%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET007-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 : Sat Jun 21 05:24:44 AM UTC 2025
% Result : Unsatisfiable 0.62s 0.85s
% Output : Proof 0.71s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SET007-1 : TPTP v9.0.0. Released v1.0.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n012.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 04:00:11 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.62/0.85 % SZS status Unsatisfiable
% 0.62/0.85 % SZS output start Proof
% 0.62/0.85 tff(member_type, type, (
% 0.62/0.85 member: ( $i * $i ) > $o)).
% 0.62/0.85 tff(aIc_type, type, (
% 0.62/0.85 aIc: $i)).
% 0.62/0.85 tff(g_type, type, (
% 0.62/0.85 g: ( $i * $i * $i ) > $i)).
% 0.62/0.85 tff(aI_bUc_type, type, (
% 0.62/0.85 aI_bUc: $i)).
% 0.62/0.85 tff(aIb_type, type, (
% 0.62/0.85 aIb: $i)).
% 0.62/0.85 tff(b_type, type, (
% 0.62/0.85 b: $i)).
% 0.62/0.85 tff(bUc_type, type, (
% 0.62/0.85 bUc: $i)).
% 0.62/0.85 tff(a_type, type, (
% 0.62/0.85 a: $i)).
% 0.62/0.85 tff(c_type, type, (
% 0.62/0.85 c: $i)).
% 0.62/0.85 tff(intersection_type, type, (
% 0.62/0.85 intersection: ( $i * $i * $i ) > $o)).
% 0.62/0.85 tff(union_type, type, (
% 0.62/0.85 union: ( $i * $i * $i ) > $o)).
% 0.62/0.85 tff(1,assumption,(~member(g(aIb, aIc, aI_bUc), aI_bUc)), introduced(assumption)).
% 0.62/0.85 tff(2,assumption,(member(g(aIb, aIc, aI_bUc), aIc)), introduced(assumption)).
% 0.62/0.85 tff(3,assumption,(~member(g(aIb, aIc, aI_bUc), a)), introduced(assumption)).
% 0.62/0.85 tff(4,plain,
% 0.62/0.85 (intersection(a, c, aIc) <=> intersection(a, c, aIc)),
% 0.62/0.85 inference(rewrite,[status(thm)],[])).
% 0.62/0.85 tff(5,axiom,(intersection(a, c, aIc)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_intersection_c')).
% 0.62/0.85 tff(6,plain,
% 0.62/0.85 (intersection(a, c, aIc)),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.62/0.85 tff(7,plain,
% 0.62/0.85 (^[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))))),
% 0.62/0.85 inference(bind,[status(th)],[])).
% 0.62/0.85 tff(8,plain,
% 0.62/0.85 (![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)))),
% 0.62/0.85 inference(quant_intro,[status(thm)],[7])).
% 0.62/0.85 tff(9,plain,
% 0.62/0.85 (![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)))),
% 0.62/0.85 inference(rewrite,[status(thm)],[])).
% 0.62/0.85 tff(10,plain,
% 0.62/0.85 (^[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)))))),
% 0.62/0.85 inference(bind,[status(th)],[])).
% 0.62/0.85 tff(11,plain,
% 0.62/0.85 (![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)))),
% 0.62/0.85 inference(quant_intro,[status(thm)],[10])).
% 0.62/0.85 tff(12,axiom,(![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set1))), file('/export/starexec/sandbox/benchmark/Axioms/SET001-2.ax','member_of_intersection_is_member_of_set1')).
% 0.62/0.85 tff(13,plain,
% 0.62/0.85 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[12, 11])).
% 0.62/0.85 tff(14,plain,
% 0.62/0.85 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[13, 9])).
% 0.62/0.85 tff(15,plain,(
% 0.62/0.85 ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.62/0.85 inference(skolemize,[status(sab)],[14])).
% 0.62/0.85 tff(16,plain,
% 0.62/0.85 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[15, 8])).
% 0.62/0.85 tff(17,plain,
% 0.62/0.85 (((~![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)))),
% 0.62/0.85 inference(rewrite,[status(thm)],[])).
% 0.62/0.85 tff(18,plain,
% 0.62/0.85 ((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)))),
% 0.62/0.85 inference(rewrite,[status(thm)],[])).
% 0.62/0.85 tff(19,plain,
% 0.62/0.85 (((~![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))))),
% 0.62/0.85 inference(monotonicity,[status(thm)],[18])).
% 0.62/0.85 tff(20,plain,
% 0.62/0.85 (((~![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)))),
% 0.62/0.85 inference(transitivity,[status(thm)],[19, 17])).
% 0.62/0.85 tff(21,plain,
% 0.62/0.85 ((~![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)))),
% 0.62/0.85 inference(quant_inst,[status(thm)],[])).
% 0.62/0.85 tff(22,plain,
% 0.62/0.85 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, c, aIc))),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.62/0.85 tff(23,plain,
% 0.62/0.85 ($false),
% 0.62/0.85 inference(unit_resolution,[status(thm)],[22, 16, 6, 2, 3])).
% 0.62/0.85 tff(24,plain,(member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIc))), inference(lemma,lemma(discharge,[]))).
% 0.62/0.85 tff(25,plain,
% 0.62/0.85 (member(g(aIb, aIc, aI_bUc), a)),
% 0.62/0.85 inference(unit_resolution,[status(thm)],[24, 2])).
% 0.62/0.85 tff(26,plain,
% 0.62/0.85 (intersection(a, bUc, aI_bUc) <=> intersection(a, bUc, aI_bUc)),
% 0.62/0.85 inference(rewrite,[status(thm)],[])).
% 0.62/0.85 tff(27,axiom,(intersection(a, bUc, aI_bUc)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_intersection_bUc')).
% 0.62/0.85 tff(28,plain,
% 0.62/0.85 (intersection(a, bUc, aI_bUc)),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.62/0.85 tff(29,plain,
% 0.62/0.85 (^[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))))),
% 0.62/0.85 inference(bind,[status(th)],[])).
% 0.62/0.85 tff(30,plain,
% 0.62/0.85 (![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)))),
% 0.62/0.85 inference(quant_intro,[status(thm)],[29])).
% 0.62/0.85 tff(31,plain,
% 0.62/0.85 (![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)))),
% 0.62/0.85 inference(rewrite,[status(thm)],[])).
% 0.62/0.85 tff(32,plain,
% 0.62/0.85 (^[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)))))),
% 0.62/0.85 inference(bind,[status(th)],[])).
% 0.62/0.85 tff(33,plain,
% 0.62/0.85 (![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)))),
% 0.62/0.85 inference(quant_intro,[status(thm)],[32])).
% 0.62/0.85 tff(34,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/sandbox/benchmark/Axioms/SET001-2.ax','member_of_both_is_member_of_intersection')).
% 0.62/0.85 tff(35,plain,
% 0.62/0.85 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.62/0.85 tff(36,plain,
% 0.62/0.85 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[35, 31])).
% 0.62/0.85 tff(37,plain,(
% 0.62/0.85 ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.62/0.85 inference(skolemize,[status(sab)],[36])).
% 0.62/0.85 tff(38,plain,
% 0.62/0.85 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Intersection) | (~member(Element, Set2)) | (~intersection(Set1, Set2, Intersection)) | (~member(Element, Set1)))),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[37, 30])).
% 0.62/0.85 tff(39,plain,
% 0.62/0.85 (((~![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)))),
% 0.62/0.85 inference(rewrite,[status(thm)],[])).
% 0.62/0.85 tff(40,plain,
% 0.62/0.85 ((~![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)))),
% 0.62/0.85 inference(quant_inst,[status(thm)],[])).
% 0.62/0.85 tff(41,plain,
% 0.62/0.85 ((~![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))),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.62/0.85 tff(42,plain,
% 0.62/0.85 ((~member(g(aIb, aIc, aI_bUc), bUc)) | (~member(g(aIb, aIc, aI_bUc), a))),
% 0.62/0.85 inference(unit_resolution,[status(thm)],[41, 38, 28, 1])).
% 0.62/0.85 tff(43,plain,
% 0.62/0.85 (~member(g(aIb, aIc, aI_bUc), bUc)),
% 0.62/0.85 inference(unit_resolution,[status(thm)],[42, 25])).
% 0.62/0.85 tff(44,plain,
% 0.62/0.85 (union(b, c, bUc) <=> union(b, c, bUc)),
% 0.62/0.85 inference(rewrite,[status(thm)],[])).
% 0.62/0.85 tff(45,axiom,(union(b, c, bUc)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','b_union_c')).
% 0.62/0.85 tff(46,plain,
% 0.62/0.85 (union(b, c, bUc)),
% 0.62/0.85 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.62/0.85 tff(47,plain,
% 0.62/0.85 (^[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))))),
% 0.62/0.85 inference(bind,[status(th)],[])).
% 0.62/0.85 tff(48,plain,
% 0.62/0.85 (![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)))),
% 0.62/0.85 inference(quant_intro,[status(thm)],[47])).
% 0.62/0.85 tff(49,plain,
% 0.62/0.85 (![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)))),
% 0.62/0.85 inference(rewrite,[status(thm)],[])).
% 0.62/0.85 tff(50,plain,
% 0.62/0.85 (^[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)))))),
% 0.62/0.86 inference(bind,[status(th)],[])).
% 0.62/0.86 tff(51,plain,
% 0.62/0.86 (![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)))),
% 0.62/0.86 inference(quant_intro,[status(thm)],[50])).
% 0.62/0.86 tff(52,axiom,(![Set1: $i, Set2: $i, Union: $i, Element: $i] : (((~union(Set1, Set2, Union)) | (~member(Element, Set2))) | member(Element, Union))), file('/export/starexec/sandbox/benchmark/Axioms/SET001-1.ax','member_of_set2_is_member_of_union')).
% 0.62/0.86 tff(53,plain,
% 0.62/0.86 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.62/0.86 tff(54,plain,
% 0.62/0.86 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.62/0.86 tff(55,plain,(
% 0.62/0.86 ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.62/0.86 inference(skolemize,[status(sab)],[54])).
% 0.62/0.86 tff(56,plain,
% 0.62/0.86 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Union) | (~union(Set1, Set2, Union)) | (~member(Element, Set2)))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[55, 48])).
% 0.62/0.86 tff(57,plain,
% 0.62/0.86 (((~![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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(58,plain,
% 0.62/0.86 ((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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(59,plain,
% 0.62/0.86 (((~![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))))),
% 0.62/0.86 inference(monotonicity,[status(thm)],[58])).
% 0.62/0.86 tff(60,plain,
% 0.62/0.86 (((~![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)))),
% 0.62/0.86 inference(transitivity,[status(thm)],[59, 57])).
% 0.62/0.86 tff(61,plain,
% 0.62/0.86 ((~![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)))),
% 0.62/0.86 inference(quant_inst,[status(thm)],[])).
% 0.62/0.86 tff(62,plain,
% 0.62/0.86 ((~![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))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.62/0.86 tff(63,plain,
% 0.62/0.86 (~member(g(aIb, aIc, aI_bUc), c)),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[62, 56, 46, 43])).
% 0.62/0.86 tff(64,plain,
% 0.62/0.86 (^[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))))),
% 0.62/0.86 inference(bind,[status(th)],[])).
% 0.62/0.86 tff(65,plain,
% 0.62/0.86 (![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)))),
% 0.62/0.86 inference(quant_intro,[status(thm)],[64])).
% 0.62/0.86 tff(66,plain,
% 0.62/0.86 (![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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(67,plain,
% 0.62/0.86 (^[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)))))),
% 0.62/0.86 inference(bind,[status(th)],[])).
% 0.62/0.86 tff(68,plain,
% 0.62/0.86 (![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)))),
% 0.62/0.86 inference(quant_intro,[status(thm)],[67])).
% 0.62/0.86 tff(69,axiom,(![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (((~intersection(Set1, Set2, Intersection)) | (~member(Element, Intersection))) | member(Element, Set2))), file('/export/starexec/sandbox/benchmark/Axioms/SET001-2.ax','member_of_intersection_is_member_of_set2')).
% 0.62/0.86 tff(70,plain,
% 0.62/0.86 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.62/0.86 tff(71,plain,
% 0.62/0.86 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[70, 66])).
% 0.62/0.86 tff(72,plain,(
% 0.62/0.86 ![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.62/0.86 inference(skolemize,[status(sab)],[71])).
% 0.62/0.86 tff(73,plain,
% 0.62/0.86 (![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[72, 65])).
% 0.62/0.86 tff(74,plain,
% 0.62/0.86 (((~![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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(75,plain,
% 0.62/0.86 ((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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(76,plain,
% 0.62/0.86 (((~![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))))),
% 0.62/0.86 inference(monotonicity,[status(thm)],[75])).
% 0.62/0.86 tff(77,plain,
% 0.62/0.86 (((~![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)))),
% 0.62/0.86 inference(transitivity,[status(thm)],[76, 74])).
% 0.62/0.86 tff(78,plain,
% 0.62/0.86 ((~![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)))),
% 0.62/0.86 inference(quant_inst,[status(thm)],[])).
% 0.62/0.86 tff(79,plain,
% 0.62/0.86 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), c) | (~intersection(a, c, aIc))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.62/0.86 tff(80,plain,
% 0.62/0.86 ($false),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[79, 73, 6, 2, 63])).
% 0.62/0.86 tff(81,plain,((~member(g(aIb, aIc, aI_bUc), aIc)) | member(g(aIb, aIc, aI_bUc), aI_bUc)), inference(lemma,lemma(discharge,[]))).
% 0.62/0.86 tff(82,plain,
% 0.62/0.86 (~member(g(aIb, aIc, aI_bUc), aIc)),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[81, 1])).
% 0.62/0.86 tff(83,plain,
% 0.62/0.86 ((~union(aIb, aIc, aI_bUc)) <=> (~union(aIb, aIc, aI_bUc))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(84,axiom,(~union(aIb, aIc, aI_bUc)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_aIb_union_aIc_is_aI_bUc')).
% 0.62/0.86 tff(85,plain,
% 0.62/0.86 (~union(aIb, aIc, aI_bUc)),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.62/0.86 tff(86,plain,
% 0.62/0.86 (^[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)))),
% 0.62/0.86 inference(bind,[status(th)],[])).
% 0.62/0.86 tff(87,plain,
% 0.62/0.86 (![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))),
% 0.62/0.86 inference(quant_intro,[status(thm)],[86])).
% 0.62/0.86 tff(88,plain,
% 0.62/0.86 (![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))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(89,plain,
% 0.62/0.86 (^[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))))),
% 0.62/0.86 inference(bind,[status(th)],[])).
% 0.62/0.86 tff(90,plain,
% 0.62/0.86 (![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))),
% 0.62/0.86 inference(quant_intro,[status(thm)],[89])).
% 0.62/0.86 tff(91,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/sandbox/benchmark/Axioms/SET001-1.ax','union_axiom1')).
% 0.62/0.86 tff(92,plain,
% 0.62/0.86 (![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))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.62/0.86 tff(93,plain,
% 0.62/0.86 (![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))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[92, 88])).
% 0.62/0.86 tff(94,plain,(
% 0.62/0.86 ![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))),
% 0.62/0.86 inference(skolemize,[status(sab)],[93])).
% 0.62/0.86 tff(95,plain,
% 0.62/0.86 (![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))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[94, 87])).
% 0.62/0.86 tff(96,plain,
% 0.62/0.86 (((~![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))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(97,plain,
% 0.62/0.86 ((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))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(98,plain,
% 0.62/0.86 (((~![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)))),
% 0.62/0.86 inference(monotonicity,[status(thm)],[97])).
% 0.62/0.86 tff(99,plain,
% 0.62/0.86 (((~![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))),
% 0.62/0.86 inference(transitivity,[status(thm)],[98, 96])).
% 0.62/0.86 tff(100,plain,
% 0.62/0.86 ((~![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))),
% 0.62/0.86 inference(quant_inst,[status(thm)],[])).
% 0.62/0.86 tff(101,plain,
% 0.62/0.86 ((~![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)),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.62/0.86 tff(102,plain,
% 0.62/0.86 (member(g(aIb, aIc, aI_bUc), aI_bUc) | member(g(aIb, aIc, aI_bUc), aIc) | member(g(aIb, aIc, aI_bUc), aIb)),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[101, 95, 85])).
% 0.62/0.86 tff(103,plain,
% 0.62/0.86 (member(g(aIb, aIc, aI_bUc), aIb)),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[102, 1, 82])).
% 0.62/0.86 tff(104,assumption,(member(g(aIb, aIc, aI_bUc), aIb)), introduced(assumption)).
% 0.62/0.86 tff(105,plain,
% 0.62/0.86 (intersection(a, b, aIb) <=> intersection(a, b, aIb)),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(106,axiom,(intersection(a, b, aIb)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_intersection_b')).
% 0.62/0.86 tff(107,plain,
% 0.62/0.86 (intersection(a, b, aIb)),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[106, 105])).
% 0.62/0.86 tff(108,plain,
% 0.62/0.86 (((~![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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(109,plain,
% 0.62/0.86 ((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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(110,plain,
% 0.62/0.86 (((~![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))))),
% 0.62/0.86 inference(monotonicity,[status(thm)],[109])).
% 0.62/0.86 tff(111,plain,
% 0.62/0.86 (((~![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)))),
% 0.62/0.86 inference(transitivity,[status(thm)],[110, 108])).
% 0.62/0.86 tff(112,plain,
% 0.62/0.86 ((~![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)))),
% 0.62/0.86 inference(quant_inst,[status(thm)],[])).
% 0.62/0.86 tff(113,plain,
% 0.62/0.86 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIb)) | member(g(aIb, aIc, aI_bUc), a) | (~intersection(a, b, aIb))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[112, 111])).
% 0.62/0.86 tff(114,plain,
% 0.62/0.86 ($false),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[113, 16, 107, 104, 3])).
% 0.62/0.86 tff(115,plain,(member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aIb))), inference(lemma,lemma(discharge,[]))).
% 0.62/0.86 tff(116,plain,
% 0.62/0.86 (member(g(aIb, aIc, aI_bUc), a)),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[115, 103])).
% 0.62/0.86 tff(117,plain,
% 0.62/0.86 (~member(g(aIb, aIc, aI_bUc), bUc)),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[41, 38, 28, 1, 116])).
% 0.62/0.86 tff(118,assumption,(member(g(aIb, aIc, aI_bUc), b)), introduced(assumption)).
% 0.62/0.86 tff(119,assumption,(~member(g(aIb, aIc, aI_bUc), bUc)), introduced(assumption)).
% 0.62/0.86 tff(120,plain,
% 0.62/0.86 (^[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))))),
% 0.62/0.86 inference(bind,[status(th)],[])).
% 0.62/0.86 tff(121,plain,
% 0.62/0.86 (![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)))),
% 0.62/0.86 inference(quant_intro,[status(thm)],[120])).
% 0.62/0.86 tff(122,plain,
% 0.62/0.86 (![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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(123,plain,
% 0.62/0.86 (^[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)))))),
% 0.62/0.86 inference(bind,[status(th)],[])).
% 0.62/0.86 tff(124,plain,
% 0.62/0.86 (![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)))),
% 0.62/0.86 inference(quant_intro,[status(thm)],[123])).
% 0.62/0.86 tff(125,axiom,(![Set1: $i, Set2: $i, Union: $i, Element: $i] : (((~union(Set1, Set2, Union)) | (~member(Element, Set1))) | member(Element, Union))), file('/export/starexec/sandbox/benchmark/Axioms/SET001-1.ax','member_of_set1_is_member_of_union')).
% 0.62/0.86 tff(126,plain,
% 0.62/0.86 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.62/0.86 tff(127,plain,
% 0.62/0.86 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[126, 122])).
% 0.62/0.86 tff(128,plain,(
% 0.62/0.86 ![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.62/0.86 inference(skolemize,[status(sab)],[127])).
% 0.62/0.86 tff(129,plain,
% 0.62/0.86 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[128, 121])).
% 0.62/0.86 tff(130,plain,
% 0.62/0.86 (((~![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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(131,plain,
% 0.62/0.86 (((~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)))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(132,plain,
% 0.62/0.86 (((~![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))))),
% 0.62/0.86 inference(monotonicity,[status(thm)],[131])).
% 0.62/0.86 tff(133,plain,
% 0.62/0.86 (((~![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)))),
% 0.62/0.86 inference(transitivity,[status(thm)],[132, 130])).
% 0.62/0.86 tff(134,plain,
% 0.62/0.86 ((~![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)))),
% 0.62/0.86 inference(quant_inst,[status(thm)],[])).
% 0.62/0.86 tff(135,plain,
% 0.62/0.86 ((~![Set1: $i, Set2: $i, Union: $i, Element: $i] : ((~member(Element, Set1)) | member(Element, Union) | (~union(Set1, Set2, Union)))) | member(g(aIb, aIc, aI_bUc), bUc) | (~union(b, c, bUc)) | (~member(g(aIb, aIc, aI_bUc), b))),
% 0.62/0.86 inference(modus_ponens,[status(thm)],[134, 133])).
% 0.62/0.86 tff(136,plain,
% 0.62/0.86 ($false),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[135, 129, 46, 119, 118])).
% 0.62/0.86 tff(137,plain,((~member(g(aIb, aIc, aI_bUc), b)) | member(g(aIb, aIc, aI_bUc), bUc)), inference(lemma,lemma(discharge,[]))).
% 0.62/0.86 tff(138,plain,
% 0.62/0.86 (~member(g(aIb, aIc, aI_bUc), b)),
% 0.62/0.86 inference(unit_resolution,[status(thm)],[137, 117])).
% 0.62/0.86 tff(139,plain,
% 0.62/0.86 (((~![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))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(140,plain,
% 0.62/0.86 ((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))),
% 0.62/0.86 inference(rewrite,[status(thm)],[])).
% 0.62/0.86 tff(141,plain,
% 0.62/0.86 (((~![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)))),
% 0.71/0.86 inference(monotonicity,[status(thm)],[140])).
% 0.71/0.86 tff(142,plain,
% 0.71/0.86 (((~![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))),
% 0.71/0.86 inference(transitivity,[status(thm)],[141, 139])).
% 0.71/0.86 tff(143,plain,
% 0.71/0.86 ((~![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)))),
% 0.71/0.86 inference(quant_inst,[status(thm)],[])).
% 0.71/0.86 tff(144,plain,
% 0.71/0.86 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aIb)) | (~intersection(a, b, aIb)) | member(g(aIb, aIc, aI_bUc), b)),
% 0.71/0.86 inference(modus_ponens,[status(thm)],[143, 142])).
% 0.71/0.86 tff(145,plain,
% 0.71/0.86 ($false),
% 0.71/0.86 inference(unit_resolution,[status(thm)],[144, 73, 107, 103, 138])).
% 0.71/0.86 tff(146,plain,(member(g(aIb, aIc, aI_bUc), aI_bUc)), inference(lemma,lemma(discharge,[]))).
% 0.71/0.86 tff(147,plain,
% 0.71/0.86 (^[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))))),
% 0.71/0.86 inference(bind,[status(th)],[])).
% 0.71/0.86 tff(148,plain,
% 0.71/0.86 (![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)))),
% 0.71/0.86 inference(quant_intro,[status(thm)],[147])).
% 0.71/0.86 tff(149,plain,
% 0.71/0.86 (![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)))),
% 0.71/0.86 inference(rewrite,[status(thm)],[])).
% 0.71/0.86 tff(150,plain,
% 0.71/0.86 (^[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)))))),
% 0.71/0.86 inference(bind,[status(th)],[])).
% 0.71/0.86 tff(151,plain,
% 0.71/0.86 (![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)))),
% 0.71/0.86 inference(quant_intro,[status(thm)],[150])).
% 0.71/0.86 tff(152,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/sandbox/benchmark/Axioms/SET001-1.ax','union_axiom3')).
% 0.71/0.86 tff(153,plain,
% 0.71/0.86 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.71/0.86 inference(modus_ponens,[status(thm)],[152, 151])).
% 0.71/0.86 tff(154,plain,
% 0.71/0.86 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.71/0.86 inference(modus_ponens,[status(thm)],[153, 149])).
% 0.71/0.86 tff(155,plain,(
% 0.71/0.86 ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.71/0.86 inference(skolemize,[status(sab)],[154])).
% 0.71/0.86 tff(156,plain,
% 0.71/0.86 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set2)))),
% 0.71/0.86 inference(modus_ponens,[status(thm)],[155, 148])).
% 0.71/0.86 tff(157,plain,
% 0.71/0.86 (((~![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)))),
% 0.71/0.86 inference(rewrite,[status(thm)],[])).
% 0.71/0.86 tff(158,plain,
% 0.71/0.86 ((~![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)))),
% 0.71/0.86 inference(quant_inst,[status(thm)],[])).
% 0.71/0.86 tff(159,plain,
% 0.71/0.86 ((~![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))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[158, 157])).
% 0.71/0.87 tff(160,plain,
% 0.71/0.87 ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIc))),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[159, 156, 85])).
% 0.71/0.87 tff(161,plain,
% 0.71/0.87 (~member(g(aIb, aIc, aI_bUc), aIc)),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[160, 146])).
% 0.71/0.87 tff(162,plain,
% 0.71/0.87 (^[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))))),
% 0.71/0.87 inference(bind,[status(th)],[])).
% 0.71/0.87 tff(163,plain,
% 0.71/0.87 (![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)))),
% 0.71/0.87 inference(quant_intro,[status(thm)],[162])).
% 0.71/0.87 tff(164,plain,
% 0.71/0.87 (![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)))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(165,plain,
% 0.71/0.87 (^[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)))))),
% 0.71/0.87 inference(bind,[status(th)],[])).
% 0.71/0.87 tff(166,plain,
% 0.71/0.87 (![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)))),
% 0.71/0.87 inference(quant_intro,[status(thm)],[165])).
% 0.71/0.87 tff(167,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/sandbox/benchmark/Axioms/SET001-1.ax','union_axiom2')).
% 0.71/0.87 tff(168,plain,
% 0.71/0.87 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[167, 166])).
% 0.71/0.87 tff(169,plain,
% 0.71/0.87 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[168, 164])).
% 0.71/0.87 tff(170,plain,(
% 0.71/0.87 ![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.71/0.87 inference(skolemize,[status(sab)],[169])).
% 0.71/0.87 tff(171,plain,
% 0.71/0.87 (![Set1: $i, Set2: $i, Union: $i] : (union(Set1, Set2, Union) | (~member(g(Set1, Set2, Union), Union)) | (~member(g(Set1, Set2, Union), Set1)))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[170, 163])).
% 0.71/0.87 tff(172,plain,
% 0.71/0.87 (((~![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)))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(173,plain,
% 0.71/0.87 ((~![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)))),
% 0.71/0.87 inference(quant_inst,[status(thm)],[])).
% 0.71/0.87 tff(174,plain,
% 0.71/0.87 ((~![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))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[173, 172])).
% 0.71/0.87 tff(175,plain,
% 0.71/0.87 ((~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~member(g(aIb, aIc, aI_bUc), aIb))),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[174, 171, 85])).
% 0.71/0.87 tff(176,plain,
% 0.71/0.87 (~member(g(aIb, aIc, aI_bUc), aIb)),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[175, 146])).
% 0.71/0.87 tff(177,assumption,(~member(g(aIb, aIc, aI_bUc), aIc)), introduced(assumption)).
% 0.71/0.87 tff(178,assumption,(~member(g(aIb, aIc, aI_bUc), aIb)), introduced(assumption)).
% 0.71/0.87 tff(179,plain,
% 0.71/0.87 (member(g(aIb, aIc, aI_bUc), aI_bUc)),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[102, 178, 177])).
% 0.71/0.87 tff(180,assumption,(member(g(aIb, aIc, aI_bUc), aI_bUc)), introduced(assumption)).
% 0.71/0.87 tff(181,plain,
% 0.71/0.87 (((~![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))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(182,plain,
% 0.71/0.87 ((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))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(183,plain,
% 0.71/0.87 (((~![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)))),
% 0.71/0.87 inference(monotonicity,[status(thm)],[182])).
% 0.71/0.87 tff(184,plain,
% 0.71/0.87 (((~![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))),
% 0.71/0.87 inference(transitivity,[status(thm)],[183, 181])).
% 0.71/0.87 tff(185,plain,
% 0.71/0.87 ((~![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)))),
% 0.71/0.87 inference(quant_inst,[status(thm)],[])).
% 0.71/0.87 tff(186,plain,
% 0.71/0.87 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set1) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | (~intersection(a, bUc, aI_bUc)) | member(g(aIb, aIc, aI_bUc), a)),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[185, 184])).
% 0.71/0.87 tff(187,plain,
% 0.71/0.87 ($false),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[186, 16, 28, 180, 3])).
% 0.71/0.87 tff(188,plain,(member(g(aIb, aIc, aI_bUc), a) | (~member(g(aIb, aIc, aI_bUc), aI_bUc))), inference(lemma,lemma(discharge,[]))).
% 0.71/0.87 tff(189,plain,
% 0.71/0.87 (member(g(aIb, aIc, aI_bUc), a)),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[188, 179])).
% 0.71/0.87 tff(190,plain,
% 0.71/0.87 (((~![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)))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(191,plain,
% 0.71/0.87 ((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)))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(192,plain,
% 0.71/0.87 (((~![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))))),
% 0.71/0.87 inference(monotonicity,[status(thm)],[191])).
% 0.71/0.87 tff(193,plain,
% 0.71/0.87 (((~![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)))),
% 0.71/0.87 inference(transitivity,[status(thm)],[192, 190])).
% 0.71/0.87 tff(194,plain,
% 0.71/0.87 ((~![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)))),
% 0.71/0.87 inference(quant_inst,[status(thm)],[])).
% 0.71/0.87 tff(195,plain,
% 0.71/0.87 ((~![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))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[194, 193])).
% 0.71/0.87 tff(196,plain,
% 0.71/0.87 ((~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), b))),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[195, 38, 107, 178])).
% 0.71/0.87 tff(197,plain,
% 0.71/0.87 (~member(g(aIb, aIc, aI_bUc), b)),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[196, 189])).
% 0.71/0.87 tff(198,plain,
% 0.71/0.87 (((~![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)))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(199,plain,
% 0.71/0.87 ((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)))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(200,plain,
% 0.71/0.87 (((~![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))))),
% 0.71/0.87 inference(monotonicity,[status(thm)],[199])).
% 0.71/0.87 tff(201,plain,
% 0.71/0.87 (((~![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)))),
% 0.71/0.87 inference(transitivity,[status(thm)],[200, 198])).
% 0.71/0.87 tff(202,plain,
% 0.71/0.87 ((~![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)))),
% 0.71/0.87 inference(quant_inst,[status(thm)],[])).
% 0.71/0.87 tff(203,plain,
% 0.71/0.87 ((~![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))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[202, 201])).
% 0.71/0.87 tff(204,plain,
% 0.71/0.87 ((~member(g(aIb, aIc, aI_bUc), a)) | (~member(g(aIb, aIc, aI_bUc), c))),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[203, 38, 6, 177])).
% 0.71/0.87 tff(205,plain,
% 0.71/0.87 (~member(g(aIb, aIc, aI_bUc), c)),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[204, 189])).
% 0.71/0.87 tff(206,plain,
% 0.71/0.87 (^[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))))),
% 0.71/0.87 inference(bind,[status(th)],[])).
% 0.71/0.87 tff(207,plain,
% 0.71/0.87 (![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)))),
% 0.71/0.87 inference(quant_intro,[status(thm)],[206])).
% 0.71/0.87 tff(208,plain,
% 0.71/0.87 (![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)))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(209,plain,
% 0.71/0.87 (^[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)))))),
% 0.71/0.87 inference(bind,[status(th)],[])).
% 0.71/0.87 tff(210,plain,
% 0.71/0.87 (![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)))),
% 0.71/0.87 inference(quant_intro,[status(thm)],[209])).
% 0.71/0.87 tff(211,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/sandbox/benchmark/Axioms/SET001-1.ax','member_of_union_is_member_of_one_set')).
% 0.71/0.87 tff(212,plain,
% 0.71/0.87 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[211, 210])).
% 0.71/0.87 tff(213,plain,
% 0.71/0.87 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[212, 208])).
% 0.71/0.87 tff(214,plain,(
% 0.71/0.87 ![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.71/0.87 inference(skolemize,[status(sab)],[213])).
% 0.71/0.87 tff(215,plain,
% 0.71/0.87 (![Set1: $i, Set2: $i, Union: $i, Element: $i] : (member(Element, Set2) | member(Element, Set1) | (~member(Element, Union)) | (~union(Set1, Set2, Union)))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[214, 207])).
% 0.71/0.87 tff(216,plain,
% 0.71/0.87 (((~![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))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(217,plain,
% 0.71/0.87 ((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))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(218,plain,
% 0.71/0.87 (((~![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)))),
% 0.71/0.87 inference(monotonicity,[status(thm)],[217])).
% 0.71/0.87 tff(219,plain,
% 0.71/0.87 (((~![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))),
% 0.71/0.87 inference(transitivity,[status(thm)],[218, 216])).
% 0.71/0.87 tff(220,plain,
% 0.71/0.87 ((~![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)))),
% 0.71/0.87 inference(quant_inst,[status(thm)],[])).
% 0.71/0.87 tff(221,plain,
% 0.71/0.87 ((~![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)),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[220, 219])).
% 0.71/0.87 tff(222,plain,
% 0.71/0.87 (~member(g(aIb, aIc, aI_bUc), bUc)),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[221, 215, 46, 205, 197])).
% 0.71/0.87 tff(223,plain,
% 0.71/0.87 (((~![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)))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(224,plain,
% 0.71/0.87 ((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)))),
% 0.71/0.87 inference(rewrite,[status(thm)],[])).
% 0.71/0.87 tff(225,plain,
% 0.71/0.87 (((~![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))))),
% 0.71/0.87 inference(monotonicity,[status(thm)],[224])).
% 0.71/0.87 tff(226,plain,
% 0.71/0.87 (((~![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)))),
% 0.71/0.87 inference(transitivity,[status(thm)],[225, 223])).
% 0.71/0.87 tff(227,plain,
% 0.71/0.87 ((~![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)))),
% 0.71/0.87 inference(quant_inst,[status(thm)],[])).
% 0.71/0.87 tff(228,plain,
% 0.71/0.87 ((~![Set1: $i, Set2: $i, Element: $i, Intersection: $i] : (member(Element, Set2) | (~member(Element, Intersection)) | (~intersection(Set1, Set2, Intersection)))) | (~member(g(aIb, aIc, aI_bUc), aI_bUc)) | member(g(aIb, aIc, aI_bUc), bUc) | (~intersection(a, bUc, aI_bUc))),
% 0.71/0.87 inference(modus_ponens,[status(thm)],[227, 226])).
% 0.71/0.87 tff(229,plain,
% 0.71/0.87 ($false),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[228, 73, 28, 179, 222])).
% 0.71/0.87 tff(230,plain,(member(g(aIb, aIc, aI_bUc), aIb) | member(g(aIb, aIc, aI_bUc), aIc)), inference(lemma,lemma(discharge,[]))).
% 0.71/0.87 tff(231,plain,
% 0.71/0.87 ($false),
% 0.71/0.87 inference(unit_resolution,[status(thm)],[230, 176, 161])).
% 0.71/0.87 % SZS output end Proof
% 0.71/0.88 % E exiting
%------------------------------------------------------------------------------