0.39 aIbIc: $i)).
% 0.19/0.39 tff(c_type, type, (
% 0.19/0.39 c: $i)).
% 0.19/0.39 tff(aIb_type, type, (
% 0.19/0.39 aIb: $i)).
% 0.19/0.39 tff(a_type, type, (
% 0.19/0.39 a: $i)).
% 0.19/0.39 tff(b_type, type, (
% 0.19/0.39 b: $i)).
% 0.19/0.39 tff(intersection_type, type, (
% 0.19/0.39 intersection: ( $i * $i * $i ) > $o)).
% 0.19/0.39 tff(1,assumption,(~member(h(aIb, c, aIbIc), a)), introduced(assumption)).
% 0.19/0.39 tff(2,assumption,(~member(h(aIb, c, aIbIc), aIb)), introduced(assumption)).
% 0.19/0.39 tff(3,plain,
% 0.19/0.39 ((~intersection(aIb, c, aIbIc)) <=> (~intersection(aIb, c, aIbIc))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(4,axiom,(~intersection(aIb, c, aIbIc)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_aIb_intersection_c_is_aIbIc')).
% 0.19/0.39 tff(5,plain,
% 0.19/0.39 (~intersection(aIb, c, aIbIc)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.39 tff(6,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, Intersection: $i] : refl((member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) <=> (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(7,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[6])).
% 0.19/0.39 tff(8,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(9,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, Intersection: $i] : rewrite(((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set1)) <=> (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(10,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set1)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[9])).
% 0.19/0.39 tff(11,axiom,(![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set1))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-2.ax','intersection_axiom1')).
% 0.19/0.39 tff(12,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.19/0.39 tff(13,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.19/0.39 tff(14,plain,(
% 0.19/0.39 ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.39 inference(skolemize,[status(sab)],[13])).
% 0.19/0.39 tff(15,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.19/0.39 tff(16,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | (member(h(aIb, c, aIbIc), aIb) | member(h(aIb, c, aIbIc), aIbIc) | intersection(aIb, c, aIbIc))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | member(h(aIb, c, aIbIc), aIb) | member(h(aIb, c, aIbIc), aIbIc) | intersection(aIb, c, aIbIc))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(17,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | (member(h(aIb, c, aIbIc), aIb) | member(h(aIb, c, aIbIc), aIbIc) | intersection(aIb, c, aIbIc))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(18,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | member(h(aIb, c, aIbIc), aIb) | member(h(aIb, c, aIbIc), aIbIc) | intersection(aIb, c, aIbIc)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.19/0.39 tff(19,plain,
% 0.19/0.39 (member(h(aIb, c, aIbIc), aIb) | member(h(aIb, c, aIbIc), aIbIc)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[18, 15, 5])).
% 0.19/0.39 tff(20,plain,
% 0.19/0.39 (member(h(aIb, c, aIbIc), aIbIc)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[19, 2])).
% 0.19/0.39 tff(21,assumption,(member(h(aIb, c, aIbIc), aIbIc)), introduced(assumption)).
% 0.19/0.39 tff(22,plain,
% 0.19/0.39 (intersection(a, bIc, aIbIc) <=> intersection(a, bIc, aIbIc)),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(23,axiom,(intersection(a, bIc, aIbIc)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_intersection_bIc')).
% 0.19/0.39 tff(24,plain,
% 0.19/0.39 (intersection(a, bIc, aIbIc)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.19/0.39 tff(25,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, Element: $i, Intersection: $i] : refl((member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) <=> (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(26,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) <=> ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[25])).
% 0.19/0.39 tff(27,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) <=> ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(28,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, Element: $i, Intersection: $i] : trans(monotonicity(rewrite(((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) <=> ((~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))), ((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set1)) <=> (((~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) | member(Element, Set1)))), rewrite((((~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) | member(Element, Set1)) <=> (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))), ((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set1)) <=> (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(29,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set1)) <=> ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[28])).
% 0.19/0.39 tff(30,axiom,(![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set1))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-2.ax','member_of_intersection_is_member_of_set1')).
% 0.19/0.39 tff(31,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.19/0.39 tff(32,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[31, 27])).
% 0.19/0.39 tff(33,plain,(
% 0.19/0.39 ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[32])).
% 0.19/0.39 tff(34,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[33, 26])).
% 0.19/0.39 tff(35,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(h(aIb, c, aIbIc), aIbIc)) | member(h(aIb, c, aIbIc), a) | (~intersection(a, bIc, aIbIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIbIc)) | member(h(aIb, c, aIbIc), a) | (~intersection(a, bIc, aIbIc)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(36,plain,
% 0.19/0.39 ((member(h(aIb, c, aIbIc), a) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc))) <=> ((~member(h(aIb, c, aIbIc), aIbIc)) | member(h(aIb, c, aIbIc), a) | (~intersection(a, bIc, aIbIc)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(37,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), a) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(h(aIb, c, aIbIc), aIbIc)) | member(h(aIb, c, aIbIc), a) | (~intersection(a, bIc, aIbIc))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[36])).
% 0.19/0.39 tff(38,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), a) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIbIc)) | member(h(aIb, c, aIbIc), a) | (~intersection(a, bIc, aIbIc)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[37, 35])).
% 0.19/0.39 tff(39,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), a) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(40,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIbIc)) | member(h(aIb, c, aIbIc), a) | (~intersection(a, bIc, aIbIc))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.19/0.40 tff(41,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[40, 34, 24, 21, 1])).
% 0.19/0.40 tff(42,plain,(member(h(aIb, c, aIbIc), a) | (~member(h(aIb, c, aIbIc), aIbIc))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(43,plain,
% 0.19/0.40 (member(h(aIb, c, aIbIc), a)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[42, 20])).
% 0.19/0.40 tff(44,plain,
% 0.19/0.40 (intersection(a, b, aIb) <=> intersection(a, b, aIb)),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(45,axiom,(intersection(a, b, aIb)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_intersection_b')).
% 0.19/0.40 tff(46,plain,
% 0.19/0.40 (intersection(a, b, aIb)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.19/0.40 tff(47,plain,
% 0.19/0.40 (^[Set1: $i, Set2: $i, Element: $i, Intersection: $i] : refl((member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1))) <=> (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(48,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1))) <=> ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[47])).
% 0.19/0.40 tff(49,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1))) <=> ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(50,plain,
% 0.19/0.40 (^[Set1: $i, Set2: $i, Element: $i, Intersection: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~intersection(Set1, Set2, Intersection)) | (~member(Element, Set2))) <=> ((~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)))), ((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Set2))) | (~member(Element, Set1))) <=> (((~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection))) | (~member(Element, Set1))))), rewrite((((~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection))) | (~member(Element, Set1))) <=> ((~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))), ((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Set2))) | (~member(Element, Set1))) <=> ((~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1))))), (((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Set2))) | (~member(Element, Set1))) | member(Element, Intersection)) <=> (((~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1))) | member(Element, Intersection)))), rewrite((((~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1))) | member(Element, Intersection)) <=> (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))), (((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Set2))) | (~member(Element, Set1))) | member(Element, Intersection)) <=> (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(51,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : ((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Set2))) | (~member(Element, Set1))) | member(Element, Intersection)) <=> ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[50])).
% 0.19/0.40 tff(52,axiom,(![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : ((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Set2))) | (~member(Element, Set1))) | member(Element, Intersection))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-2.ax','member_of_both_is_member_of_intersection')).
% 0.19/0.40 tff(53,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.19/0.40 tff(54,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.19/0.40 tff(55,plain,(
% 0.19/0.40 ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[54])).
% 0.19/0.40 tff(56,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[55, 48])).
% 0.19/0.40 tff(57,plain,
% 0.19/0.40 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), aIb) | (~member(h(aIb, c, aIbIc), b)) | (~intersection(a, b, aIb)) | (~member(h(aIb, c, aIbIc), a)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(h(aIb, c, aIbIc), aIb) | (~member(h(aIb, c, aIbIc), b)) | (~intersection(a, b, aIb)) | (~member(h(aIb, c, aIbIc), a)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(58,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), aIb) | (~member(h(aIb, c, aIbIc), b)) | (~intersection(a, b, aIb)) | (~member(h(aIb, c, aIbIc), a)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(59,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(h(aIb, c, aIbIc), aIb) | (~member(h(aIb, c, aIbIc), b)) | (~intersection(a, b, aIb)) | (~member(h(aIb, c, aIbIc), a))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.19/0.40 tff(60,plain,
% 0.19/0.40 ((~member(h(aIb, c, aIbIc), b)) | (~member(h(aIb, c, aIbIc), a))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[59, 56, 46, 2])).
% 0.19/0.40 tff(61,plain,
% 0.19/0.40 (~member(h(aIb, c, aIbIc), b)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[60, 43])).
% 0.19/0.40 tff(62,plain,
% 0.19/0.40 (intersection(b, c, bIc) <=> intersection(b, c, bIc)),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(63,axiom,(intersection(b, c, bIc)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','b_intersection_c')).
% 0.19/0.40 tff(64,plain,
% 0.19/0.40 (intersection(b, c, bIc)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[63, 62])).
% 0.19/0.40 tff(65,plain,
% 0.19/0.40 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), b) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(b, c, bIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | member(h(aIb, c, aIbIc), b) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(b, c, bIc)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(66,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), b) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(b, c, bIc)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(67,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | member(h(aIb, c, aIbIc), b) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(b, c, bIc))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.19/0.40 tff(68,plain,
% 0.19/0.40 (~member(h(aIb, c, aIbIc), bIc)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[67, 34, 64, 61])).
% 0.19/0.40 tff(69,plain,
% 0.19/0.40 (^[Set1: $i, Set2: $i, Element: $i, Intersection: $i] : refl((member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) <=> (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(70,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) <=> ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[69])).
% 0.19/0.40 tff(71,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) <=> ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(72,plain,
% 0.19/0.40 (^[Set1: $i, Set2: $i, Element: $i, Intersection: $i] : trans(monotonicity(rewrite(((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) <=> ((~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))), ((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set2)) <=> (((~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) | member(Element, Set2)))), rewrite((((~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection))) | member(Element, Set2)) <=> (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))), ((((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set2)) <=> (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(73,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set2)) <=> ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[72])).
% 0.19/0.40 tff(74,axiom,(![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set2))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-2.ax','member_of_intersection_is_member_of_set2')).
% 0.19/0.40 tff(75,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.19/0.40 tff(76,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[75, 71])).
% 0.19/0.40 tff(77,plain,(
% 0.19/0.40 ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[76])).
% 0.19/0.40 tff(78,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[77, 70])).
% 0.19/0.40 tff(79,plain,
% 0.19/0.40 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)) | member(h(aIb, c, aIbIc), bIc))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)) | member(h(aIb, c, aIbIc), bIc))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(80,plain,
% 0.19/0.40 ((member(h(aIb, c, aIbIc), bIc) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc))) <=> ((~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)) | member(h(aIb, c, aIbIc), bIc))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(81,plain,
% 0.19/0.40 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), bIc) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)) | member(h(aIb, c, aIbIc), bIc)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[80])).
% 0.19/0.40 tff(82,plain,
% 0.19/0.40 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), bIc) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)) | member(h(aIb, c, aIbIc), bIc))),
% 0.19/0.40 inference(transitivity,[status(thm)],[81, 79])).
% 0.19/0.40 tff(83,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), bIc) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(84,plain,
% 0.19/0.41 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIbIc)) | (~intersection(a, bIc, aIbIc)) | member(h(aIb, c, aIbIc), bIc)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.19/0.41 tff(85,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[84, 78, 24, 20, 68])).
% 0.19/0.41 tff(86,plain,(member(h(aIb, c, aIbIc), aIb)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(87,plain,
% 0.19/0.41 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)) | member(h(aIb, c, aIbIc), a))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)) | member(h(aIb, c, aIbIc), a))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(88,plain,
% 0.19/0.41 ((member(h(aIb, c, aIbIc), a) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb))) <=> ((~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)) | member(h(aIb, c, aIbIc), a))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(89,plain,
% 0.19/0.41 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), a) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)) | member(h(aIb, c, aIbIc), a)))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[88])).
% 0.19/0.41 tff(90,plain,
% 0.19/0.41 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), a) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)) | member(h(aIb, c, aIbIc), a))),
% 0.19/0.41 inference(transitivity,[status(thm)],[89, 87])).
% 0.19/0.41 tff(91,plain,
% 0.19/0.41 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), a) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(92,plain,
% 0.19/0.41 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)) | member(h(aIb, c, aIbIc), a)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.19/0.41 tff(93,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[92, 34, 46, 86, 1])).
% 0.19/0.41 tff(94,plain,(member(h(aIb, c, aIbIc), a)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(95,assumption,(~member(h(aIb, c, aIbIc), aIbIc)), introduced(assumption)).
% 0.19/0.41 tff(96,plain,
% 0.19/0.41 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), a)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), bIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), a)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), bIc)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(97,plain,
% 0.19/0.41 ((member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), a))) <=> (member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), a)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), bIc)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(98,plain,
% 0.19/0.41 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), a)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), a)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), bIc))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[97])).
% 0.19/0.41 tff(99,plain,
% 0.19/0.41 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), a)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), a)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), bIc)))),
% 0.19/0.41 inference(transitivity,[status(thm)],[98, 96])).
% 0.19/0.41 tff(100,plain,
% 0.19/0.41 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), a)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(101,plain,
% 0.19/0.41 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(h(aIb, c, aIbIc), aIbIc) | (~member(h(aIb, c, aIbIc), a)) | (~intersection(a, bIc, aIbIc)) | (~member(h(aIb, c, aIbIc), bIc))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.19/0.41 tff(102,plain,
% 0.19/0.41 (~member(h(aIb, c, aIbIc), bIc)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[101, 56, 24, 95, 94])).
% 0.19/0.41 tff(103,assumption,(~member(h(aIb, c, aIbIc), b)), introduced(assumption)).
% 0.19/0.41 tff(104,plain,
% 0.19/0.41 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(h(aIb, c, aIbIc), aIb)) | member(h(aIb, c, aIbIc), b) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIb)) | member(h(aIb, c, aIbIc), b) | (~intersection(a, b, aIb)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(105,plain,
% 0.19/0.41 ((member(h(aIb, c, aIbIc), b) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb))) <=> ((~member(h(aIb, c, aIbIc), aIb)) | member(h(aIb, c, aIbIc), b) | (~intersection(a, b, aIb)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(106,plain,
% 0.19/0.41 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), b) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(h(aIb, c, aIbIc), aIb)) | member(h(aIb, c, aIbIc), b) | (~intersection(a, b, aIb))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[105])).
% 0.19/0.41 tff(107,plain,
% 0.19/0.41 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), b) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIb)) | member(h(aIb, c, aIbIc), b) | (~intersection(a, b, aIb)))),
% 0.19/0.41 inference(transitivity,[status(thm)],[106, 104])).
% 0.19/0.41 tff(108,plain,
% 0.19/0.41 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), b) | (~member(h(aIb, c, aIbIc), aIb)) | (~intersection(a, b, aIb)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(109,plain,
% 0.19/0.42 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(h(aIb, c, aIbIc), aIb)) | member(h(aIb, c, aIbIc), b) | (~intersection(a, b, aIb))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.19/0.42 tff(110,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[109, 78, 46, 86, 103])).
% 0.19/0.42 tff(111,plain,(member(h(aIb, c, aIbIc), b)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(112,plain,
% 0.19/0.42 (^[Set1: $i, Set2: $i, Intersection: $i] : refl((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)) <=> (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(113,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[112])).
% 0.19/0.42 tff(114,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(115,plain,
% 0.19/0.42 (^[Set1: $i, Set2: $i, Intersection: $i] : rewrite(((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set2)) <=> (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(116,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set2)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[115])).
% 0.19/0.42 tff(117,axiom,(![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set2))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-2.ax','intersection_axiom2')).
% 0.19/0.42 tff(118,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[117, 116])).
% 0.19/0.42 tff(119,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[118, 114])).
% 0.19/0.42 tff(120,plain,(
% 0.19/0.42 ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.42 inference(skolemize,[status(sab)],[119])).
% 0.19/0.42 tff(121,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[120, 113])).
% 0.19/0.42 tff(122,plain,
% 0.19/0.42 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c) | member(h(aIb, c, aIbIc), aIbIc))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c) | member(h(aIb, c, aIbIc), aIbIc))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(123,plain,
% 0.19/0.42 ((member(h(aIb, c, aIbIc), aIbIc) | intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c)) <=> (intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c) | member(h(aIb, c, aIbIc), aIbIc))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(124,plain,
% 0.19/0.42 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (member(h(aIb, c, aIbIc), aIbIc) | intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c) | member(h(aIb, c, aIbIc), aIbIc)))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[123])).
% 0.19/0.42 tff(125,plain,
% 0.19/0.42 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (member(h(aIb, c, aIbIc), aIbIc) | intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c) | member(h(aIb, c, aIbIc), aIbIc))),
% 0.19/0.42 inference(transitivity,[status(thm)],[124, 122])).
% 0.19/0.42 tff(126,plain,
% 0.19/0.42 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (member(h(aIb, c, aIbIc), aIbIc) | intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(127,plain,
% 0.19/0.42 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | intersection(aIb, c, aIbIc) | member(h(aIb, c, aIbIc), c) | member(h(aIb, c, aIbIc), aIbIc)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.19/0.42 tff(128,plain,
% 0.19/0.42 (member(h(aIb, c, aIbIc), c) | member(h(aIb, c, aIbIc), aIbIc)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[127, 121, 5])).
% 0.19/0.42 tff(129,plain,
% 0.19/0.42 (member(h(aIb, c, aIbIc), c)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[128, 95])).
% 0.19/0.42 tff(130,plain,
% 0.19/0.42 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | ((~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), b)) | member(h(aIb, c, aIbIc), bIc) | (~intersection(b, c, bIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), b)) | member(h(aIb, c, aIbIc), bIc) | (~intersection(b, c, bIc)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(131,plain,
% 0.19/0.42 ((member(h(aIb, c, aIbIc), bIc) | (~member(h(aIb, c, aIbIc), c)) | (~intersection(b, c, bIc)) | (~member(h(aIb, c, aIbIc), b))) <=> ((~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), b)) | member(h(aIb, c, aIbIc), bIc) | (~intersection(b, c, bIc)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(132,plain,
% 0.19/0.42 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), bIc) | (~member(h(aIb, c, aIbIc), c)) | (~intersection(b, c, bIc)) | (~member(h(aIb, c, aIbIc), b)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | ((~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), b)) | member(h(aIb, c, aIbIc), bIc) | (~intersection(b, c, bIc))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[131])).
% 0.19/0.42 tff(133,plain,
% 0.19/0.42 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), bIc) | (~member(h(aIb, c, aIbIc), c)) | (~intersection(b, c, bIc)) | (~member(h(aIb, c, aIbIc), b)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), b)) | member(h(aIb, c, aIbIc), bIc) | (~intersection(b, c, bIc)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[132, 130])).
% 0.19/0.42 tff(134,plain,
% 0.19/0.42 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(h(aIb, c, aIbIc), bIc) | (~member(h(aIb, c, aIbIc), c)) | (~intersection(b, c, bIc)) | (~member(h(aIb, c, aIbIc), b)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(135,plain,
% 0.19/0.42 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), b)) | member(h(aIb, c, aIbIc), bIc) | (~intersection(b, c, bIc))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[134, 133])).
% 0.19/0.42 tff(136,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[135, 56, 64, 129, 111, 102])).
% 0.19/0.42 tff(137,plain,(member(h(aIb, c, aIbIc), aIbIc)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(138,plain,
% 0.19/0.42 (^[Set1: $i, Set2: $i, Intersection: $i] : refl((intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) <=> (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(139,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[138])).
% 0.19/0.42 tff(140,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(141,plain,
% 0.19/0.42 (^[Set1: $i, Set2: $i, Intersection: $i] : trans(monotonicity(rewrite((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) <=> ((~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))), (((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) | intersection(Set1, Set2, Intersection)) <=> (((~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) | intersection(Set1, Set2, Intersection)))), rewrite((((~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) | intersection(Set1, Set2, Intersection)) <=> (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))), (((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) | intersection(Set1, Set2, Intersection)) <=> (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(142,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : ((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) | intersection(Set1, Set2, Intersection)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[141])).
% 0.19/0.42 tff(143,axiom,(![Set1: $i, Set2: $i, Intersection: $i] : ((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) | intersection(Set1, Set2, Intersection))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-2.ax','intersection_axiom3')).
% 0.19/0.42 tff(144,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[143, 142])).
% 0.19/0.42 tff(145,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[144, 140])).
% 0.19/0.42 tff(146,plain,(
% 0.19/0.42 ![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[145])).
% 0.19/0.42 tff(147,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[146, 139])).
% 0.19/0.42 tff(148,plain,
% 0.19/0.42 (((~![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))) | (intersection(aIb, c, aIbIc) | (~member(h(aIb, c, aIbIc), aIb)) | (~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), aIbIc)))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))) | intersection(aIb, c, aIbIc) | (~member(h(aIb, c, aIbIc), aIb)) | (~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), aIbIc)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(149,plain,
% 0.19/0.43 ((~![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))) | (intersection(aIb, c, aIbIc) | (~member(h(aIb, c, aIbIc), aIb)) | (~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), aIbIc)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(150,plain,
% 0.19/0.43 ((~![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))) | intersection(aIb, c, aIbIc) | (~member(h(aIb, c, aIbIc), aIb)) | (~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), aIbIc))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[149, 148])).
% 0.19/0.43 tff(151,plain,
% 0.19/0.43 ((~member(h(aIb, c, aIbIc), aIb)) | (~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), aIbIc))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[150, 147, 5])).
% 0.19/0.43 tff(152,plain,
% 0.19/0.43 ((~member(h(aIb, c, aIbIc), c)) | (~member(h(aIb, c, aIbIc), aIbIc))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[151, 86])).
% 0.19/0.43 tff(153,plain,
% 0.19/0.43 (~member(h(aIb, c, aIbIc), c)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[152, 137])).
% 0.19/0.43 tff(154,plain,
% 0.19/0.43 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), c) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(b, c, bIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | member(h(aIb, c, aIbIc), c) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(b, c, bIc)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(155,plain,
% 0.19/0.43 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(h(aIb, c, aIbIc), c) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(b, c, bIc)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(156,plain,
% 0.19/0.43 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | member(h(aIb, c, aIbIc), c) | (~member(h(aIb, c, aIbIc), bIc)) | (~intersection(b, c, bIc))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[155, 154])).
% 0.19/0.43 tff(157,plain,
% 0.19/0.43 (~member(h(aIb, c, aIbIc), bIc)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[156, 78, 64, 153])).
% 0.19/0.43 tff(158,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[84, 78, 24, 137, 157])).
% 0.19/0.43 % SZS output end Proof
%------------------------------------------------------------------------------