%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET013-1 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 05:24:46 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.37s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET013-1 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n015.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 04:01:26 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.20/0.37 % SZS status Unsatisfiable
% 0.20/0.37 % SZS output start Proof
% 0.20/0.37 tff(member_type, type, (
% 0.20/0.37 member: ( $i * $i ) > $o)).
% 0.20/0.37 tff(bs_type, type, (
% 0.20/0.37 bs: $i)).
% 0.20/0.37 tff(member_of_1_not_of_2_type, type, (
% 0.20/0.37 member_of_1_not_of_2: ( $i * $i ) > $i)).
% 0.20/0.37 tff(intersection_type, type, (
% 0.20/0.37 intersection: ( $i * $i ) > $i)).
% 0.20/0.37 tff(as_type, type, (
% 0.20/0.37 as: $i)).
% 0.20/0.37 tff(subset_type, type, (
% 0.20/0.37 subset: ( $i * $i ) > $o)).
% 0.20/0.37 tff(equal_sets_type, type, (
% 0.20/0.37 equal_sets: ( $i * $i ) > $o)).
% 0.20/0.37 tff(1,assumption,(~subset(intersection(bs, as), intersection(as, bs))), introduced(assumption)).
% 0.20/0.37 tff(2,plain,
% 0.20/0.37 (^[Subset: $i, Superset: $i] : refl((subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset)) <=> (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset)))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(3,plain,
% 0.20/0.37 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset)) <=> ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[2])).
% 0.20/0.37 tff(4,plain,
% 0.20/0.37 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset)) <=> ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(5,axiom,(![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))), file('/export/starexec/sandbox/benchmark/Axioms/SET002-0.ax','subsets_axiom1')).
% 0.20/0.37 tff(6,plain,
% 0.20/0.37 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.20/0.37 tff(7,plain,(
% 0.20/0.37 ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.37 inference(skolemize,[status(sab)],[6])).
% 0.20/0.37 tff(8,plain,
% 0.20/0.37 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.20/0.37 tff(9,plain,
% 0.20/0.37 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(intersection(bs, as), intersection(as, bs)) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as)))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(intersection(bs, as), intersection(as, bs)) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as)))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(10,plain,
% 0.20/0.37 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(intersection(bs, as), intersection(as, bs)) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as)))),
% 0.20/0.37 inference(quant_inst,[status(thm)],[])).
% 0.20/0.37 tff(11,plain,
% 0.20/0.37 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(intersection(bs, as), intersection(as, bs)) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.20/0.37 tff(12,plain,
% 0.20/0.37 (member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[11, 8, 1])).
% 0.20/0.37 tff(13,plain,
% 0.20/0.37 (^[Xs: $i, X: $i, Ys: $i] : refl(((~member(X, intersection(Xs, Ys))) | member(X, Ys)) <=> ((~member(X, intersection(Xs, Ys))) | member(X, Ys)))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(14,plain,
% 0.20/0.37 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys)) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[13])).
% 0.20/0.37 tff(15,plain,
% 0.20/0.37 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys)) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(16,axiom,(![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))), file('/export/starexec/sandbox/benchmark/Axioms/SET002-0.ax','member_of_intersection_is_member_of_set2')).
% 0.20/0.38 tff(17,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.20/0.38 tff(18,plain,(
% 0.20/0.38 ![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))),
% 0.20/0.38 inference(skolemize,[status(sab)],[17])).
% 0.20/0.38 tff(19,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.20/0.38 tff(20,plain,
% 0.20/0.38 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))) | ((~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(21,plain,
% 0.20/0.38 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))) | ((~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(22,plain,
% 0.20/0.38 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as)),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.20/0.38 tff(23,plain,
% 0.20/0.38 (member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as)),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[22, 19, 12])).
% 0.20/0.38 tff(24,plain,
% 0.20/0.38 (^[Xs: $i, X: $i, Ys: $i] : refl(((~member(X, intersection(Xs, Ys))) | member(X, Xs)) <=> ((~member(X, intersection(Xs, Ys))) | member(X, Xs)))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(25,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs)) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[24])).
% 0.20/0.38 tff(26,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs)) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(27,axiom,(![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))), file('/export/starexec/sandbox/benchmark/Axioms/SET002-0.ax','member_of_intersection_is_member_of_set1')).
% 0.20/0.38 tff(28,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.38 tff(29,plain,(
% 0.20/0.38 ![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))),
% 0.20/0.38 inference(skolemize,[status(sab)],[28])).
% 0.20/0.38 tff(30,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.20/0.38 tff(31,plain,
% 0.20/0.38 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))) | ((~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(32,plain,
% 0.20/0.38 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))) | ((~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(33,plain,
% 0.20/0.38 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(bs, as))) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs)),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.20/0.38 tff(34,plain,
% 0.20/0.38 (member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs)),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[33, 30, 12])).
% 0.20/0.38 tff(35,plain,
% 0.20/0.38 (^[Subset: $i, Superset: $i] : refl(((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset)) <=> ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset)))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(36,plain,
% 0.20/0.38 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[35])).
% 0.20/0.38 tff(37,plain,
% 0.20/0.38 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset)) <=> ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(38,axiom,(![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))), file('/export/starexec/sandbox/benchmark/Axioms/SET002-0.ax','subsets_axiom2')).
% 0.20/0.38 tff(39,plain,
% 0.20/0.38 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.20/0.38 tff(40,plain,(
% 0.20/0.38 ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.20/0.38 inference(skolemize,[status(sab)],[39])).
% 0.20/0.38 tff(41,plain,
% 0.20/0.38 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[40, 36])).
% 0.20/0.38 tff(42,plain,
% 0.20/0.38 (((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(as, bs))) | subset(intersection(bs, as), intersection(as, bs)))) <=> ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(as, bs))) | subset(intersection(bs, as), intersection(as, bs)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(43,plain,
% 0.20/0.38 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(as, bs))) | subset(intersection(bs, as), intersection(as, bs)))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(44,plain,
% 0.20/0.38 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(as, bs))) | subset(intersection(bs, as), intersection(as, bs))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.20/0.38 tff(45,plain,
% 0.20/0.38 (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(as, bs))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[44, 41, 1])).
% 0.20/0.38 tff(46,plain,
% 0.20/0.38 (^[Xs: $i, X: $i, Ys: $i] : refl(((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys))) <=> ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(47,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys))) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[46])).
% 0.20/0.38 tff(48,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys))) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(49,plain,
% 0.20/0.38 (^[Xs: $i, X: $i, Ys: $i] : trans(monotonicity(rewrite(((~member(X, Xs)) | (~member(X, Ys))) <=> ((~member(X, Xs)) | (~member(X, Ys)))), ((((~member(X, Xs)) | (~member(X, Ys))) | member(X, intersection(Xs, Ys))) <=> (((~member(X, Xs)) | (~member(X, Ys))) | member(X, intersection(Xs, Ys))))), rewrite((((~member(X, Xs)) | (~member(X, Ys))) | member(X, intersection(Xs, Ys))) <=> ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))), ((((~member(X, Xs)) | (~member(X, Ys))) | member(X, intersection(Xs, Ys))) <=> ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(50,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : (((~member(X, Xs)) | (~member(X, Ys))) | member(X, intersection(Xs, Ys))) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[49])).
% 0.20/0.38 tff(51,axiom,(![Xs: $i, X: $i, Ys: $i] : (((~member(X, Xs)) | (~member(X, Ys))) | member(X, intersection(Xs, Ys)))), file('/export/starexec/sandbox/benchmark/Axioms/SET002-0.ax','member_of_both_is_member_of_intersection')).
% 0.20/0.38 tff(52,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.20/0.38 tff(53,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.20/0.38 tff(54,plain,(
% 0.20/0.38 ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))),
% 0.20/0.38 inference(skolemize,[status(sab)],[53])).
% 0.20/0.38 tff(55,plain,
% 0.20/0.38 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[54, 47])).
% 0.20/0.38 tff(56,plain,
% 0.20/0.38 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | ((~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as)) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs)) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(as, bs)))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as)) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs)) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(as, bs)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(57,plain,
% 0.20/0.38 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | ((~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as)) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs)) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(as, bs)))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(58,plain,
% 0.20/0.38 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as)) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs)) | member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), intersection(as, bs))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.20/0.38 tff(59,plain,
% 0.20/0.38 ((~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), as)) | (~member(member_of_1_not_of_2(intersection(bs, as), intersection(as, bs)), bs))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[58, 55, 45])).
% 0.20/0.38 tff(60,plain,
% 0.20/0.38 ($false),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[59, 34, 23])).
% 0.20/0.38 tff(61,plain,(subset(intersection(bs, as), intersection(as, bs))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.38 tff(62,plain,
% 0.20/0.38 ((~equal_sets(intersection(as, bs), intersection(bs, as))) <=> (~equal_sets(intersection(as, bs), intersection(bs, as)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(63,axiom,(~equal_sets(intersection(as, bs), intersection(bs, as))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_commutativity')).
% 0.20/0.38 tff(64,plain,
% 0.20/0.38 (~equal_sets(intersection(as, bs), intersection(bs, as))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[63, 62])).
% 0.20/0.38 tff(65,plain,
% 0.20/0.38 (^[Set1: $i, Set2: $i] : refl((equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2))) <=> (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(66,plain,
% 0.20/0.38 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2))) <=> ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[65])).
% 0.20/0.38 tff(67,plain,
% 0.20/0.38 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2))) <=> ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(68,plain,
% 0.20/0.38 (^[Set1: $i, Set2: $i] : trans(monotonicity(rewrite(((~subset(Set1, Set2)) | (~subset(Set2, Set1))) <=> ((~subset(Set2, Set1)) | (~subset(Set1, Set2)))), ((((~subset(Set1, Set2)) | (~subset(Set2, Set1))) | equal_sets(Set2, Set1)) <=> (((~subset(Set2, Set1)) | (~subset(Set1, Set2))) | equal_sets(Set2, Set1)))), rewrite((((~subset(Set2, Set1)) | (~subset(Set1, Set2))) | equal_sets(Set2, Set1)) <=> (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))), ((((~subset(Set1, Set2)) | (~subset(Set2, Set1))) | equal_sets(Set2, Set1)) <=> (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(69,plain,
% 0.20/0.38 (![Set1: $i, Set2: $i] : (((~subset(Set1, Set2)) | (~subset(Set2, Set1))) | equal_sets(Set2, Set1)) <=> ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[68])).
% 0.20/0.38 tff(70,axiom,(![Set1: $i, Set2: $i] : (((~subset(Set1, Set2)) | (~subset(Set2, Set1))) | equal_sets(Set2, Set1))), file('/export/starexec/sandbox/benchmark/Axioms/SET002-0.ax','subsets_are_set_equal_sets')).
% 0.20/0.38 tff(71,plain,
% 0.20/0.38 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.20/0.38 tff(72,plain,
% 0.20/0.38 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[71, 67])).
% 0.20/0.38 tff(73,plain,(
% 0.20/0.38 ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.38 inference(skolemize,[status(sab)],[72])).
% 0.20/0.38 tff(74,plain,
% 0.20/0.38 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[73, 66])).
% 0.20/0.38 tff(75,plain,
% 0.20/0.38 (((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(intersection(as, bs), intersection(bs, as)) | (~subset(intersection(as, bs), intersection(bs, as))) | (~subset(intersection(bs, as), intersection(as, bs))))) <=> ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(intersection(as, bs), intersection(bs, as)) | (~subset(intersection(as, bs), intersection(bs, as))) | (~subset(intersection(bs, as), intersection(as, bs))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(76,plain,
% 0.20/0.39 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(intersection(as, bs), intersection(bs, as)) | (~subset(intersection(as, bs), intersection(bs, as))) | (~subset(intersection(bs, as), intersection(as, bs))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(77,plain,
% 0.20/0.39 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(intersection(as, bs), intersection(bs, as)) | (~subset(intersection(as, bs), intersection(bs, as))) | (~subset(intersection(bs, as), intersection(as, bs)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.20/0.39 tff(78,plain,
% 0.20/0.39 ((~subset(intersection(as, bs), intersection(bs, as))) | (~subset(intersection(bs, as), intersection(as, bs)))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[77, 74, 64])).
% 0.20/0.39 tff(79,plain,
% 0.20/0.39 (~subset(intersection(as, bs), intersection(bs, as))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[78, 61])).
% 0.20/0.39 tff(80,plain,
% 0.20/0.39 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(intersection(as, bs), intersection(bs, as)) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs)))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(intersection(as, bs), intersection(bs, as)) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(81,plain,
% 0.20/0.39 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(intersection(as, bs), intersection(bs, as)) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(82,plain,
% 0.20/0.39 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(intersection(as, bs), intersection(bs, as)) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.20/0.39 tff(83,plain,
% 0.20/0.39 (member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[82, 8, 79])).
% 0.20/0.39 tff(84,plain,
% 0.20/0.39 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))) | ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(85,plain,
% 0.20/0.39 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))) | ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(86,plain,
% 0.20/0.39 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Ys))) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.20/0.39 tff(87,plain,
% 0.20/0.39 (member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[86, 19, 83])).
% 0.20/0.39 tff(88,plain,
% 0.20/0.39 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))) | ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(89,plain,
% 0.20/0.39 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))) | ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(90,plain,
% 0.20/0.39 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, intersection(Xs, Ys))) | member(X, Xs))) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(as, bs))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.20/0.39 tff(91,plain,
% 0.20/0.39 (member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[90, 30, 83])).
% 0.20/0.39 tff(92,plain,
% 0.20/0.39 (((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as))) | subset(intersection(as, bs), intersection(bs, as)))) <=> ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as))) | subset(intersection(as, bs), intersection(bs, as)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(93,plain,
% 0.20/0.39 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as))) | subset(intersection(as, bs), intersection(bs, as)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(94,plain,
% 0.20/0.39 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as))) | subset(intersection(as, bs), intersection(bs, as))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.20/0.39 tff(95,plain,
% 0.20/0.39 (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[94, 41, 79])).
% 0.20/0.39 tff(96,plain,
% 0.20/0.39 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | (member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(97,plain,
% 0.20/0.39 (((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as))) <=> (member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(98,plain,
% 0.20/0.39 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as)))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | (member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as))))),
% 0.20/0.39 inference(monotonicity,[status(thm)],[97])).
% 0.20/0.39 tff(99,plain,
% 0.20/0.39 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as)))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)))),
% 0.20/0.39 inference(transitivity,[status(thm)],[98, 96])).
% 0.20/0.39 tff(100,plain,
% 0.20/0.39 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as)) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(101,plain,
% 0.20/0.39 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | (~member(X, Ys)) | member(X, intersection(Xs, Ys)))) | member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), intersection(bs, as)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.20/0.39 tff(102,plain,
% 0.20/0.39 ((~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), bs)) | (~member(member_of_1_not_of_2(intersection(as, bs), intersection(bs, as)), as))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[101, 55, 95])).
% 0.20/0.39 tff(103,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[102, 91, 87])).
% 0.20/0.39 % SZS output end Proof
% 0.20/0.40 % E exiting
%------------------------------------------------------------------------------