TSTP Solution File: SET007-1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET007-1 : 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 0.42s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET007-1 : 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 01:22:09 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.19/0.42 % SZS status Unsatisfiable
% 0.19/0.42 % SZS output start Proof
% 0.19/0.42 tff(member_type, type, (
% 0.19/0.42 member: ( $i * $i ) > $o)).
% 0.19/0.42 tff(aIc_type, type, (
% 0.19/0.42 aIc: $i)).
% 0.19/0.42 tff(g_type, type, (
% 0.19/0.42 g: ( $i * $i * $i ) > $i)).
% 0.19/0.42 tff(aI_bUc_type, type, (
% 0.19/0.42 aI_bUc: $i)).
% 0.19/0.42 tff(aIb_type, type, (
% 0.19/0.42 aIb: $i)).
% 0.19/0.42 tff(bUc_type, type, (
% 0.19/0.42 bUc: $i)).
% 0.19/0.42 tff(c_type, type, (
% 0.19/0.42 c: $i)).
% 0.19/0.42 tff(a_type, type, (
% 0.19/0.42 a: $i)).
% 0.19/0.42 tff(b_type, type, (
% 0.19/0.42 b: $i)).
% 0.19/0.42 tff(union_type, type, (
% 0.19/0.42 union: ( $i * $i * $i ) > $o)).
% 0.19/0.42 tff(intersection_type, type, (
% 0.19/0.42 intersection: ( $i * $i * $i ) > $o)).
% 0.19/0.42 tff(1,assumption,(~member(g(aIb, aIc, aI_bUc), aIc)), introduced(assumption)).
% 0.19/0.42 tff(2,assumption,(~member(g(aIb, aIc, aI_bUc), aI_bUc)), introduced(assumption)).
% 0.19/0.42 tff(3,plain,
% 0.19/0.42 ((~union(aIb, aIc, aI_bUc)) <=> (~union(aIb, aIc, aI_bUc))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(4,axiom,(~union(aIb, aIc, aI_bUc)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_aIb_union_aIc_is_aI_bUc')).
% 0.19/0.42 tff(5,plain,
% 0.19/0.42 (~union(aIb, aIc, aI_bUc)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.42 tff(6,plain,
% 0.19/0.42 (^[Set1: $i, Set2: $i, Union: $i] : refl((member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union)) <=> (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union)))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(7,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union)) <=> ![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[6])).
% 0.19/0.42 tff(8,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union)) <=> ![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(9,plain,
% 0.19/0.42 (^[Set1: $i, Set2: $i, Union: $i] : trans(monotonicity(trans(monotonicity(rewrite((union(Set1, Set2, Union) | member(g(Set1, Set2, Union), Set1)) <=> (member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))), (((union(Set1, Set2, Union) | member(g(Set1, Set2, Union), Set1)) | member(g(Set1, Set2, Union), Set2)) <=> ((member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union)) | member(g(Set1, Set2, Union), Set2)))), rewrite(((member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union)) | member(g(Set1, Set2, Union), Set2)) <=> (member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))), (((union(Set1, Set2, Union) | member(g(Set1, Set2, Union), Set1)) | member(g(Set1, Set2, Union), Set2)) <=> (member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union)))), ((((union(Set1, Set2, Union) | member(g(Set1, Set2, Union), Set1)) | member(g(Set1, Set2, Union), Set2)) | member(g(Set1, Set2, Union), Union)) <=> ((member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union)) | member(g(Set1, Set2, Union), Union)))), rewrite(((member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union)) | member(g(Set1, Set2, Union), Union)) <=> (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))), ((((union(Set1, Set2, Union) | member(g(Set1, Set2, Union), Set1)) | member(g(Set1, Set2, Union), Set2)) | member(g(Set1, Set2, Union), Union)) <=> (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(10,plain,
% 0.19/0.42 (![Set1: $i, Set2: $i, Union: $i] : (((union(Set1, Set2, Union) | member(g(Set1, Set2, Union), Set1)) | member(g(Set1, Set2, Union), Set2)) | member(g(Set1, Set2, Union), Union)) <=> ![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[9])).
% 0.19/0.43 tff(11,axiom,(![Set1: $i, Set2: $i, Union: $i] : (((union(Set1, Set2, Union) | member(g(Set1, Set2, Union), Set1)) | member(g(Set1, Set2, Union), Set2)) | member(g(Set1, Set2, Union), Union))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-1.ax','union_axiom1')).
% 0.19/0.43 tff(12,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.19/0.43 tff(13,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.19/0.43 tff(14,plain,(
% 0.19/0.43 ![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))),
% 0.19/0.43 inference(skolemize,[status(sab)],[13])).
% 0.19/0.43 tff(15,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.19/0.43 tff(16,plain,
% 0.19/0.43 (((~![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))) | (union(aIb, aIc, aI_bUc) | member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb))) <=> ((~![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))) | union(aIb, aIc, aI_bUc) | member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(17,plain,
% 0.19/0.43 ((member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb) | union(aIb, aIc, aI_bUc)) <=> (union(aIb, aIc, aI_bUc) | member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(18,plain,
% 0.19/0.43 (((~![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))) | (member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb) | union(aIb, aIc, aI_bUc))) <=> ((~![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))) | (union(aIb, aIc, aI_bUc) | member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb)))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[17])).
% 0.19/0.43 tff(19,plain,
% 0.19/0.43 (((~![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))) | (member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb) | union(aIb, aIc, aI_bUc))) <=> ((~![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))) | union(aIb, aIc, aI_bUc) | member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb))),
% 0.19/0.43 inference(transitivity,[status(thm)],[18, 16])).
% 0.19/0.43 tff(20,plain,
% 0.19/0.43 ((~![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))) | (member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb) | union(aIb, aIc, aI_bUc))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(21,plain,
% 0.19/0.43 ((~![Set1: $i, Set2: $i, Union: $i] : (member(g(Set1, Set2, Union), Union) | member(g(Set1, Set2, Union), Set2) | member(g(Set1, Set2, Union), Set1) | union(Set1, Set2, Union))) | union(aIb, aIc, aI_bUc) | member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.19/0.43 tff(22,plain,
% 0.19/0.43 (member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[21, 15, 5])).
% 0.19/0.43 tff(23,plain,
% 0.19/0.43 (member(g(aIb, aIc, aI_bUc), aIb)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[22, 2, 1])).
% 0.19/0.43 tff(24,plain,
% 0.19/0.43 (intersection(a, b, aIb) <=> intersection(a, b, aIb)),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(25,axiom,(intersection(a, b, aIb)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_intersection_b')).
% 0.19/0.43 tff(26,plain,
% 0.19/0.43 (intersection(a, b, aIb)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.19/0.43 tff(27,plain,
% 0.19/0.43 (^[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))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(28,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[27])).
% 0.19/0.43 tff(29,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(30,plain,
% 0.19/0.43 (^[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)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(31,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[30])).
% 0.19/0.43 tff(32,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')).
% 0.19/0.43 tff(33,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.19/0.43 tff(34,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[33, 29])).
% 0.19/0.43 tff(35,plain,(
% 0.19/0.43 ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[34])).
% 0.19/0.43 tff(36,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[35, 28])).
% 0.19/0.43 tff(37,plain,
% 0.19/0.43 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)) | member(g(aIb, aIc, aI_bUc), b))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)) | member(g(aIb, aIc, aI_bUc), b))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(38,plain,
% 0.19/0.43 ((member(g(aIb, aIc, aI_bUc), b) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb))) <=> ((~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)) | member(g(aIb, aIc, aI_bUc), b))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(39,plain,
% 0.19/0.43 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), b) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)) | member(g(aIb, aIc, aI_bUc), b)))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[38])).
% 0.19/0.43 tff(40,plain,
% 0.19/0.43 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), b) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)) | member(g(aIb, aIc, aI_bUc), b))),
% 0.19/0.43 inference(transitivity,[status(thm)],[39, 37])).
% 0.19/0.43 tff(41,plain,
% 0.19/0.43 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), b) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(42,plain,
% 0.19/0.43 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)) | member(g(aIb, aIc, aI_bUc), b)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.19/0.43 tff(43,plain,
% 0.19/0.43 ((~member(g(aIb, aIc, aI_bUc), aIb)) | member(g(aIb, aIc, aI_bUc), b)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[42, 36, 26])).
% 0.19/0.43 tff(44,plain,
% 0.19/0.43 (member(g(aIb, aIc, aI_bUc), b)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[43, 23])).
% 0.19/0.43 tff(45,plain,
% 0.19/0.43 (union(b, c, bUc) <=> union(b, c, bUc)),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(46,axiom,(union(b, c, bUc)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','b_union_c')).
% 0.19/0.43 tff(47,plain,
% 0.19/0.43 (union(b, c, bUc)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.19/0.43 tff(48,plain,
% 0.19/0.43 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : refl(((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union))) <=> ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(49,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[48])).
% 0.19/0.43 tff(50,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(51,plain,
% 0.19/0.43 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : trans(monotonicity(rewrite(((~union(Set1, Set2, Union)) | (~member(Element, Set1))) <=> ((~member(Element, Set1)) | (~union(Set1, Set2, Union)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Set1))) | member(Element, Union)) <=> (((~member(Element, Set1)) | (~union(Set1, Set2, Union))) | member(Element, Union)))), rewrite((((~member(Element, Set1)) | (~union(Set1, Set2, Union))) | member(Element, Union)) <=> ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Set1))) | member(Element, Union)) <=> ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(52,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (((~union(Set1, Set2, Union)) | (~member(Element, Set1))) | member(Element, Union)) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[51])).
% 0.19/0.43 tff(53,axiom,(![Set1: $i, Set2: $i, Union: $i, Element: $i] : (((~union(Set1, Set2, Union)) | (~member(Element, Set1))) | member(Element, Union))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-1.ax','member_of_set1_is_member_of_union')).
% 0.19/0.43 tff(54,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.19/0.43 tff(55,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.19/0.43 tff(56,plain,(
% 0.19/0.43 ![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[55])).
% 0.19/0.43 tff(57,plain,
% 0.19/0.43 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[56, 49])).
% 0.19/0.43 tff(58,plain,
% 0.19/0.43 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), b)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))) | member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), b)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(59,plain,
% 0.19/0.43 (((~member(g(aIb, aIc, aI_bUc), b)) | member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc))) <=> (member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), b)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(60,plain,
% 0.19/0.43 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))) | ((~member(g(aIb, aIc, aI_bUc), b)) | member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), b))))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[59])).
% 0.19/0.44 tff(61,plain,
% 0.19/0.44 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))) | ((~member(g(aIb, aIc, aI_bUc), b)) | member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))) | member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), b)))),
% 0.19/0.44 inference(transitivity,[status(thm)],[60, 58])).
% 0.19/0.44 tff(62,plain,
% 0.19/0.44 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))) | ((~member(g(aIb, aIc, aI_bUc), b)) | member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(63,plain,
% 0.19/0.44 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))) | member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), b))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.19/0.44 tff(64,plain,
% 0.19/0.44 (member(g(aIb, aIc, aI_bUc), bUc)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[63, 57, 47, 44])).
% 0.19/0.44 tff(65,assumption,(~member(g(aIb, aIc, aI_bUc), a)), introduced(assumption)).
% 0.19/0.44 tff(66,assumption,(member(g(aIb, aIc, aI_bUc), aIb)), introduced(assumption)).
% 0.19/0.44 tff(67,plain,
% 0.19/0.44 (^[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))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(68,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[67])).
% 0.19/0.44 tff(69,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(70,plain,
% 0.19/0.44 (^[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)))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(71,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[70])).
% 0.19/0.44 tff(72,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')).
% 0.19/0.44 tff(73,plain,
% 0.19/0.44 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.19/0.44 tff(74,plain,
% 0.19/0.44 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.19/0.44 tff(75,plain,(
% 0.19/0.44 ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.44 inference(skolemize,[status(sab)],[74])).
% 0.19/0.44 tff(76,plain,
% 0.19/0.44 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[75, 68])).
% 0.19/0.44 tff(77,plain,
% 0.19/0.44 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aIb)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIb)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, b, aIb)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(78,plain,
% 0.19/0.44 ((member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb))) <=> ((~member(g(aIb, aIc, aI_bUc), aIb)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, b, aIb)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(79,plain,
% 0.19/0.44 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aIb)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, b, aIb))))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[78])).
% 0.19/0.44 tff(80,plain,
% 0.19/0.44 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIb)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, b, aIb)))),
% 0.19/0.44 inference(transitivity,[status(thm)],[79, 77])).
% 0.19/0.44 tff(81,plain,
% 0.19/0.44 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(82,plain,
% 0.19/0.44 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIb)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, b, aIb))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.19/0.44 tff(83,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[82, 76, 26, 66, 65])).
% 0.19/0.44 tff(84,plain,(member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIb))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.44 tff(85,plain,
% 0.19/0.44 (member(g(aIb, aIc, aI_bUc), a)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[84, 23])).
% 0.19/0.44 tff(86,plain,
% 0.19/0.44 (intersection(a, bUc, aI_bUc) <=> intersection(a, bUc, aI_bUc)),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(87,axiom,(intersection(a, bUc, aI_bUc)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_intersection_bUc')).
% 0.19/0.44 tff(88,plain,
% 0.19/0.44 (intersection(a, bUc, aI_bUc)),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.19/0.44 tff(89,plain,
% 0.19/0.44 (^[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))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(90,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[89])).
% 0.19/0.44 tff(91,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(92,plain,
% 0.19/0.44 (^[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)))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(93,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[92])).
% 0.19/0.45 tff(94,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')).
% 0.19/0.45 tff(95,plain,
% 0.19/0.45 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.19/0.45 tff(96,plain,
% 0.19/0.45 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[95, 91])).
% 0.19/0.45 tff(97,plain,(
% 0.19/0.45 ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.45 inference(skolemize,[status(sab)],[96])).
% 0.19/0.45 tff(98,plain,
% 0.19/0.45 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[97, 90])).
% 0.19/0.45 tff(99,plain,
% 0.19/0.45 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aI_bUc) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~intersection(a, bUc, aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), a)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(g(aIb, aIc, aI_bUc), aI_bUc) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~intersection(a, bUc, aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), a)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(100,plain,
% 0.19/0.45 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aI_bUc) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~intersection(a, bUc, aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), a)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(101,plain,
% 0.19/0.45 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(g(aIb, aIc, aI_bUc), aI_bUc) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~intersection(a, bUc, aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), a))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.19/0.45 tff(102,plain,
% 0.19/0.45 ($false),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[101, 98, 88, 2, 85, 64])).
% 0.19/0.45 tff(103,plain,(member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.45 tff(104,plain,
% 0.19/0.45 (member(g(aIb, aIc, aI_bUc), aI_bUc)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[103, 1])).
% 0.19/0.45 tff(105,assumption,(member(g(aIb, aIc, aI_bUc), aI_bUc)), introduced(assumption)).
% 0.19/0.45 tff(106,plain,
% 0.19/0.45 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)) | member(g(aIb, aIc, aI_bUc), a))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)) | member(g(aIb, aIc, aI_bUc), a))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(107,plain,
% 0.19/0.45 ((member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc))) <=> ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)) | member(g(aIb, aIc, aI_bUc), a))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(108,plain,
% 0.19/0.45 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)) | member(g(aIb, aIc, aI_bUc), a)))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[107])).
% 0.19/0.45 tff(109,plain,
% 0.19/0.45 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)) | member(g(aIb, aIc, aI_bUc), a))),
% 0.19/0.45 inference(transitivity,[status(thm)],[108, 106])).
% 0.19/0.45 tff(110,plain,
% 0.19/0.45 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(111,plain,
% 0.19/0.45 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)) | member(g(aIb, aIc, aI_bUc), a)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[110, 109])).
% 0.19/0.45 tff(112,plain,
% 0.19/0.45 ($false),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[111, 76, 88, 105, 65])).
% 0.19/0.45 tff(113,plain,(member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aI_bUc))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.45 tff(114,plain,
% 0.19/0.45 (member(g(aIb, aIc, aI_bUc), a)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[113, 104])).
% 0.19/0.45 tff(115,plain,
% 0.19/0.45 (intersection(a, c, aIc) <=> intersection(a, c, aIc)),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(116,axiom,(intersection(a, c, aIc)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_intersection_c')).
% 0.19/0.45 tff(117,plain,
% 0.19/0.45 (intersection(a, c, aIc)),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[116, 115])).
% 0.19/0.45 tff(118,plain,
% 0.19/0.45 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(119,plain,
% 0.19/0.45 ((member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc)) | (~member(g(aIb, aIc, aI_bUc), a))) <=> (member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(120,plain,
% 0.19/0.45 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc)) | (~member(g(aIb, aIc, aI_bUc), a)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[119])).
% 0.19/0.45 tff(121,plain,
% 0.19/0.45 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc)) | (~member(g(aIb, aIc, aI_bUc), a)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc)))),
% 0.19/0.45 inference(transitivity,[status(thm)],[120, 118])).
% 0.19/0.45 tff(122,plain,
% 0.19/0.45 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc)) | (~member(g(aIb, aIc, aI_bUc), a)))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(123,plain,
% 0.19/0.45 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(g(aIb, aIc, aI_bUc), aIc) | (~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), c)) | (~intersection(a, c, aIc))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[122, 121])).
% 0.19/0.45 tff(124,plain,
% 0.19/0.45 ((~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), c))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[123, 98, 117, 1])).
% 0.19/0.45 tff(125,plain,
% 0.19/0.45 (~member(g(aIb, aIc, aI_bUc), c)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[124, 114])).
% 0.19/0.45 tff(126,plain,
% 0.19/0.45 (^[Set1: $i, Set2: $i, Union: $i] : refl((union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1))) <=> (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1))))),
% 0.19/0.45 inference(bind,[status(th)],[])).
% 0.19/0.45 tff(127,plain,
% 0.19/0.45 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1))) <=> ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.19/0.45 inference(quant_intro,[status(thm)],[126])).
% 0.19/0.45 tff(128,plain,
% 0.19/0.45 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1))) <=> ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(129,plain,
% 0.19/0.45 (^[Set1: $i, Set2: $i, Union: $i] : trans(monotonicity(rewrite(((~member(g(Set1, Set2, Union), Set1)) | (~member(g(Set1, Set2, Union), Union))) <=> ((~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))), ((((~member(g(Set1, Set2, Union), Set1)) | (~member(g(Set1, Set2, Union), Union))) | union(Set1, Set2, Union)) <=> (((~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1))) | union(Set1, Set2, Union)))), rewrite((((~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1))) | union(Set1, Set2, Union)) <=> (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))), ((((~member(g(Set1, Set2, Union), Set1)) | (~member(g(Set1, Set2, Union), Union))) | union(Set1, Set2, Union)) <=> (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(130,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i] : (((~member(g(Set1, Set2, Union), Set1)) | (~member(g(Set1, Set2, Union), Union))) | union(Set1, Set2, Union)) <=> ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[129])).
% 0.19/0.46 tff(131,axiom,(![Set1: $i, Set2: $i, Union: $i] : (((~member(g(Set1, Set2, Union), Set1)) | (~member(g(Set1, Set2, Union), Union))) | union(Set1, Set2, Union))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-1.ax','union_axiom2')).
% 0.19/0.46 tff(132,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.19/0.46 tff(133,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[132, 128])).
% 0.19/0.46 tff(134,plain,(
% 0.19/0.46 ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.19/0.46 inference(skolemize,[status(sab)],[133])).
% 0.19/0.46 tff(135,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[134, 127])).
% 0.19/0.46 tff(136,plain,
% 0.19/0.46 (((~![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))) | (union(aIb, aIc, aI_bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIb)))) <=> ((~![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))) | union(aIb, aIc, aI_bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIb)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(137,plain,
% 0.19/0.46 ((~![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))) | (union(aIb, aIc, aI_bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIb)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(138,plain,
% 0.19/0.46 ((~![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))) | union(aIb, aIc, aI_bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIb))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[137, 136])).
% 0.19/0.46 tff(139,plain,
% 0.19/0.46 ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIb))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[138, 135, 5])).
% 0.19/0.46 tff(140,plain,
% 0.19/0.46 (~member(g(aIb, aIc, aI_bUc), aIb)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[139, 104])).
% 0.19/0.46 tff(141,plain,
% 0.19/0.46 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), a)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), b)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), a)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), b)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(142,plain,
% 0.19/0.46 ((member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), b)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), a))) <=> (member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), a)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), b)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(143,plain,
% 0.19/0.46 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), b)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), a)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), a)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), b))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[142])).
% 0.19/0.46 tff(144,plain,
% 0.19/0.46 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), b)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), a)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), a)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), b)))),
% 0.19/0.46 inference(transitivity,[status(thm)],[143, 141])).
% 0.19/0.46 tff(145,plain,
% 0.19/0.46 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | (member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), b)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), a)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(146,plain,
% 0.19/0.46 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))) | member(g(aIb, aIc, aI_bUc), aIb) | (~member(g(aIb, aIc, aI_bUc), a)) | (~intersection(a, b, aIb)) | (~member(g(aIb, aIc, aI_bUc), b))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[145, 144])).
% 0.19/0.46 tff(147,plain,
% 0.19/0.46 ((~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), b))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[146, 98, 26, 140])).
% 0.19/0.46 tff(148,plain,
% 0.19/0.46 (~member(g(aIb, aIc, aI_bUc), b)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[147, 114])).
% 0.19/0.46 tff(149,plain,
% 0.19/0.46 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : refl((member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) <=> (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(150,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[149])).
% 0.19/0.46 tff(151,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(152,plain,
% 0.19/0.46 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~union(Set1, Set2, Union)) | (~member(Element, Union))) <=> ((~member(Element, Union)) | (~union(Set1, Set2, Union)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) <=> (((~member(Element, Union)) | (~union(Set1, Set2, Union))) | member(Element, Set1)))), rewrite((((~member(Element, Union)) | (~union(Set1, Set2, Union))) | member(Element, Set1)) <=> (member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) <=> (member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))))), (((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) | member(Element, Set2)) <=> ((member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) | member(Element, Set2)))), rewrite(((member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union))) | member(Element, Set2)) <=> (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))), (((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) | member(Element, Set2)) <=> (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(153,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) | member(Element, Set2)) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[152])).
% 0.19/0.46 tff(154,axiom,(![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((((~union(Set1, Set2, Union)) | (~member(Element, Union))) | member(Element, Set1)) | member(Element, Set2))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-1.ax','member_of_union_is_member_of_one_set')).
% 0.19/0.46 tff(155,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[154, 153])).
% 0.19/0.46 tff(156,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[155, 151])).
% 0.19/0.46 tff(157,plain,(
% 0.19/0.46 ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.19/0.46 inference(skolemize,[status(sab)],[156])).
% 0.19/0.46 tff(158,plain,
% 0.19/0.46 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[157, 150])).
% 0.19/0.46 tff(159,plain,
% 0.19/0.46 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | ((~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc)) | member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc)) | member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(160,plain,
% 0.19/0.46 ((member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc))) <=> ((~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc)) | member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(161,plain,
% 0.19/0.46 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | ((~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc)) | member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b)))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[160])).
% 0.19/0.47 tff(162,plain,
% 0.19/0.47 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc)) | member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b))),
% 0.19/0.47 inference(transitivity,[status(thm)],[161, 159])).
% 0.19/0.47 tff(163,plain,
% 0.19/0.47 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc)))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(164,plain,
% 0.19/0.47 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))) | (~member(g(aIb, aIc, aI_bUc), bUc)) | (~union(b, c, bUc)) | member(g(aIb, aIc, aI_bUc), c) | member(g(aIb, aIc, aI_bUc), b)),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[163, 162])).
% 0.19/0.47 tff(165,plain,
% 0.19/0.47 (~member(g(aIb, aIc, aI_bUc), bUc)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[164, 158, 47, 148, 125])).
% 0.19/0.47 tff(166,plain,
% 0.19/0.47 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | member(g(aIb, aIc, aI_bUc), bUc) | (~intersection(a, bUc, aI_bUc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | member(g(aIb, aIc, aI_bUc), bUc) | (~intersection(a, bUc, aI_bUc)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(167,plain,
% 0.19/0.47 ((member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc))) <=> ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | member(g(aIb, aIc, aI_bUc), bUc) | (~intersection(a, bUc, aI_bUc)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(168,plain,
% 0.19/0.47 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | member(g(aIb, aIc, aI_bUc), bUc) | (~intersection(a, bUc, aI_bUc))))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[167])).
% 0.19/0.47 tff(169,plain,
% 0.19/0.47 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | member(g(aIb, aIc, aI_bUc), bUc) | (~intersection(a, bUc, aI_bUc)))),
% 0.19/0.47 inference(transitivity,[status(thm)],[168, 166])).
% 0.19/0.47 tff(170,plain,
% 0.19/0.47 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(171,plain,
% 0.19/0.47 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | member(g(aIb, aIc, aI_bUc), bUc) | (~intersection(a, bUc, aI_bUc))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[170, 169])).
% 0.19/0.47 tff(172,plain,
% 0.19/0.47 ($false),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[171, 36, 88, 104, 165])).
% 0.19/0.47 tff(173,plain,(member(g(aIb, aIc, aI_bUc), aIc)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.47 tff(174,plain,
% 0.19/0.47 (^[Set1: $i, Set2: $i, Union: $i] : refl((union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2))) <=> (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(175,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2))) <=> ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[174])).
% 0.19/0.47 tff(176,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2))) <=> ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(177,plain,
% 0.19/0.47 (^[Set1: $i, Set2: $i, Union: $i] : trans(monotonicity(rewrite(((~member(g(Set1, Set2, Union), Set2)) | (~member(g(Set1, Set2, Union), Union))) <=> ((~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))), ((((~member(g(Set1, Set2, Union), Set2)) | (~member(g(Set1, Set2, Union), Union))) | union(Set1, Set2, Union)) <=> (((~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2))) | union(Set1, Set2, Union)))), rewrite((((~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2))) | union(Set1, Set2, Union)) <=> (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))), ((((~member(g(Set1, Set2, Union), Set2)) | (~member(g(Set1, Set2, Union), Union))) | union(Set1, Set2, Union)) <=> (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(178,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i] : (((~member(g(Set1, Set2, Union), Set2)) | (~member(g(Set1, Set2, Union), Union))) | union(Set1, Set2, Union)) <=> ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[177])).
% 0.19/0.47 tff(179,axiom,(![Set1: $i, Set2: $i, Union: $i] : (((~member(g(Set1, Set2, Union), Set2)) | (~member(g(Set1, Set2, Union), Union))) | union(Set1, Set2, Union))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-1.ax','union_axiom3')).
% 0.19/0.47 tff(180,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[179, 178])).
% 0.19/0.47 tff(181,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[180, 176])).
% 0.19/0.47 tff(182,plain,(
% 0.19/0.47 ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.19/0.47 inference(skolemize,[status(sab)],[181])).
% 0.19/0.47 tff(183,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[182, 175])).
% 0.19/0.47 tff(184,plain,
% 0.19/0.47 (((~![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))) | (union(aIb, aIc, aI_bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIc)))) <=> ((~![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))) | union(aIb, aIc, aI_bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIc)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(185,plain,
% 0.19/0.47 ((~![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))) | (union(aIb, aIc, aI_bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIc)))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(186,plain,
% 0.19/0.47 ((~![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))) | union(aIb, aIc, aI_bUc) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIc))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[185, 184])).
% 0.19/0.47 tff(187,plain,
% 0.19/0.47 ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIc))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[186, 183, 5])).
% 0.19/0.47 tff(188,plain,
% 0.19/0.47 (~member(g(aIb, aIc, aI_bUc), aI_bUc)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[187, 173])).
% 0.19/0.47 tff(189,assumption,(member(g(aIb, aIc, aI_bUc), aIc)), introduced(assumption)).
% 0.19/0.47 tff(190,plain,
% 0.19/0.47 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, c, aIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, c, aIc)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(191,plain,
% 0.19/0.47 ((member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIc)) | (~intersection(a, c, aIc))) <=> ((~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, c, aIc)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(192,plain,
% 0.19/0.47 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIc)) | (~intersection(a, c, aIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, c, aIc))))),
% 0.19/0.47 inference(monotonicity,[status(thm)],[191])).
% 0.19/0.47 tff(193,plain,
% 0.19/0.47 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIc)) | (~intersection(a, c, aIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, c, aIc)))),
% 0.19/0.47 inference(transitivity,[status(thm)],[192, 190])).
% 0.19/0.47 tff(194,plain,
% 0.19/0.47 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIc)) | (~intersection(a, c, aIc)))),
% 0.19/0.47 inference(quant_inst,[status(thm)],[])).
% 0.19/0.47 tff(195,plain,
% 0.19/0.47 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, c, aIc))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[194, 193])).
% 0.19/0.47 tff(196,plain,
% 0.19/0.47 ($false),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[195, 76, 117, 189, 65])).
% 0.19/0.47 tff(197,plain,(member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIc))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.47 tff(198,plain,
% 0.19/0.47 (member(g(aIb, aIc, aI_bUc), a)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[197, 189])).
% 0.19/0.47 tff(199,plain,
% 0.19/0.47 ((~member(g(aIb, aIc, aI_bUc), bUc)) | (~member(g(aIb, aIc, aI_bUc), a))),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[101, 98, 88, 2])).
% 0.19/0.47 tff(200,plain,
% 0.19/0.47 (~member(g(aIb, aIc, aI_bUc), bUc)),
% 0.19/0.47 inference(unit_resolution,[status(thm)],[199, 198])).
% 0.19/0.47 tff(201,plain,
% 0.19/0.47 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : refl((member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2))) <=> (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(202,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[201])).
% 0.19/0.47 tff(203,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2))) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.19/0.47 inference(rewrite,[status(thm)],[])).
% 0.19/0.47 tff(204,plain,
% 0.19/0.47 (^[Set1: $i, Set2: $i, Union: $i, Element: $i] : trans(monotonicity(rewrite(((~union(Set1, Set2, Union)) | (~member(Element, Set2))) <=> ((~union(Set1, Set2, Union)) | (~member(Element, Set2)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)) <=> (((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)))), rewrite((((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)) <=> (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))), ((((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)) <=> (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))))),
% 0.19/0.47 inference(bind,[status(th)],[])).
% 0.19/0.47 tff(205,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union)) <=> ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.19/0.47 inference(quant_intro,[status(thm)],[204])).
% 0.19/0.47 tff(206,axiom,(![Set1: $i, Set2: $i, Union: $i, Element: $i] : (((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-1.ax','member_of_set2_is_member_of_union')).
% 0.19/0.47 tff(207,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[206, 205])).
% 0.19/0.47 tff(208,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[207, 203])).
% 0.19/0.47 tff(209,plain,(
% 0.19/0.47 ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.19/0.47 inference(skolemize,[status(sab)],[208])).
% 0.19/0.47 tff(210,plain,
% 0.19/0.47 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.19/0.47 inference(modus_ponens,[status(thm)],[209, 202])).
% 0.19/0.48 tff(211,plain,
% 0.19/0.48 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~union(b, c, bUc)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~union(b, c, bUc)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(212,plain,
% 0.19/0.48 ((member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), c))) <=> (member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~union(b, c, bUc)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(213,plain,
% 0.19/0.48 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), c)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~union(b, c, bUc))))),
% 0.19/0.48 inference(monotonicity,[status(thm)],[212])).
% 0.19/0.48 tff(214,plain,
% 0.19/0.48 (((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), c)))) <=> ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~union(b, c, bUc)))),
% 0.19/0.48 inference(transitivity,[status(thm)],[213, 211])).
% 0.19/0.48 tff(215,plain,
% 0.19/0.48 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | (member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), c)))),
% 0.19/0.48 inference(quant_inst,[status(thm)],[])).
% 0.19/0.48 tff(216,plain,
% 0.19/0.48 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))) | member(g(aIb, aIc, aI_bUc), bUc) | (~member(g(aIb, aIc, aI_bUc), c)) | (~union(b, c, bUc))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[215, 214])).
% 0.19/0.48 tff(217,plain,
% 0.19/0.48 (~member(g(aIb, aIc, aI_bUc), c)),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[216, 210, 47, 200])).
% 0.19/0.48 tff(218,plain,
% 0.19/0.48 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), c) | (~intersection(a, c, aIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), c) | (~intersection(a, c, aIc)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(219,plain,
% 0.19/0.48 ((member(g(aIb, aIc, aI_bUc), c) | (~member(g(aIb, aIc, aI_bUc), aIc)) | (~intersection(a, c, aIc))) <=> ((~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), c) | (~intersection(a, c, aIc)))),
% 0.19/0.48 inference(rewrite,[status(thm)],[])).
% 0.19/0.48 tff(220,plain,
% 0.19/0.48 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), c) | (~member(g(aIb, aIc, aI_bUc), aIc)) | (~intersection(a, c, aIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | ((~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), c) | (~intersection(a, c, aIc))))),
% 0.19/0.48 inference(monotonicity,[status(thm)],[219])).
% 0.19/0.48 tff(221,plain,
% 0.19/0.48 (((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), c) | (~member(g(aIb, aIc, aI_bUc), aIc)) | (~intersection(a, c, aIc)))) <=> ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), c) | (~intersection(a, c, aIc)))),
% 0.19/0.48 inference(transitivity,[status(thm)],[220, 218])).
% 0.19/0.48 tff(222,plain,
% 0.19/0.48 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (member(g(aIb, aIc, aI_bUc), c) | (~member(g(aIb, aIc, aI_bUc), aIc)) | (~intersection(a, c, aIc)))),
% 0.19/0.48 inference(quant_inst,[status(thm)],[])).
% 0.19/0.48 tff(223,plain,
% 0.19/0.48 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), c) | (~intersection(a, c, aIc))),
% 0.19/0.48 inference(modus_ponens,[status(thm)],[222, 221])).
% 0.19/0.48 tff(224,plain,
% 0.19/0.48 ($false),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[223, 36, 117, 189, 217])).
% 0.19/0.48 tff(225,plain,((~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), aI_bUc)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.48 tff(226,plain,
% 0.19/0.48 ($false),
% 0.19/0.48 inference(unit_resolution,[status(thm)],[225, 188, 173])).
% 0.19/0.48 % SZS output end Proof
%------------------------------------------------------------------------------