TSTP Solution File: SET015-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET015-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n004.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 20 05:04:47 EDT 2022
% Result : Unsatisfiable 0.20s 0.41s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : SET015-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.04/0.14 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.35 % Computer : n004.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat Sep 3 01:03:19 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.14/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.35 Usage: tptp [options] [-file:]file
% 0.14/0.35 -h, -? prints this message.
% 0.14/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.35 -m, -model generate model.
% 0.14/0.35 -p, -proof generate proof.
% 0.14/0.35 -c, -core generate unsat core of named formulas.
% 0.14/0.35 -st, -statistics display statistics.
% 0.14/0.35 -t:timeout set timeout (in second).
% 0.14/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.35 -<param>:<value> configuration parameter and value.
% 0.14/0.35 -o:<output-file> file to place output in.
% 0.20/0.41 % SZS status Unsatisfiable
% 0.20/0.41 % SZS output start Proof
% 0.20/0.41 tff(member_type, type, (
% 0.20/0.41 member: ( $i * $i ) > $o)).
% 0.20/0.41 tff(as_type, type, (
% 0.20/0.41 as: $i)).
% 0.20/0.41 tff(member_of_1_not_of_2_type, type, (
% 0.20/0.41 member_of_1_not_of_2: ( $i * $i ) > $i)).
% 0.20/0.41 tff(union_type, type, (
% 0.20/0.41 union: ( $i * $i ) > $i)).
% 0.20/0.41 tff(bs_type, type, (
% 0.20/0.41 bs: $i)).
% 0.20/0.41 tff(subset_type, type, (
% 0.20/0.41 subset: ( $i * $i ) > $o)).
% 0.20/0.41 tff(equal_sets_type, type, (
% 0.20/0.41 equal_sets: ( $i * $i ) > $o)).
% 0.20/0.41 tff(1,assumption,(~subset(union(bs, as), union(as, bs))), introduced(assumption)).
% 0.20/0.41 tff(2,plain,
% 0.20/0.41 (^[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)))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(3,plain,
% 0.20/0.41 (![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))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[2])).
% 0.20/0.41 tff(4,plain,
% 0.20/0.41 (![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))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(5,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')).
% 0.20/0.41 tff(6,plain,
% 0.20/0.41 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.20/0.41 tff(7,plain,(
% 0.20/0.41 ![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.20/0.41 inference(skolemize,[status(sab)],[6])).
% 0.20/0.41 tff(8,plain,
% 0.20/0.41 (![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.20/0.41 tff(9,plain,
% 0.20/0.41 (((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs))) | subset(union(bs, as), union(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(union(bs, as), union(as, bs)), union(as, bs))) | subset(union(bs, as), union(as, bs)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(10,plain,
% 0.20/0.41 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs))) | subset(union(bs, as), union(as, bs)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(11,plain,
% 0.20/0.41 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs))) | subset(union(bs, as), union(as, bs))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[10, 9])).
% 0.20/0.41 tff(12,plain,
% 0.20/0.41 (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[11, 8, 1])).
% 0.20/0.41 tff(13,plain,
% 0.20/0.41 (^[Xs: $i, X: $i, Ys: $i] : refl(((~member(X, Ys)) | member(X, union(Xs, Ys))) <=> ((~member(X, Ys)) | member(X, union(Xs, Ys))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(14,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys))) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[13])).
% 0.20/0.41 tff(15,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys))) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(16,axiom,(![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))), file('/export/starexec/sandbox/benchmark/Axioms/SET002-0.ax','member_of_set2_is_member_of_union')).
% 0.20/0.41 tff(17,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.20/0.41 tff(18,plain,(
% 0.20/0.41 ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[17])).
% 0.20/0.41 tff(19,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.20/0.41 tff(20,plain,
% 0.20/0.41 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))) | ((~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), bs)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs)))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))) | (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), bs)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(21,plain,
% 0.20/0.41 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))) | ((~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), bs)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(22,plain,
% 0.20/0.41 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))) | (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), bs)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.20/0.41 tff(23,plain,
% 0.20/0.41 (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), bs)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[22, 19, 12])).
% 0.20/0.41 tff(24,plain,
% 0.20/0.41 (^[Xs: $i, X: $i, Ys: $i] : refl(((~member(X, Xs)) | member(X, union(Xs, Ys))) <=> ((~member(X, Xs)) | member(X, union(Xs, Ys))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(25,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys))) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[24])).
% 0.20/0.41 tff(26,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys))) <=> ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(27,axiom,(![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))), file('/export/starexec/sandbox/benchmark/Axioms/SET002-0.ax','member_of_set1_is_member_of_union')).
% 0.20/0.41 tff(28,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.41 tff(29,plain,(
% 0.20/0.41 ![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[28])).
% 0.20/0.41 tff(30,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.20/0.41 tff(31,plain,
% 0.20/0.41 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))) | ((~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), as)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs)))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))) | (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), as)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(32,plain,
% 0.20/0.41 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))) | ((~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), as)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(33,plain,
% 0.20/0.41 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))) | (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), as)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(as, bs))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.20/0.41 tff(34,plain,
% 0.20/0.41 (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), as)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[33, 30, 12])).
% 0.20/0.41 tff(35,plain,
% 0.20/0.41 (^[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)))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(36,plain,
% 0.20/0.41 (![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))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[35])).
% 0.20/0.41 tff(37,plain,
% 0.20/0.41 (![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))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(38,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')).
% 0.20/0.41 tff(39,plain,
% 0.20/0.41 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.20/0.41 tff(40,plain,(
% 0.20/0.41 ![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.41 inference(skolemize,[status(sab)],[39])).
% 0.20/0.41 tff(41,plain,
% 0.20/0.41 (![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[40, 36])).
% 0.20/0.41 tff(42,plain,
% 0.20/0.41 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(union(bs, as), union(as, bs)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(bs, as)))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(union(bs, as), union(as, bs)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(bs, as)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(43,plain,
% 0.20/0.41 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(union(bs, as), union(as, bs)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(bs, as)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(44,plain,
% 0.20/0.41 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(union(bs, as), union(as, bs)) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(bs, as))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.20/0.41 tff(45,plain,
% 0.20/0.41 (member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(bs, as))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[44, 41, 1])).
% 0.20/0.41 tff(46,plain,
% 0.20/0.41 (^[Xs: $i, X: $i, Ys: $i] : refl((member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys)))) <=> (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(47,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys)))) <=> ![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[46])).
% 0.20/0.41 tff(48,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys)))) <=> ![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(49,plain,
% 0.20/0.41 (^[Xs: $i, X: $i, Ys: $i] : rewrite((((~member(X, union(Xs, Ys))) | member(X, Xs)) | member(X, Ys)) <=> (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(50,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : (((~member(X, union(Xs, Ys))) | member(X, Xs)) | member(X, Ys)) <=> ![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[49])).
% 0.20/0.41 tff(51,axiom,(![Xs: $i, X: $i, Ys: $i] : (((~member(X, union(Xs, Ys))) | member(X, Xs)) | member(X, Ys))), file('/export/starexec/sandbox/benchmark/Axioms/SET002-0.ax','member_of_union_is_member_of_one_set')).
% 0.20/0.41 tff(52,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.20/0.41 tff(53,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.20/0.41 tff(54,plain,(
% 0.20/0.41 ![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[53])).
% 0.20/0.41 tff(55,plain,
% 0.20/0.41 (![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[54, 47])).
% 0.20/0.41 tff(56,plain,
% 0.20/0.41 (((~![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))) | (member(member_of_1_not_of_2(union(bs, as), union(as, bs)), as) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), bs) | (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(bs, as))))) <=> ((~![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), as) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), bs) | (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(bs, as))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(57,plain,
% 0.20/0.41 ((~![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))) | (member(member_of_1_not_of_2(union(bs, as), union(as, bs)), as) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), bs) | (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(bs, as))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(58,plain,
% 0.20/0.41 ((~![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), as) | member(member_of_1_not_of_2(union(bs, as), union(as, bs)), bs) | (~member(member_of_1_not_of_2(union(bs, as), union(as, bs)), union(bs, as)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.20/0.41 tff(59,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[58, 55, 45, 34, 23])).
% 0.20/0.41 tff(60,plain,(subset(union(bs, as), union(as, bs))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 ((~equal_sets(union(as, bs), union(bs, as))) <=> (~equal_sets(union(as, bs), union(bs, as)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(62,axiom,(~equal_sets(union(as, bs), union(bs, as))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_commutativity')).
% 0.20/0.41 tff(63,plain,
% 0.20/0.41 (~equal_sets(union(as, bs), union(bs, as))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.20/0.41 tff(64,plain,
% 0.20/0.41 (^[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))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 (![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)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[64])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 (![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)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(67,plain,
% 0.20/0.41 (^[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)))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(68,plain,
% 0.20/0.42 (![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)))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[67])).
% 0.20/0.42 tff(69,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')).
% 0.20/0.42 tff(70,plain,
% 0.20/0.42 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.20/0.42 tff(71,plain,
% 0.20/0.42 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[70, 66])).
% 0.20/0.42 tff(72,plain,(
% 0.20/0.42 ![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.42 inference(skolemize,[status(sab)],[71])).
% 0.20/0.42 tff(73,plain,
% 0.20/0.42 (![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[72, 65])).
% 0.20/0.42 tff(74,plain,
% 0.20/0.42 (((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(union(as, bs), union(bs, as)) | (~subset(union(as, bs), union(bs, as))) | (~subset(union(bs, as), union(as, bs))))) <=> ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(union(as, bs), union(bs, as)) | (~subset(union(as, bs), union(bs, as))) | (~subset(union(bs, as), union(as, bs))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(75,plain,
% 0.20/0.42 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | (equal_sets(union(as, bs), union(bs, as)) | (~subset(union(as, bs), union(bs, as))) | (~subset(union(bs, as), union(as, bs))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(76,plain,
% 0.20/0.42 ((~![Set1: $i, Set2: $i] : (equal_sets(Set2, Set1) | (~subset(Set2, Set1)) | (~subset(Set1, Set2)))) | equal_sets(union(as, bs), union(bs, as)) | (~subset(union(as, bs), union(bs, as))) | (~subset(union(bs, as), union(as, bs)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.20/0.42 tff(77,plain,
% 0.20/0.42 ((~subset(union(as, bs), union(bs, as))) | (~subset(union(bs, as), union(as, bs)))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[76, 73, 63])).
% 0.20/0.42 tff(78,plain,
% 0.20/0.42 (~subset(union(as, bs), union(bs, as))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[77, 60])).
% 0.20/0.42 tff(79,plain,
% 0.20/0.42 (((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as))) | subset(union(as, bs), union(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(union(as, bs), union(bs, as)), union(bs, as))) | subset(union(as, bs), union(bs, as)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(80,plain,
% 0.20/0.42 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | ((~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as))) | subset(union(as, bs), union(bs, as)))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(81,plain,
% 0.20/0.42 ((~![Subset: $i, Superset: $i] : ((~member(member_of_1_not_of_2(Subset, Superset), Superset)) | subset(Subset, Superset))) | (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as))) | subset(union(as, bs), union(bs, as))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.20/0.42 tff(82,plain,
% 0.20/0.42 (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[81, 8, 78])).
% 0.20/0.42 tff(83,plain,
% 0.20/0.42 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))) | ((~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), as)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as)))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))) | (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), as)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(84,plain,
% 0.20/0.42 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))) | ((~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), as)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as)))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(85,plain,
% 0.20/0.42 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Ys)) | member(X, union(Xs, Ys)))) | (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), as)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.20/0.42 tff(86,plain,
% 0.20/0.42 (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), as)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[85, 19, 82])).
% 0.20/0.42 tff(87,plain,
% 0.20/0.42 (((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))) | ((~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), bs)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as)))) <=> ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))) | (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), bs)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(88,plain,
% 0.20/0.42 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))) | ((~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), bs)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as)))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(89,plain,
% 0.20/0.42 ((~![Xs: $i, X: $i, Ys: $i] : ((~member(X, Xs)) | member(X, union(Xs, Ys)))) | (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), bs)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(bs, as))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.20/0.42 tff(90,plain,
% 0.20/0.42 (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), bs)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[89, 30, 82])).
% 0.20/0.42 tff(91,plain,
% 0.20/0.42 (((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(union(as, bs), union(bs, as)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(as, bs)))) <=> ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(union(as, bs), union(bs, as)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(as, bs)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(92,plain,
% 0.20/0.42 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | (subset(union(as, bs), union(bs, as)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(as, bs)))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(93,plain,
% 0.20/0.42 ((~![Subset: $i, Superset: $i] : (subset(Subset, Superset) | member(member_of_1_not_of_2(Subset, Superset), Subset))) | subset(union(as, bs), union(bs, as)) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(as, bs))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.20/0.42 tff(94,plain,
% 0.20/0.42 (member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(as, bs))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[93, 41, 78])).
% 0.20/0.43 tff(95,plain,
% 0.20/0.43 (((~![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))) | (member(member_of_1_not_of_2(union(as, bs), union(bs, as)), bs) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), as) | (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(as, bs))))) <=> ((~![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), bs) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), as) | (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(as, bs))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(96,plain,
% 0.20/0.43 ((~![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))) | (member(member_of_1_not_of_2(union(as, bs), union(bs, as)), bs) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), as) | (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(as, bs))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(97,plain,
% 0.20/0.43 ((~![Xs: $i, X: $i, Ys: $i] : (member(X, Ys) | member(X, Xs) | (~member(X, union(Xs, Ys))))) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), bs) | member(member_of_1_not_of_2(union(as, bs), union(bs, as)), as) | (~member(member_of_1_not_of_2(union(as, bs), union(bs, as)), union(as, bs)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.20/0.43 tff(98,plain,
% 0.20/0.43 ($false),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[97, 55, 94, 90, 86])).
% 0.20/0.43 % SZS output end Proof
%------------------------------------------------------------------------------