%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU327+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.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 05:27:37 AM UTC 2025
% Result : Theorem 0.21s 0.41s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.14 % Problem : SEU327+1 : TPTP v9.0.0. Released v3.3.0.
% 0.09/0.14 % Command : run_E %s %d THM
% 0.14/0.35 % Computer : n011.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 : Fri Jun 20 06:45:26 EDT 2025
% 0.14/0.35 % CPUTime :
% 0.21/0.41 % SZS status Theorem
% 0.21/0.41 % SZS output start Proof
% 0.21/0.41 tff(element_type, type, (
% 0.21/0.41 element: ( $i * $i ) > $o)).
% 0.21/0.41 tff(powerset_type, type, (
% 0.21/0.41 powerset: $i > $i)).
% 0.21/0.41 tff(tptp_fun_A_5_type, type, (
% 0.21/0.41 tptp_fun_A_5: $i)).
% 0.21/0.41 tff(union_type, type, (
% 0.21/0.41 union: $i > $i)).
% 0.21/0.41 tff(tptp_fun_B_4_type, type, (
% 0.21/0.41 tptp_fun_B_4: $i)).
% 0.21/0.41 tff(union_of_subsets_type, type, (
% 0.21/0.41 union_of_subsets: ( $i * $i ) > $i)).
% 0.21/0.41 tff(empty_set_type, type, (
% 0.21/0.41 empty_set: $i)).
% 0.21/0.41 tff(subset_complement_type, type, (
% 0.21/0.41 subset_complement: ( $i * $i ) > $i)).
% 0.21/0.41 tff(meet_of_subsets_type, type, (
% 0.21/0.41 meet_of_subsets: ( $i * $i ) > $i)).
% 0.21/0.41 tff(complements_of_subsets_type, type, (
% 0.21/0.41 complements_of_subsets: ( $i * $i ) > $i)).
% 0.21/0.41 tff(set_difference_type, type, (
% 0.21/0.41 set_difference: ( $i * $i ) > $i)).
% 0.21/0.41 tff(cast_to_subset_type, type, (
% 0.21/0.41 cast_to_subset: $i > $i)).
% 0.21/0.41 tff(subset_difference_type, type, (
% 0.21/0.41 subset_difference: ( $i * $i * $i ) > $i)).
% 0.21/0.41 tff(1,plain,
% 0.21/0.41 ((~((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))))) <=> (~((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5))))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(2,plain,
% 0.21/0.41 ((~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))) <=> (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A))))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(3,plain,
% 0.21/0.41 ((~![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B)))))) <=> (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A))))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(4,axiom,(~![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t11_tops_2')).
% 0.21/0.41 tff(5,plain,
% 0.21/0.41 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.21/0.41 tff(6,plain,
% 0.21/0.41 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.21/0.41 tff(7,plain,
% 0.21/0.41 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.21/0.41 tff(8,plain,
% 0.21/0.41 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.21/0.41 tff(9,plain,(
% 0.21/0.41 ~((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))))),
% 0.21/0.41 inference(skolemize,[status(sab)],[8])).
% 0.21/0.41 tff(10,plain,
% 0.21/0.41 (~((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[9, 1])).
% 0.21/0.41 tff(11,plain,
% 0.21/0.41 (element(B!4, powerset(powerset(A!5)))),
% 0.21/0.41 inference(or_elim,[status(thm)],[10])).
% 0.21/0.41 tff(12,plain,
% 0.21/0.41 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))) <=> ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(13,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[12])).
% 0.21/0.42 tff(14,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(15,plain,
% 0.21/0.42 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (union_of_subsets(A, B) = union(B))) <=> ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(16,plain,
% 0.21/0.42 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (union_of_subsets(A, B) = union(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[15])).
% 0.21/0.42 tff(17,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (union_of_subsets(A, B) = union(B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_k5_setfam_1')).
% 0.21/0.42 tff(18,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.21/0.42 tff(19,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.21/0.42 tff(20,plain,(
% 0.21/0.42 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.42 inference(skolemize,[status(sab)],[19])).
% 0.21/0.42 tff(21,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[20, 13])).
% 0.21/0.42 tff(22,plain,
% 0.21/0.42 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | ((~element(B!4, powerset(powerset(A!5)))) | (union_of_subsets(A!5, B!4) = union(B!4)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | (~element(B!4, powerset(powerset(A!5)))) | (union_of_subsets(A!5, B!4) = union(B!4)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(23,plain,
% 0.21/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | ((~element(B!4, powerset(powerset(A!5)))) | (union_of_subsets(A!5, B!4) = union(B!4)))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(24,plain,
% 0.21/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | (~element(B!4, powerset(powerset(A!5)))) | (union_of_subsets(A!5, B!4) = union(B!4))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.21/0.42 tff(25,plain,
% 0.21/0.42 (union_of_subsets(A!5, B!4) = union(B!4)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[24, 21, 11])).
% 0.21/0.42 tff(26,plain,
% 0.21/0.42 (union(B!4) = union_of_subsets(A!5, B!4)),
% 0.21/0.42 inference(symmetry,[status(thm)],[25])).
% 0.21/0.42 tff(27,plain,
% 0.21/0.42 (element(union(B!4), powerset(A!5)) <=> element(union_of_subsets(A!5, B!4), powerset(A!5))),
% 0.21/0.42 inference(monotonicity,[status(thm)],[26])).
% 0.21/0.42 tff(28,plain,
% 0.21/0.42 (element(union_of_subsets(A!5, B!4), powerset(A!5)) <=> element(union(B!4), powerset(A!5))),
% 0.21/0.42 inference(symmetry,[status(thm)],[27])).
% 0.21/0.42 tff(29,plain,
% 0.21/0.42 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(30,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[29])).
% 0.21/0.42 tff(31,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(32,plain,
% 0.21/0.42 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => element(union_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(33,plain,
% 0.21/0.42 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(union_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[32])).
% 0.21/0.42 tff(34,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(union_of_subsets(A, B), powerset(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k5_setfam_1')).
% 0.21/0.42 tff(35,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.21/0.42 tff(36,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[35, 31])).
% 0.21/0.42 tff(37,plain,(
% 0.21/0.42 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.42 inference(skolemize,[status(sab)],[36])).
% 0.21/0.42 tff(38,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[37, 30])).
% 0.21/0.42 tff(39,plain,
% 0.21/0.42 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | ((~element(B!4, powerset(powerset(A!5)))) | element(union_of_subsets(A!5, B!4), powerset(A!5)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | (~element(B!4, powerset(powerset(A!5)))) | element(union_of_subsets(A!5, B!4), powerset(A!5)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(40,plain,
% 0.21/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | ((~element(B!4, powerset(powerset(A!5)))) | element(union_of_subsets(A!5, B!4), powerset(A!5)))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(41,plain,
% 0.21/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | (~element(B!4, powerset(powerset(A!5)))) | element(union_of_subsets(A!5, B!4), powerset(A!5))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.21/0.42 tff(42,plain,
% 0.21/0.42 (element(union_of_subsets(A!5, B!4), powerset(A!5))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[41, 38, 11])).
% 0.21/0.42 tff(43,plain,
% 0.21/0.42 (element(union(B!4), powerset(A!5))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[42, 28])).
% 0.21/0.42 tff(44,plain,
% 0.21/0.42 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))) <=> ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(45,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[44])).
% 0.21/0.42 tff(46,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(47,plain,
% 0.21/0.42 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => (subset_complement(A, B) = set_difference(A, B))) <=> ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(48,plain,
% 0.21/0.42 (![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, B) = set_difference(A, B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[47])).
% 0.21/0.42 tff(49,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, B) = set_difference(A, B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d5_subset_1')).
% 0.21/0.42 tff(50,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.21/0.42 tff(51,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[50, 46])).
% 0.21/0.42 tff(52,plain,(
% 0.21/0.42 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.42 inference(skolemize,[status(sab)],[51])).
% 0.21/0.42 tff(53,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[52, 45])).
% 0.21/0.42 tff(54,plain,
% 0.21/0.42 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(union_of_subsets(A!5, B!4), powerset(A!5))) | (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(union_of_subsets(A!5, B!4), powerset(A!5))) | (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(55,plain,
% 0.21/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(union_of_subsets(A!5, B!4), powerset(A!5))) | (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(56,plain,
% 0.21/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(union_of_subsets(A!5, B!4), powerset(A!5))) | (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.21/0.42 tff(57,plain,
% 0.21/0.42 (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[56, 53, 42])).
% 0.21/0.42 tff(58,plain,
% 0.21/0.42 (set_difference(A!5, union_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))),
% 0.21/0.42 inference(symmetry,[status(thm)],[57])).
% 0.21/0.42 tff(59,plain,
% 0.21/0.42 (^[A: $i] : refl((cast_to_subset(A) = A) <=> (cast_to_subset(A) = A))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(60,plain,
% 0.21/0.42 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.42 inference(quant_intro,[status(thm)],[59])).
% 0.21/0.42 tff(61,plain,
% 0.21/0.42 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(62,axiom,(![A: $i] : (cast_to_subset(A) = A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d4_subset_1')).
% 0.21/0.42 tff(63,plain,
% 0.21/0.42 (![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.21/0.42 tff(64,plain,(
% 0.21/0.42 ![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.42 inference(skolemize,[status(sab)],[63])).
% 0.21/0.42 tff(65,plain,
% 0.21/0.42 (![A: $i] : (cast_to_subset(A) = A)),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[64, 60])).
% 0.21/0.42 tff(66,plain,
% 0.21/0.42 ((~![A: $i] : (cast_to_subset(A) = A)) | (cast_to_subset(A!5) = A!5)),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(67,plain,
% 0.21/0.42 (cast_to_subset(A!5) = A!5),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[66, 65])).
% 0.21/0.42 tff(68,plain,
% 0.21/0.42 (set_difference(cast_to_subset(A!5), union(B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))),
% 0.21/0.42 inference(monotonicity,[status(thm)],[67, 26])).
% 0.21/0.42 tff(69,plain,
% 0.21/0.42 (set_difference(cast_to_subset(A!5), union(B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))),
% 0.21/0.42 inference(transitivity,[status(thm)],[68, 58])).
% 0.21/0.42 tff(70,plain,
% 0.21/0.42 (^[A: $i, B: $i] : refl(((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(71,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[70])).
% 0.21/0.42 tff(72,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(73,plain,
% 0.21/0.42 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B)))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))), ((element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> (element(B, powerset(powerset(A))) => ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))))), rewrite((element(B, powerset(powerset(A))) => ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))), ((element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(74,plain,
% 0.21/0.42 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[73])).
% 0.21/0.42 tff(75,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t47_setfam_1')).
% 0.21/0.42 tff(76,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.21/0.42 tff(77,plain,
% 0.21/0.42 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[76, 72])).
% 0.21/0.43 tff(78,plain,(
% 0.21/0.43 ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.43 inference(skolemize,[status(sab)],[77])).
% 0.21/0.43 tff(79,plain,
% 0.21/0.43 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[78, 71])).
% 0.21/0.43 tff(80,plain,
% 0.21/0.43 (~(B!4 = empty_set)),
% 0.21/0.43 inference(or_elim,[status(thm)],[10])).
% 0.21/0.43 tff(81,plain,
% 0.21/0.43 (((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(82,plain,
% 0.21/0.43 (((B!4 = empty_set) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))) | (~element(B!4, powerset(powerset(A!5))))) <=> ((B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(83,plain,
% 0.21/0.43 (((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))) | (~element(B!4, powerset(powerset(A!5)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[82])).
% 0.21/0.43 tff(84,plain,
% 0.21/0.43 (((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))) | (~element(B!4, powerset(powerset(A!5)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[83, 81])).
% 0.21/0.43 tff(85,plain,
% 0.21/0.43 ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))) | (~element(B!4, powerset(powerset(A!5)))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(86,plain,
% 0.21/0.43 ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.21/0.43 tff(87,plain,
% 0.21/0.43 (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[86, 80, 11, 79])).
% 0.21/0.43 tff(88,plain,
% 0.21/0.43 (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[26])).
% 0.21/0.43 tff(89,plain,
% 0.21/0.43 (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))),
% 0.21/0.43 inference(transitivity,[status(thm)],[88, 87])).
% 0.21/0.43 tff(90,plain,
% 0.21/0.43 ((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) <=> (meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4)))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[89, 69])).
% 0.21/0.43 tff(91,plain,
% 0.21/0.43 ((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) <=> (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))),
% 0.21/0.43 inference(symmetry,[status(thm)],[90])).
% 0.21/0.43 tff(92,plain,
% 0.21/0.43 ((~(meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4)))) <=> (~(subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[91])).
% 0.21/0.43 tff(93,plain,
% 0.21/0.43 (~(meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4)))),
% 0.21/0.43 inference(or_elim,[status(thm)],[10])).
% 0.21/0.43 tff(94,plain,
% 0.21/0.43 (~(subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.21/0.43 tff(95,plain,
% 0.21/0.43 (^[A: $i] : refl(element(cast_to_subset(A), powerset(A)) <=> element(cast_to_subset(A), powerset(A)))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(96,plain,
% 0.21/0.43 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[95])).
% 0.21/0.43 tff(97,plain,
% 0.21/0.43 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(98,axiom,(![A: $i] : element(cast_to_subset(A), powerset(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k2_subset_1')).
% 0.21/0.43 tff(99,plain,
% 0.21/0.43 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.21/0.43 tff(100,plain,(
% 0.21/0.43 ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.43 inference(skolemize,[status(sab)],[99])).
% 0.21/0.43 tff(101,plain,
% 0.21/0.43 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[100, 96])).
% 0.21/0.43 tff(102,plain,
% 0.21/0.43 ((~![A: $i] : element(cast_to_subset(A), powerset(A))) | element(cast_to_subset(A!5), powerset(A!5))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(103,plain,
% 0.21/0.43 (element(cast_to_subset(A!5), powerset(A!5))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[102, 101])).
% 0.21/0.43 tff(104,plain,
% 0.21/0.43 (^[A: $i, B: $i, C: $i] : refl(((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A)))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A)))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(105,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A)))) <=> ![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[104])).
% 0.21/0.43 tff(106,plain,
% 0.21/0.43 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((element(B, powerset(A)) & element(C, powerset(A))) <=> (~((~element(B, powerset(A))) | (~element(C, powerset(A)))))), ((~(element(B, powerset(A)) & element(C, powerset(A)))) <=> (~(~((~element(B, powerset(A))) | (~element(C, powerset(A)))))))), rewrite((~(~((~element(B, powerset(A))) | (~element(C, powerset(A)))))) <=> ((~element(B, powerset(A))) | (~element(C, powerset(A))))), ((~(element(B, powerset(A)) & element(C, powerset(A)))) <=> ((~element(B, powerset(A))) | (~element(C, powerset(A)))))), (((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> (((~element(B, powerset(A))) | (~element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))))), rewrite((((~element(B, powerset(A))) | (~element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))), (((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(107,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[106])).
% 0.21/0.43 tff(108,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(109,plain,
% 0.21/0.43 (^[A: $i, B: $i, C: $i] : rewrite(((element(B, powerset(A)) & element(C, powerset(A))) => (subset_difference(A, B, C) = set_difference(B, C))) <=> ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(110,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((element(B, powerset(A)) & element(C, powerset(A))) => (subset_difference(A, B, C) = set_difference(B, C))) <=> ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[109])).
% 0.21/0.43 tff(111,axiom,(![A: $i, B: $i, C: $i] : ((element(B, powerset(A)) & element(C, powerset(A))) => (subset_difference(A, B, C) = set_difference(B, C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_k6_subset_1')).
% 0.21/0.43 tff(112,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.21/0.43 tff(113,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[112, 108])).
% 0.21/0.43 tff(114,plain,(
% 0.21/0.43 ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.21/0.43 inference(skolemize,[status(sab)],[113])).
% 0.21/0.43 tff(115,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[114, 107])).
% 0.21/0.43 tff(116,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[115, 105])).
% 0.21/0.44 tff(117,plain,
% 0.21/0.44 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(118,plain,
% 0.21/0.44 (((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5)))) <=> ((~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(119,plain,
% 0.21/0.44 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[118])).
% 0.21/0.44 tff(120,plain,
% 0.21/0.44 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))),
% 0.21/0.44 inference(transitivity,[status(thm)],[119, 117])).
% 0.21/0.44 tff(121,plain,
% 0.21/0.44 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(122,plain,
% 0.21/0.44 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[121, 120])).
% 0.21/0.44 tff(123,plain,
% 0.21/0.44 ((~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[122, 116, 103])).
% 0.21/0.44 tff(124,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[123, 94, 43])).
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44 % E exiting
%------------------------------------------------------------------------------