%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET011-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n013.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 05:24:45 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET011-1 : TPTP v9.0.0. Released v1.0.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n013.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
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 04:01:08 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.19/0.38 % SZS status Unsatisfiable
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 tff(member_type, type, (
% 0.19/0.38 member: ( $i * $i ) > $o)).
% 0.19/0.38 tff(aDb_type, type, (
% 0.19/0.38 aDb: $i)).
% 0.19/0.38 tff(h_type, type, (
% 0.19/0.38 h: ( $i * $i * $i ) > $i)).
% 0.19/0.38 tff(aD_aDb_type, type, (
% 0.19/0.38 aD_aDb: $i)).
% 0.19/0.38 tff(b_type, type, (
% 0.19/0.38 b: $i)).
% 0.19/0.38 tff(a_type, type, (
% 0.19/0.38 a: $i)).
% 0.19/0.38 tff(intersection_type, type, (
% 0.19/0.38 intersection: ( $i * $i * $i ) > $o)).
% 0.19/0.38 tff(difference_type, type, (
% 0.19/0.38 difference: ( $i * $i * $i ) > $o)).
% 0.19/0.38 tff(1,assumption,(~member(h(a, b, aD_aDb), aD_aDb)), introduced(assumption)).
% 0.19/0.38 tff(2,assumption,(~member(h(a, b, aD_aDb), a)), introduced(assumption)).
% 0.19/0.38 tff(3,plain,
% 0.19/0.38 ((~intersection(a, b, aD_aDb)) <=> (~intersection(a, b, aD_aDb))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(4,axiom,(~intersection(a, b, aD_aDb)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_a_intersection_b_is_aD_aDb')).
% 0.19/0.38 tff(5,plain,
% 0.19/0.38 (~intersection(a, b, aD_aDb)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.38 tff(6,plain,
% 0.19/0.38 (^[Set1: $i, Set2: $i, Intersection: $i] : refl((member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) <=> (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(7,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[6])).
% 0.19/0.38 tff(8,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(9,plain,
% 0.19/0.38 (^[Set1: $i, Set2: $i, Intersection: $i] : rewrite(((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set1)) <=> (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(10,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set1)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[9])).
% 0.19/0.38 tff(11,axiom,(![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set1))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-2.ax','intersection_axiom1')).
% 0.19/0.38 tff(12,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.19/0.38 tff(13,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.19/0.38 tff(14,plain,(
% 0.19/0.38 ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.38 inference(skolemize,[status(sab)],[13])).
% 0.19/0.38 tff(15,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.19/0.38 tff(16,plain,
% 0.19/0.38 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | (member(h(a, b, aD_aDb), a) | member(h(a, b, aD_aDb), aD_aDb) | intersection(a, b, aD_aDb))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | member(h(a, b, aD_aDb), a) | member(h(a, b, aD_aDb), aD_aDb) | intersection(a, b, aD_aDb))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(17,plain,
% 0.19/0.38 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | (member(h(a, b, aD_aDb), a) | member(h(a, b, aD_aDb), aD_aDb) | intersection(a, b, aD_aDb))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(18,plain,
% 0.19/0.38 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | member(h(a, b, aD_aDb), a) | member(h(a, b, aD_aDb), aD_aDb) | intersection(a, b, aD_aDb)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.19/0.38 tff(19,plain,
% 0.19/0.38 (member(h(a, b, aD_aDb), a) | member(h(a, b, aD_aDb), aD_aDb)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[18, 15, 5])).
% 0.19/0.38 tff(20,plain,
% 0.19/0.38 (member(h(a, b, aD_aDb), aD_aDb)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[19, 2])).
% 0.19/0.38 tff(21,plain,
% 0.19/0.38 (difference(a, aDb, aD_aDb) <=> difference(a, aDb, aD_aDb)),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(22,axiom,(difference(a, aDb, aD_aDb)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_minus_aDb')).
% 0.19/0.38 tff(23,plain,
% 0.19/0.38 (difference(a, aDb, aD_aDb)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.19/0.38 tff(24,plain,
% 0.19/0.38 (^[Set1: $i, Set2: $i, Difference: $i, Element: $i] : refl((member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference))) <=> (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(25,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference))) <=> ![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[24])).
% 0.19/0.38 tff(26,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference))) <=> ![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 (^[Set1: $i, Set2: $i, Difference: $i, Element: $i] : trans(monotonicity(rewrite(((~difference(Set1, Set2, Difference)) | (~member(Element, Difference))) <=> ((~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))), ((((~difference(Set1, Set2, Difference)) | (~member(Element, Difference))) | member(Element, Set1)) <=> (((~member(Element, Difference)) | (~difference(Set1, Set2, Difference))) | member(Element, Set1)))), rewrite((((~member(Element, Difference)) | (~difference(Set1, Set2, Difference))) | member(Element, Set1)) <=> (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))), ((((~difference(Set1, Set2, Difference)) | (~member(Element, Difference))) | member(Element, Set1)) <=> (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(28,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (((~difference(Set1, Set2, Difference)) | (~member(Element, Difference))) | member(Element, Set1)) <=> ![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[27])).
% 0.19/0.38 tff(29,axiom,(![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (((~difference(Set1, Set2, Difference)) | (~member(Element, Difference))) | member(Element, Set1))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-3.ax','member_of_difference')).
% 0.19/0.38 tff(30,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.19/0.38 tff(31,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.19/0.38 tff(32,plain,(
% 0.19/0.38 ![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[31])).
% 0.19/0.38 tff(33,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[32, 25])).
% 0.19/0.38 tff(34,plain,
% 0.19/0.38 (((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))) | (member(h(a, b, aD_aDb), a) | (~member(h(a, b, aD_aDb), aD_aDb)) | (~difference(a, aDb, aD_aDb)))) <=> ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))) | member(h(a, b, aD_aDb), a) | (~member(h(a, b, aD_aDb), aD_aDb)) | (~difference(a, aDb, aD_aDb)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(35,plain,
% 0.19/0.38 ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))) | (member(h(a, b, aD_aDb), a) | (~member(h(a, b, aD_aDb), aD_aDb)) | (~difference(a, aDb, aD_aDb)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(36,plain,
% 0.19/0.38 ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Set1) | (~member(Element, Difference)) | (~difference(Set1, Set2, Difference)))) | member(h(a, b, aD_aDb), a) | (~member(h(a, b, aD_aDb), aD_aDb)) | (~difference(a, aDb, aD_aDb))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.19/0.38 tff(37,plain,
% 0.19/0.38 ($false),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[36, 33, 23, 2, 20])).
% 0.19/0.38 tff(38,plain,(member(h(a, b, aD_aDb), a)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.38 tff(39,plain,
% 0.19/0.38 (^[Set1: $i, Set2: $i, Difference: $i, Element: $i] : refl((member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1))) <=> (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(40,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1))) <=> ![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[39])).
% 0.19/0.38 tff(41,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1))) <=> ![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(42,plain,
% 0.19/0.38 (^[Set1: $i, Set2: $i, Difference: $i, Element: $i] : trans(monotonicity(rewrite((((~member(Element, Set1)) | (~difference(Set1, Set2, Difference))) | member(Element, Difference)) <=> (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | (~member(Element, Set1)))), (((((~member(Element, Set1)) | (~difference(Set1, Set2, Difference))) | member(Element, Difference)) | member(Element, Set2)) <=> ((member(Element, Difference) | (~difference(Set1, Set2, Difference)) | (~member(Element, Set1))) | member(Element, Set2)))), rewrite(((member(Element, Difference) | (~difference(Set1, Set2, Difference)) | (~member(Element, Set1))) | member(Element, Set2)) <=> (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))), (((((~member(Element, Set1)) | (~difference(Set1, Set2, Difference))) | member(Element, Difference)) | member(Element, Set2)) <=> (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(43,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : ((((~member(Element, Set1)) | (~difference(Set1, Set2, Difference))) | member(Element, Difference)) | member(Element, Set2)) <=> ![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[42])).
% 0.19/0.38 tff(44,axiom,(![Set1: $i, Set2: $i, Difference: $i, Element: $i] : ((((~member(Element, Set1)) | (~difference(Set1, Set2, Difference))) | member(Element, Difference)) | member(Element, Set2))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-3.ax','member_of_difference_or_set2')).
% 0.19/0.38 tff(45,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.19/0.38 tff(46,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.19/0.38 tff(47,plain,(
% 0.19/0.38 ![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[46])).
% 0.19/0.38 tff(48,plain,
% 0.19/0.38 (![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[47, 40])).
% 0.19/0.38 tff(49,plain,
% 0.19/0.38 (((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | ((~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb))) <=> ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(50,plain,
% 0.19/0.38 ((member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb) | (~member(h(a, b, aD_aDb), a))) <=> ((~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(51,plain,
% 0.19/0.38 (((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb) | (~member(h(a, b, aD_aDb), a)))) <=> ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | ((~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb)))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[50])).
% 0.19/0.39 tff(52,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb) | (~member(h(a, b, aD_aDb), a)))) <=> ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb))),
% 0.19/0.39 inference(transitivity,[status(thm)],[51, 49])).
% 0.19/0.39 tff(53,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb) | (~member(h(a, b, aD_aDb), a)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(54,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), aD_aDb) | (~difference(a, aDb, aD_aDb)) | member(h(a, b, aD_aDb), aDb)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.19/0.39 tff(55,plain,
% 0.19/0.39 (member(h(a, b, aD_aDb), aDb)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[54, 48, 23, 38, 1])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, Intersection: $i] : refl((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)) <=> (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(57,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[56])).
% 0.19/0.39 tff(58,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(59,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, Intersection: $i] : rewrite(((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set2)) <=> (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(60,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set2)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[59])).
% 0.19/0.39 tff(61,axiom,(![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set2))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-2.ax','intersection_axiom2')).
% 0.19/0.39 tff(62,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.19/0.39 tff(63,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[62, 58])).
% 0.19/0.39 tff(64,plain,(
% 0.19/0.39 ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.39 inference(skolemize,[status(sab)],[63])).
% 0.19/0.39 tff(65,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[64, 57])).
% 0.19/0.39 tff(66,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aD_aDb))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aD_aDb))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(67,plain,
% 0.19/0.39 ((member(h(a, b, aD_aDb), aD_aDb) | intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b)) <=> (intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aD_aDb))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(68,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (member(h(a, b, aD_aDb), aD_aDb) | intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aD_aDb)))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[67])).
% 0.19/0.39 tff(69,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (member(h(a, b, aD_aDb), aD_aDb) | intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aD_aDb))),
% 0.19/0.39 inference(transitivity,[status(thm)],[68, 66])).
% 0.19/0.39 tff(70,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (member(h(a, b, aD_aDb), aD_aDb) | intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(71,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | intersection(a, b, aD_aDb) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aD_aDb)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 (member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aD_aDb)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[71, 65, 5])).
% 0.19/0.39 tff(73,plain,
% 0.19/0.39 (member(h(a, b, aD_aDb), b)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[72, 1])).
% 0.19/0.39 tff(74,plain,
% 0.19/0.39 (difference(a, b, aDb) <=> difference(a, b, aDb)),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(75,axiom,(difference(a, b, aDb)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_minus_b')).
% 0.19/0.39 tff(76,plain,
% 0.19/0.39 (difference(a, b, aDb)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.19/0.39 tff(77,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, A_set: $i, Element: $i] : refl(((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1))) <=> ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(78,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1))) <=> ![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[77])).
% 0.19/0.39 tff(79,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1))) <=> ![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(80,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, A_set: $i, Element: $i] : trans(monotonicity(rewrite(((~member(Element, Set1)) | (~member(Element, Set2))) <=> ((~member(Element, Set2)) | (~member(Element, Set1)))), ((((~member(Element, Set1)) | (~member(Element, Set2))) | (~difference(A_set, Set1, Set2))) <=> (((~member(Element, Set2)) | (~member(Element, Set1))) | (~difference(A_set, Set1, Set2))))), rewrite((((~member(Element, Set2)) | (~member(Element, Set1))) | (~difference(A_set, Set1, Set2))) <=> ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))), ((((~member(Element, Set1)) | (~member(Element, Set2))) | (~difference(A_set, Set1, Set2))) <=> ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : (((~member(Element, Set1)) | (~member(Element, Set2))) | (~difference(A_set, Set1, Set2))) <=> ![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[80])).
% 0.19/0.39 tff(82,axiom,(![Set1: $i, Set2: $i, A_set: $i, Element: $i] : (((~member(Element, Set1)) | (~member(Element, Set2))) | (~difference(A_set, Set1, Set2)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-3.ax','not_member_of_difference')).
% 0.19/0.39 tff(83,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.19/0.39 tff(84,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[83, 79])).
% 0.19/0.39 tff(85,plain,(
% 0.19/0.39 ![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[84])).
% 0.19/0.39 tff(86,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[85, 78])).
% 0.19/0.39 tff(87,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | ((~difference(a, b, aDb)) | (~member(h(a, b, aD_aDb), aDb)) | (~member(h(a, b, aD_aDb), b)))) <=> ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | (~difference(a, b, aDb)) | (~member(h(a, b, aD_aDb), aDb)) | (~member(h(a, b, aD_aDb), b)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(88,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | ((~difference(a, b, aDb)) | (~member(h(a, b, aD_aDb), aDb)) | (~member(h(a, b, aD_aDb), b)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(89,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | (~difference(a, b, aDb)) | (~member(h(a, b, aD_aDb), aDb)) | (~member(h(a, b, aD_aDb), b))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.19/0.39 tff(90,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[89, 86, 76, 73, 55])).
% 0.19/0.39 tff(91,plain,(member(h(a, b, aD_aDb), aD_aDb)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(92,plain,
% 0.19/0.39 (((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | ((~difference(a, aDb, aD_aDb)) | (~member(h(a, b, aD_aDb), aD_aDb)) | (~member(h(a, b, aD_aDb), aDb)))) <=> ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | (~difference(a, aDb, aD_aDb)) | (~member(h(a, b, aD_aDb), aD_aDb)) | (~member(h(a, b, aD_aDb), aDb)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(93,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | ((~difference(a, aDb, aD_aDb)) | (~member(h(a, b, aD_aDb), aD_aDb)) | (~member(h(a, b, aD_aDb), aDb)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(94,plain,
% 0.19/0.39 ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | (~difference(a, aDb, aD_aDb)) | (~member(h(a, b, aD_aDb), aD_aDb)) | (~member(h(a, b, aD_aDb), aDb))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.19/0.39 tff(95,plain,
% 0.19/0.39 (~member(h(a, b, aD_aDb), aDb)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[94, 86, 23, 91])).
% 0.19/0.39 tff(96,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, Intersection: $i] : refl((intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) <=> (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(97,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[96])).
% 0.19/0.39 tff(98,plain,
% 0.19/0.39 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(99,plain,
% 0.19/0.39 (^[Set1: $i, Set2: $i, Intersection: $i] : trans(monotonicity(rewrite((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) <=> ((~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))), (((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) | intersection(Set1, Set2, Intersection)) <=> (((~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) | intersection(Set1, Set2, Intersection)))), rewrite((((~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection))) | intersection(Set1, Set2, Intersection)) <=> (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))), (((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) | intersection(Set1, Set2, Intersection)) <=> (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(100,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Intersection: $i] : ((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) | intersection(Set1, Set2, Intersection)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[99])).
% 0.19/0.40 tff(101,axiom,(![Set1: $i, Set2: $i, Intersection: $i] : ((((~member(h(Set1, Set2, Intersection), Intersection)) | (~member(h(Set1, Set2, Intersection), Set2))) | (~member(h(Set1, Set2, Intersection), Set1))) | intersection(Set1, Set2, Intersection))), file('/export/starexec/sandbox2/benchmark/Axioms/SET001-2.ax','intersection_axiom3')).
% 0.19/0.40 tff(102,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.19/0.40 tff(103,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[102, 98])).
% 0.19/0.40 tff(104,plain,(
% 0.19/0.40 ![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[103])).
% 0.19/0.40 tff(105,plain,
% 0.19/0.40 (![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[104, 97])).
% 0.19/0.40 tff(106,plain,
% 0.19/0.40 (((~![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))) | (intersection(a, b, aD_aDb) | (~member(h(a, b, aD_aDb), a)) | (~member(h(a, b, aD_aDb), b)) | (~member(h(a, b, aD_aDb), aD_aDb)))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))) | intersection(a, b, aD_aDb) | (~member(h(a, b, aD_aDb), a)) | (~member(h(a, b, aD_aDb), b)) | (~member(h(a, b, aD_aDb), aD_aDb)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(107,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))) | (intersection(a, b, aD_aDb) | (~member(h(a, b, aD_aDb), a)) | (~member(h(a, b, aD_aDb), b)) | (~member(h(a, b, aD_aDb), aD_aDb)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(108,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Intersection: $i] : (intersection(Set1, Set2, Intersection) | (~member(h(Set1, Set2, Intersection), Set1)) | (~member(h(Set1, Set2, Intersection), Set2)) | (~member(h(Set1, Set2, Intersection), Intersection)))) | intersection(a, b, aD_aDb) | (~member(h(a, b, aD_aDb), a)) | (~member(h(a, b, aD_aDb), b)) | (~member(h(a, b, aD_aDb), aD_aDb))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.19/0.40 tff(109,plain,
% 0.19/0.40 ((~member(h(a, b, aD_aDb), a)) | (~member(h(a, b, aD_aDb), b)) | (~member(h(a, b, aD_aDb), aD_aDb))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[108, 105, 5])).
% 0.19/0.40 tff(110,plain,
% 0.19/0.40 (~member(h(a, b, aD_aDb), b)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[109, 91, 38])).
% 0.19/0.40 tff(111,plain,
% 0.19/0.40 (((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | ((~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb)))) <=> ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(112,plain,
% 0.19/0.40 ((member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb)) | member(h(a, b, aD_aDb), b) | (~member(h(a, b, aD_aDb), a))) <=> ((~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(113,plain,
% 0.19/0.40 (((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb)) | member(h(a, b, aD_aDb), b) | (~member(h(a, b, aD_aDb), a)))) <=> ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | ((~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[112])).
% 0.19/0.40 tff(114,plain,
% 0.19/0.40 (((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb)) | member(h(a, b, aD_aDb), b) | (~member(h(a, b, aD_aDb), a)))) <=> ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[113, 111])).
% 0.19/0.40 tff(115,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb)) | member(h(a, b, aD_aDb), b) | (~member(h(a, b, aD_aDb), a)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(116,plain,
% 0.19/0.40 ((~![Set1: $i, Set2: $i, Difference: $i, Element: $i] : (member(Element, Difference) | (~difference(Set1, Set2, Difference)) | member(Element, Set2) | (~member(Element, Set1)))) | (~member(h(a, b, aD_aDb), a)) | member(h(a, b, aD_aDb), b) | member(h(a, b, aD_aDb), aDb) | (~difference(a, b, aDb))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.19/0.40 tff(117,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[116, 48, 76, 38, 110, 95])).
% 0.19/0.40 % SZS output end Proof
% 0.19/0.40 % E exiting
%------------------------------------------------------------------------------