TSTP Solution File: SEU176+2 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SEU176+2 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 20 07:28:00 EDT 2022
% Result : Theorem 0.62s 0.67s
% Output : Proof 0.62s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.10 % Problem : SEU176+2 : TPTP v8.1.0. Released v3.3.0.
% 0.10/0.11 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.10/0.32 % Computer : n012.cluster.edu
% 0.10/0.32 % Model : x86_64 x86_64
% 0.10/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.32 % Memory : 8042.1875MB
% 0.10/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.32 % CPULimit : 300
% 0.10/0.32 % WCLimit : 300
% 0.10/0.32 % DateTime : Sat Sep 3 09:55:33 EDT 2022
% 0.10/0.32 % CPUTime :
% 0.10/0.32 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.10/0.32 Usage: tptp [options] [-file:]file
% 0.10/0.32 -h, -? prints this message.
% 0.10/0.32 -smt2 print SMT-LIB2 benchmark.
% 0.10/0.32 -m, -model generate model.
% 0.10/0.32 -p, -proof generate proof.
% 0.10/0.32 -c, -core generate unsat core of named formulas.
% 0.10/0.32 -st, -statistics display statistics.
% 0.10/0.32 -t:timeout set timeout (in second).
% 0.10/0.32 -smt2status display status in smt2 format instead of SZS.
% 0.10/0.32 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.10/0.32 -<param>:<value> configuration parameter and value.
% 0.10/0.32 -o:<output-file> file to place output in.
% 0.62/0.67 % SZS status Theorem
% 0.62/0.67 % SZS output start Proof
% 0.62/0.67 tff(subset_difference_type, type, (
% 0.62/0.67 subset_difference: ( $i * $i * $i ) > $i)).
% 0.62/0.67 tff(meet_of_subsets_type, type, (
% 0.62/0.67 meet_of_subsets: ( $i * $i ) > $i)).
% 0.62/0.67 tff(tptp_fun_B_28_type, type, (
% 0.62/0.67 tptp_fun_B_28: $i)).
% 0.62/0.67 tff(tptp_fun_A_29_type, type, (
% 0.62/0.67 tptp_fun_A_29: $i)).
% 0.62/0.67 tff(cast_to_subset_type, type, (
% 0.62/0.67 cast_to_subset: $i > $i)).
% 0.62/0.67 tff(union_of_subsets_type, type, (
% 0.62/0.67 union_of_subsets: ( $i * $i ) > $i)).
% 0.62/0.67 tff(complements_of_subsets_type, type, (
% 0.62/0.67 complements_of_subsets: ( $i * $i ) > $i)).
% 0.62/0.67 tff(set_difference_type, type, (
% 0.62/0.67 set_difference: ( $i * $i ) > $i)).
% 0.62/0.67 tff(element_type, type, (
% 0.62/0.67 element: ( $i * $i ) > $o)).
% 0.62/0.67 tff(powerset_type, type, (
% 0.62/0.67 powerset: $i > $i)).
% 0.62/0.67 tff(empty_set_type, type, (
% 0.62/0.67 empty_set: $i)).
% 0.62/0.67 tff(set_meet_type, type, (
% 0.62/0.67 set_meet: $i > $i)).
% 0.62/0.67 tff(subset_complement_type, type, (
% 0.62/0.67 subset_complement: ( $i * $i ) > $i)).
% 0.62/0.67 tff(union_type, type, (
% 0.62/0.67 union: $i > $i)).
% 0.62/0.67 tff(1,plain,
% 0.62/0.67 (^[A: $i] : refl(element(cast_to_subset(A), powerset(A)) <=> element(cast_to_subset(A), powerset(A)))),
% 0.62/0.67 inference(bind,[status(th)],[])).
% 0.62/0.67 tff(2,plain,
% 0.62/0.67 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.62/0.67 inference(quant_intro,[status(thm)],[1])).
% 0.62/0.67 tff(3,plain,
% 0.62/0.67 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.62/0.67 inference(rewrite,[status(thm)],[])).
% 0.62/0.67 tff(4,axiom,(![A: $i] : element(cast_to_subset(A), powerset(A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k2_subset_1')).
% 0.62/0.67 tff(5,plain,
% 0.62/0.67 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.62/0.67 tff(6,plain,(
% 0.62/0.67 ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.62/0.67 inference(skolemize,[status(sab)],[5])).
% 0.62/0.67 tff(7,plain,
% 0.62/0.67 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.62/0.67 tff(8,plain,
% 0.62/0.67 ((~![A: $i] : element(cast_to_subset(A), powerset(A))) | element(cast_to_subset(A!29), powerset(A!29))),
% 0.62/0.67 inference(quant_inst,[status(thm)],[])).
% 0.62/0.67 tff(9,plain,
% 0.62/0.67 (element(cast_to_subset(A!29), powerset(A!29))),
% 0.62/0.67 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.62/0.67 tff(10,plain,
% 0.62/0.67 ((~((B!28 = empty_set) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(B!28, powerset(powerset(A!29)))))) <=> (~((B!28 = empty_set) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(B!28, powerset(powerset(A!29))))))),
% 0.62/0.67 inference(rewrite,[status(thm)],[])).
% 0.62/0.67 tff(11,plain,
% 0.62/0.67 ((~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) <=> (~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 0.62/0.67 inference(rewrite,[status(thm)],[])).
% 0.62/0.67 tff(12,plain,
% 0.62/0.67 ((~![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B)))))) <=> (~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 0.62/0.67 inference(rewrite,[status(thm)],[])).
% 0.62/0.67 tff(13,axiom,(~![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t48_setfam_1')).
% 0.62/0.67 tff(14,plain,
% 0.62/0.67 (~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.62/0.67 tff(15,plain,
% 0.62/0.67 (~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[14, 11])).
% 0.62/0.67 tff(16,plain,
% 0.62/0.67 (~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.62/0.67 tff(17,plain,
% 0.62/0.67 (~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[16, 11])).
% 0.62/0.67 tff(18,plain,
% 0.62/0.67 (~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[17, 11])).
% 0.62/0.67 tff(19,plain,
% 0.62/0.67 (~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.62/0.67 tff(20,plain,
% 0.62/0.67 (~![A: $i, B: $i] : ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[19, 11])).
% 0.62/0.67 tff(21,plain,(
% 0.62/0.67 ~((B!28 = empty_set) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(B!28, powerset(powerset(A!29)))))),
% 0.62/0.67 inference(skolemize,[status(sab)],[20])).
% 0.62/0.67 tff(22,plain,
% 0.62/0.67 (~((B!28 = empty_set) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(B!28, powerset(powerset(A!29)))))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[21, 10])).
% 0.62/0.67 tff(23,plain,
% 0.62/0.67 (element(B!28, powerset(powerset(A!29)))),
% 0.62/0.67 inference(or_elim,[status(thm)],[22])).
% 0.62/0.67 tff(24,plain,
% 0.62/0.67 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))))),
% 0.62/0.67 inference(bind,[status(th)],[])).
% 0.62/0.67 tff(25,plain,
% 0.62/0.67 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.62/0.67 inference(quant_intro,[status(thm)],[24])).
% 0.62/0.67 tff(26,plain,
% 0.62/0.67 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.62/0.67 inference(rewrite,[status(thm)],[])).
% 0.62/0.67 tff(27,plain,
% 0.62/0.67 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => element(meet_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A))))),
% 0.62/0.67 inference(bind,[status(th)],[])).
% 0.62/0.67 tff(28,plain,
% 0.62/0.67 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(meet_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.62/0.67 inference(quant_intro,[status(thm)],[27])).
% 0.62/0.67 tff(29,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(meet_of_subsets(A, B), powerset(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k6_setfam_1')).
% 0.62/0.67 tff(30,plain,
% 0.62/0.67 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.62/0.67 tff(31,plain,
% 0.62/0.67 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.62/0.67 tff(32,plain,(
% 0.62/0.67 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.62/0.67 inference(skolemize,[status(sab)],[31])).
% 0.62/0.67 tff(33,plain,
% 0.62/0.67 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[32, 25])).
% 0.62/0.67 tff(34,plain,
% 0.62/0.67 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | ((~element(B!28, powerset(powerset(A!29)))) | element(meet_of_subsets(A!29, B!28), powerset(A!29)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | (~element(B!28, powerset(powerset(A!29)))) | element(meet_of_subsets(A!29, B!28), powerset(A!29)))),
% 0.62/0.67 inference(rewrite,[status(thm)],[])).
% 0.62/0.67 tff(35,plain,
% 0.62/0.67 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | ((~element(B!28, powerset(powerset(A!29)))) | element(meet_of_subsets(A!29, B!28), powerset(A!29)))),
% 0.62/0.67 inference(quant_inst,[status(thm)],[])).
% 0.62/0.67 tff(36,plain,
% 0.62/0.67 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | (~element(B!28, powerset(powerset(A!29)))) | element(meet_of_subsets(A!29, B!28), powerset(A!29))),
% 0.62/0.67 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.62/0.67 tff(37,plain,
% 0.62/0.67 (element(meet_of_subsets(A!29, B!28), powerset(A!29))),
% 0.62/0.67 inference(unit_resolution,[status(thm)],[36, 33, 23])).
% 0.62/0.67 tff(38,plain,
% 0.62/0.67 (^[A: $i, B: $i, C: $i] : refl(((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A)))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A)))))),
% 0.62/0.67 inference(bind,[status(th)],[])).
% 0.62/0.67 tff(39,plain,
% 0.62/0.67 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A)))) <=> ![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 0.62/0.67 inference(quant_intro,[status(thm)],[38])).
% 0.62/0.67 tff(40,plain,
% 0.62/0.67 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((element(B, powerset(A)) & element(C, powerset(A))) <=> (~((~element(C, powerset(A))) | (~element(B, powerset(A)))))), ((~(element(B, powerset(A)) & element(C, powerset(A)))) <=> (~(~((~element(C, powerset(A))) | (~element(B, powerset(A)))))))), rewrite((~(~((~element(C, powerset(A))) | (~element(B, powerset(A)))))) <=> ((~element(C, powerset(A))) | (~element(B, powerset(A))))), ((~(element(B, powerset(A)) & element(C, powerset(A)))) <=> ((~element(C, powerset(A))) | (~element(B, powerset(A)))))), (((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> (((~element(C, powerset(A))) | (~element(B, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))))), rewrite((((~element(C, powerset(A))) | (~element(B, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, 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(C, powerset(A))) | (~element(B, powerset(A))))))),
% 0.62/0.67 inference(bind,[status(th)],[])).
% 0.62/0.67 tff(41,plain,
% 0.62/0.67 (![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(C, powerset(A))) | (~element(B, powerset(A))))),
% 0.62/0.67 inference(quant_intro,[status(thm)],[40])).
% 0.62/0.67 tff(42,plain,
% 0.62/0.67 (![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)))),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(43,plain,
% 0.62/0.68 (^[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))))),
% 0.62/0.68 inference(bind,[status(th)],[])).
% 0.62/0.68 tff(44,plain,
% 0.62/0.68 (![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)))),
% 0.62/0.68 inference(quant_intro,[status(thm)],[43])).
% 0.62/0.68 tff(45,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/sandbox/benchmark/theBenchmark.p','redefinition_k6_subset_1')).
% 0.62/0.68 tff(46,plain,
% 0.62/0.68 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.62/0.68 tff(47,plain,
% 0.62/0.68 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.62/0.68 tff(48,plain,(
% 0.62/0.68 ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.62/0.68 inference(skolemize,[status(sab)],[47])).
% 0.62/0.68 tff(49,plain,
% 0.62/0.68 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[48, 41])).
% 0.62/0.68 tff(50,plain,
% 0.62/0.68 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[49, 39])).
% 0.62/0.68 tff(51,plain,
% 0.62/0.68 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))))),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(52,plain,
% 0.62/0.68 (((subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29)))) <=> ((~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))))),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(53,plain,
% 0.62/0.68 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28)))))),
% 0.62/0.68 inference(monotonicity,[status(thm)],[52])).
% 0.62/0.68 tff(54,plain,
% 0.62/0.68 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))))),
% 0.62/0.68 inference(transitivity,[status(thm)],[53, 51])).
% 0.62/0.68 tff(55,plain,
% 0.62/0.68 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))))),
% 0.62/0.68 inference(quant_inst,[status(thm)],[])).
% 0.62/0.68 tff(56,plain,
% 0.62/0.68 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(meet_of_subsets(A!29, B!28), powerset(A!29))) | (~element(cast_to_subset(A!29), powerset(A!29))) | (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.62/0.68 tff(57,plain,
% 0.62/0.68 (subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))),
% 0.62/0.68 inference(unit_resolution,[status(thm)],[56, 50, 37, 9])).
% 0.62/0.68 tff(58,plain,
% 0.62/0.68 (set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))),
% 0.62/0.68 inference(symmetry,[status(thm)],[57])).
% 0.62/0.68 tff(59,plain,
% 0.62/0.68 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))) <=> ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))))),
% 0.62/0.68 inference(bind,[status(th)],[])).
% 0.62/0.68 tff(60,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.62/0.68 inference(quant_intro,[status(thm)],[59])).
% 0.62/0.68 tff(61,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(62,plain,
% 0.62/0.68 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (meet_of_subsets(A, B) = set_meet(B))) <=> ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B))))),
% 0.62/0.68 inference(bind,[status(th)],[])).
% 0.62/0.68 tff(63,plain,
% 0.62/0.68 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (meet_of_subsets(A, B) = set_meet(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.62/0.68 inference(quant_intro,[status(thm)],[62])).
% 0.62/0.68 tff(64,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (meet_of_subsets(A, B) = set_meet(B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','redefinition_k6_setfam_1')).
% 0.62/0.68 tff(65,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.62/0.68 tff(66,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[65, 61])).
% 0.62/0.68 tff(67,plain,(
% 0.62/0.68 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.62/0.68 inference(skolemize,[status(sab)],[66])).
% 0.62/0.68 tff(68,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[67, 60])).
% 0.62/0.68 tff(69,plain,
% 0.62/0.68 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))) | ((~element(B!28, powerset(powerset(A!29)))) | (meet_of_subsets(A!29, B!28) = set_meet(B!28)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))) | (~element(B!28, powerset(powerset(A!29)))) | (meet_of_subsets(A!29, B!28) = set_meet(B!28)))),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(70,plain,
% 0.62/0.68 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))) | ((~element(B!28, powerset(powerset(A!29)))) | (meet_of_subsets(A!29, B!28) = set_meet(B!28)))),
% 0.62/0.68 inference(quant_inst,[status(thm)],[])).
% 0.62/0.68 tff(71,plain,
% 0.62/0.68 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (meet_of_subsets(A, B) = set_meet(B)))) | (~element(B!28, powerset(powerset(A!29)))) | (meet_of_subsets(A!29, B!28) = set_meet(B!28))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.62/0.68 tff(72,plain,
% 0.62/0.68 (meet_of_subsets(A!29, B!28) = set_meet(B!28)),
% 0.62/0.68 inference(unit_resolution,[status(thm)],[71, 68, 23])).
% 0.62/0.68 tff(73,plain,
% 0.62/0.68 (set_meet(B!28) = meet_of_subsets(A!29, B!28)),
% 0.62/0.68 inference(symmetry,[status(thm)],[72])).
% 0.62/0.68 tff(74,plain,
% 0.62/0.68 (^[A: $i] : refl((cast_to_subset(A) = A) <=> (cast_to_subset(A) = A))),
% 0.62/0.68 inference(bind,[status(th)],[])).
% 0.62/0.68 tff(75,plain,
% 0.62/0.68 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 0.62/0.68 inference(quant_intro,[status(thm)],[74])).
% 0.62/0.68 tff(76,plain,
% 0.62/0.68 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(77,axiom,(![A: $i] : (cast_to_subset(A) = A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d4_subset_1')).
% 0.62/0.68 tff(78,plain,
% 0.62/0.68 (![A: $i] : (cast_to_subset(A) = A)),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.62/0.68 tff(79,plain,(
% 0.62/0.68 ![A: $i] : (cast_to_subset(A) = A)),
% 0.62/0.68 inference(skolemize,[status(sab)],[78])).
% 0.62/0.68 tff(80,plain,
% 0.62/0.68 (![A: $i] : (cast_to_subset(A) = A)),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[79, 75])).
% 0.62/0.68 tff(81,plain,
% 0.62/0.68 ((~![A: $i] : (cast_to_subset(A) = A)) | (cast_to_subset(A!29) = A!29)),
% 0.62/0.68 inference(quant_inst,[status(thm)],[])).
% 0.62/0.68 tff(82,plain,
% 0.62/0.68 (cast_to_subset(A!29) = A!29),
% 0.62/0.68 inference(unit_resolution,[status(thm)],[81, 80])).
% 0.62/0.68 tff(83,plain,
% 0.62/0.68 (A!29 = cast_to_subset(A!29)),
% 0.62/0.68 inference(symmetry,[status(thm)],[82])).
% 0.62/0.68 tff(84,plain,
% 0.62/0.68 (set_difference(A!29, set_meet(B!28)) = set_difference(cast_to_subset(A!29), meet_of_subsets(A!29, B!28))),
% 0.62/0.68 inference(monotonicity,[status(thm)],[83, 73])).
% 0.62/0.68 tff(85,plain,
% 0.62/0.68 (element(set_meet(B!28), powerset(A!29)) <=> element(meet_of_subsets(A!29, B!28), powerset(A!29))),
% 0.62/0.68 inference(monotonicity,[status(thm)],[73])).
% 0.62/0.68 tff(86,plain,
% 0.62/0.68 (element(meet_of_subsets(A!29, B!28), powerset(A!29)) <=> element(set_meet(B!28), powerset(A!29))),
% 0.62/0.68 inference(symmetry,[status(thm)],[85])).
% 0.62/0.68 tff(87,plain,
% 0.62/0.68 (element(set_meet(B!28), powerset(A!29))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[37, 86])).
% 0.62/0.68 tff(88,plain,
% 0.62/0.68 (^[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))))),
% 0.62/0.68 inference(bind,[status(th)],[])).
% 0.62/0.68 tff(89,plain,
% 0.62/0.68 (![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)))),
% 0.62/0.68 inference(quant_intro,[status(thm)],[88])).
% 0.62/0.68 tff(90,plain,
% 0.62/0.68 (![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)))),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(91,plain,
% 0.62/0.68 (^[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))))),
% 0.62/0.68 inference(bind,[status(th)],[])).
% 0.62/0.68 tff(92,plain,
% 0.62/0.68 (![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)))),
% 0.62/0.68 inference(quant_intro,[status(thm)],[91])).
% 0.62/0.68 tff(93,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, B) = set_difference(A, B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d5_subset_1')).
% 0.62/0.68 tff(94,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.62/0.68 tff(95,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[94, 90])).
% 0.62/0.68 tff(96,plain,(
% 0.62/0.68 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.62/0.68 inference(skolemize,[status(sab)],[95])).
% 0.62/0.68 tff(97,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[96, 89])).
% 0.62/0.68 tff(98,plain,
% 0.62/0.68 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28))))),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(99,plain,
% 0.62/0.68 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28))))),
% 0.62/0.68 inference(quant_inst,[status(thm)],[])).
% 0.62/0.68 tff(100,plain,
% 0.62/0.68 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28)))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.62/0.68 tff(101,plain,
% 0.62/0.68 ((~element(set_meet(B!28), powerset(A!29))) | (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28)))),
% 0.62/0.68 inference(unit_resolution,[status(thm)],[100, 97])).
% 0.62/0.68 tff(102,plain,
% 0.62/0.68 (subset_complement(A!29, set_meet(B!28)) = set_difference(A!29, set_meet(B!28))),
% 0.62/0.68 inference(unit_resolution,[status(thm)],[101, 87])).
% 0.62/0.68 tff(103,plain,
% 0.62/0.68 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)))),
% 0.62/0.68 inference(bind,[status(th)],[])).
% 0.62/0.68 tff(104,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.62/0.68 inference(quant_intro,[status(thm)],[103])).
% 0.62/0.68 tff(105,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(106,plain,
% 0.62/0.68 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B)))),
% 0.62/0.68 inference(bind,[status(th)],[])).
% 0.62/0.68 tff(107,plain,
% 0.62/0.68 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.62/0.68 inference(quant_intro,[status(thm)],[106])).
% 0.62/0.68 tff(108,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (complements_of_subsets(A, complements_of_subsets(A, B)) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','involutiveness_k7_setfam_1')).
% 0.62/0.68 tff(109,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.62/0.68 tff(110,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[109, 105])).
% 0.62/0.68 tff(111,plain,(
% 0.62/0.68 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.62/0.68 inference(skolemize,[status(sab)],[110])).
% 0.62/0.68 tff(112,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[111, 104])).
% 0.62/0.68 tff(113,plain,
% 0.62/0.68 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!28, powerset(powerset(A!29)))) | (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!28, powerset(powerset(A!29)))) | (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28))),
% 0.62/0.68 inference(rewrite,[status(thm)],[])).
% 0.62/0.68 tff(114,plain,
% 0.62/0.68 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!28, powerset(powerset(A!29)))) | (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28))),
% 0.62/0.68 inference(quant_inst,[status(thm)],[])).
% 0.62/0.68 tff(115,plain,
% 0.62/0.68 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!28, powerset(powerset(A!29)))) | (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28)),
% 0.62/0.68 inference(modus_ponens,[status(thm)],[114, 113])).
% 0.62/0.68 tff(116,plain,
% 0.62/0.68 (complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = B!28),
% 0.62/0.68 inference(unit_resolution,[status(thm)],[115, 112, 23])).
% 0.62/0.68 tff(117,plain,
% 0.62/0.68 (meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, B!28)),
% 0.62/0.68 inference(monotonicity,[status(thm)],[116])).
% 0.62/0.68 tff(118,plain,
% 0.62/0.68 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))))),
% 0.62/0.68 inference(bind,[status(th)],[])).
% 0.62/0.68 tff(119,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.62/0.68 inference(quant_intro,[status(thm)],[118])).
% 0.62/0.68 tff(120,plain,
% 0.62/0.68 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.62/0.69 inference(rewrite,[status(thm)],[])).
% 0.62/0.69 tff(121,plain,
% 0.62/0.69 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A)))))),
% 0.62/0.69 inference(bind,[status(th)],[])).
% 0.62/0.69 tff(122,plain,
% 0.62/0.69 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A)))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.62/0.69 inference(quant_intro,[status(thm)],[121])).
% 0.62/0.69 tff(123,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(complements_of_subsets(A, B), powerset(powerset(A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k7_setfam_1')).
% 0.62/0.69 tff(124,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.62/0.69 tff(125,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[124, 120])).
% 0.62/0.69 tff(126,plain,(
% 0.62/0.69 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.62/0.69 inference(skolemize,[status(sab)],[125])).
% 0.62/0.69 tff(127,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[126, 119])).
% 0.62/0.69 tff(128,plain,
% 0.62/0.69 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!28, powerset(powerset(A!29)))) | element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!28, powerset(powerset(A!29)))) | element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29))))),
% 0.62/0.69 inference(rewrite,[status(thm)],[])).
% 0.62/0.69 tff(129,plain,
% 0.62/0.69 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!28, powerset(powerset(A!29)))) | element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29))))),
% 0.62/0.69 inference(quant_inst,[status(thm)],[])).
% 0.62/0.69 tff(130,plain,
% 0.62/0.69 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!28, powerset(powerset(A!29)))) | element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[129, 128])).
% 0.62/0.69 tff(131,plain,
% 0.62/0.69 (element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))),
% 0.62/0.69 inference(unit_resolution,[status(thm)],[130, 127, 23])).
% 0.62/0.69 tff(132,plain,
% 0.62/0.69 (~(B!28 = empty_set)),
% 0.62/0.69 inference(or_elim,[status(thm)],[22])).
% 0.62/0.69 tff(133,plain,
% 0.62/0.69 (^[A: $i, B: $i] : refl(((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set))) <=> ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set))))),
% 0.62/0.69 inference(bind,[status(th)],[])).
% 0.62/0.69 tff(134,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set))) <=> ![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))),
% 0.62/0.69 inference(quant_intro,[status(thm)],[133])).
% 0.62/0.69 tff(135,plain,
% 0.62/0.69 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set)) <=> (~((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))), ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) <=> (~(~((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))))), rewrite((~(~((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)))), ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))))), (((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> (((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))))), rewrite((((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))), (((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))))),
% 0.62/0.69 inference(bind,[status(th)],[])).
% 0.62/0.69 tff(136,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))),
% 0.62/0.69 inference(quant_intro,[status(thm)],[135])).
% 0.62/0.69 tff(137,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.69 inference(rewrite,[status(thm)],[])).
% 0.62/0.69 tff(138,plain,
% 0.62/0.69 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set)))) <=> ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A))))))),
% 0.62/0.69 inference(bind,[status(th)],[])).
% 0.62/0.69 tff(139,plain,
% 0.62/0.69 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set)))) <=> ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.69 inference(quant_intro,[status(thm)],[138])).
% 0.62/0.69 tff(140,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t46_setfam_1')).
% 0.62/0.69 tff(141,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[140, 139])).
% 0.62/0.69 tff(142,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[141, 137])).
% 0.62/0.69 tff(143,plain,(
% 0.62/0.69 ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 0.62/0.69 inference(skolemize,[status(sab)],[142])).
% 0.62/0.69 tff(144,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[143, 136])).
% 0.62/0.69 tff(145,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[144, 134])).
% 0.62/0.69 tff(146,plain,
% 0.62/0.69 (((~![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))) | ((B!28 = empty_set) | (~element(B!28, powerset(powerset(A!29)))) | (~(complements_of_subsets(A!29, B!28) = empty_set)))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))) | (B!28 = empty_set) | (~element(B!28, powerset(powerset(A!29)))) | (~(complements_of_subsets(A!29, B!28) = empty_set)))),
% 0.62/0.69 inference(rewrite,[status(thm)],[])).
% 0.62/0.69 tff(147,plain,
% 0.62/0.69 ((~![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))) | ((B!28 = empty_set) | (~element(B!28, powerset(powerset(A!29)))) | (~(complements_of_subsets(A!29, B!28) = empty_set)))),
% 0.62/0.69 inference(quant_inst,[status(thm)],[])).
% 0.62/0.69 tff(148,plain,
% 0.62/0.69 ((~![A: $i, B: $i] : ((B = empty_set) | (~element(B, powerset(powerset(A)))) | (~(complements_of_subsets(A, B) = empty_set)))) | (B!28 = empty_set) | (~element(B!28, powerset(powerset(A!29)))) | (~(complements_of_subsets(A!29, B!28) = empty_set))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[147, 146])).
% 0.62/0.69 tff(149,plain,
% 0.62/0.69 (~(complements_of_subsets(A!29, B!28) = empty_set)),
% 0.62/0.69 inference(unit_resolution,[status(thm)],[148, 145, 132, 23])).
% 0.62/0.69 tff(150,plain,
% 0.62/0.69 (^[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))))))),
% 0.62/0.69 inference(bind,[status(th)],[])).
% 0.62/0.69 tff(151,plain,
% 0.62/0.69 (![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)))))),
% 0.62/0.69 inference(quant_intro,[status(thm)],[150])).
% 0.62/0.69 tff(152,plain,
% 0.62/0.69 (![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)))))),
% 0.62/0.69 inference(rewrite,[status(thm)],[])).
% 0.62/0.69 tff(153,plain,
% 0.62/0.69 (^[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)))))))),
% 0.62/0.69 inference(bind,[status(th)],[])).
% 0.62/0.69 tff(154,plain,
% 0.62/0.69 (![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)))))),
% 0.62/0.69 inference(quant_intro,[status(thm)],[153])).
% 0.62/0.69 tff(155,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/sandbox/benchmark/theBenchmark.p','t47_setfam_1')).
% 0.62/0.69 tff(156,plain,
% 0.62/0.69 (![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)))))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[155, 154])).
% 0.62/0.69 tff(157,plain,
% 0.62/0.69 (![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)))))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[156, 152])).
% 0.62/0.69 tff(158,plain,(
% 0.62/0.69 ![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)))))),
% 0.62/0.69 inference(skolemize,[status(sab)],[157])).
% 0.62/0.69 tff(159,plain,
% 0.62/0.69 (![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)))))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[158, 151])).
% 0.62/0.69 tff(160,plain,
% 0.62/0.69 (((~![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)))))) | ((complements_of_subsets(A!29, B!28) = empty_set) | (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))))) <=> ((~![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)))))) | (complements_of_subsets(A!29, B!28) = empty_set) | (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))))),
% 0.62/0.69 inference(rewrite,[status(thm)],[])).
% 0.62/0.69 tff(161,plain,
% 0.62/0.69 ((~![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)))))) | ((complements_of_subsets(A!29, B!28) = empty_set) | (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))))),
% 0.62/0.69 inference(quant_inst,[status(thm)],[])).
% 0.62/0.69 tff(162,plain,
% 0.62/0.69 ((~![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)))))) | (complements_of_subsets(A!29, B!28) = empty_set) | (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29))))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[161, 160])).
% 0.62/0.69 tff(163,plain,
% 0.62/0.69 (subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28))) = meet_of_subsets(A!29, complements_of_subsets(A!29, complements_of_subsets(A!29, B!28)))),
% 0.62/0.69 inference(unit_resolution,[status(thm)],[162, 159, 149, 131])).
% 0.62/0.69 tff(164,plain,
% 0.62/0.69 (^[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))))),
% 0.62/0.69 inference(bind,[status(th)],[])).
% 0.62/0.69 tff(165,plain,
% 0.62/0.69 (![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)))),
% 0.62/0.69 inference(quant_intro,[status(thm)],[164])).
% 0.62/0.69 tff(166,plain,
% 0.62/0.69 (![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)))),
% 0.62/0.69 inference(rewrite,[status(thm)],[])).
% 0.62/0.69 tff(167,plain,
% 0.62/0.69 (^[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))))),
% 0.62/0.69 inference(bind,[status(th)],[])).
% 0.62/0.69 tff(168,plain,
% 0.62/0.69 (![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)))),
% 0.62/0.69 inference(quant_intro,[status(thm)],[167])).
% 0.62/0.69 tff(169,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (union_of_subsets(A, B) = union(B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','redefinition_k5_setfam_1')).
% 0.62/0.69 tff(170,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[169, 168])).
% 0.62/0.69 tff(171,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[170, 166])).
% 0.62/0.69 tff(172,plain,(
% 0.62/0.69 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.62/0.69 inference(skolemize,[status(sab)],[171])).
% 0.62/0.69 tff(173,plain,
% 0.62/0.69 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[172, 165])).
% 0.62/0.69 tff(174,plain,
% 0.62/0.69 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | ((~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28))))),
% 0.62/0.69 inference(rewrite,[status(thm)],[])).
% 0.62/0.69 tff(175,plain,
% 0.62/0.69 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | ((~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28))))),
% 0.62/0.69 inference(quant_inst,[status(thm)],[])).
% 0.62/0.69 tff(176,plain,
% 0.62/0.69 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28)))),
% 0.62/0.69 inference(modus_ponens,[status(thm)],[175, 174])).
% 0.62/0.69 tff(177,plain,
% 0.62/0.69 (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = union(complements_of_subsets(A!29, B!28))),
% 0.62/0.69 inference(unit_resolution,[status(thm)],[176, 173, 131])).
% 0.62/0.69 tff(178,plain,
% 0.62/0.69 (union(complements_of_subsets(A!29, B!28)) = union_of_subsets(A!29, complements_of_subsets(A!29, B!28))),
% 0.62/0.69 inference(symmetry,[status(thm)],[177])).
% 0.62/0.69 tff(179,plain,
% 0.62/0.69 (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = subset_difference(A!29, cast_to_subset(A!29), union_of_subsets(A!29, complements_of_subsets(A!29, B!28)))),
% 0.62/0.69 inference(monotonicity,[status(thm)],[83, 178])).
% 0.62/0.69 tff(180,plain,
% 0.62/0.69 (element(union(complements_of_subsets(A!29, B!28)), powerset(A!29)) <=> element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.62/0.70 inference(monotonicity,[status(thm)],[178])).
% 0.62/0.70 tff(181,plain,
% 0.62/0.70 (element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29)) <=> element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.62/0.70 inference(symmetry,[status(thm)],[180])).
% 0.62/0.70 tff(182,plain,
% 0.62/0.70 (^[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))))),
% 0.62/0.70 inference(bind,[status(th)],[])).
% 0.62/0.70 tff(183,plain,
% 0.62/0.70 (![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)))),
% 0.62/0.70 inference(quant_intro,[status(thm)],[182])).
% 0.62/0.70 tff(184,plain,
% 0.62/0.70 (![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)))),
% 0.62/0.70 inference(rewrite,[status(thm)],[])).
% 0.62/0.70 tff(185,plain,
% 0.62/0.70 (^[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))))),
% 0.62/0.70 inference(bind,[status(th)],[])).
% 0.62/0.70 tff(186,plain,
% 0.62/0.70 (![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)))),
% 0.62/0.70 inference(quant_intro,[status(thm)],[185])).
% 0.62/0.70 tff(187,axiom,(![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(union_of_subsets(A, B), powerset(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k5_setfam_1')).
% 0.62/0.70 tff(188,plain,
% 0.62/0.70 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[187, 186])).
% 0.62/0.70 tff(189,plain,
% 0.62/0.70 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[188, 184])).
% 0.62/0.70 tff(190,plain,(
% 0.62/0.70 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.62/0.70 inference(skolemize,[status(sab)],[189])).
% 0.62/0.70 tff(191,plain,
% 0.62/0.70 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[190, 183])).
% 0.62/0.70 tff(192,plain,
% 0.62/0.70 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | ((~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29)))),
% 0.62/0.70 inference(rewrite,[status(thm)],[])).
% 0.62/0.70 tff(193,plain,
% 0.62/0.70 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | ((~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29)))),
% 0.62/0.70 inference(quant_inst,[status(thm)],[])).
% 0.62/0.70 tff(194,plain,
% 0.62/0.70 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | (~element(complements_of_subsets(A!29, B!28), powerset(powerset(A!29)))) | element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[193, 192])).
% 0.62/0.70 tff(195,plain,
% 0.62/0.70 (element(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.62/0.70 inference(unit_resolution,[status(thm)],[194, 191, 131])).
% 0.62/0.70 tff(196,plain,
% 0.62/0.70 (element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[195, 181])).
% 0.62/0.70 tff(197,plain,
% 0.62/0.70 (element(A!29, powerset(A!29)) <=> element(cast_to_subset(A!29), powerset(A!29))),
% 0.62/0.70 inference(monotonicity,[status(thm)],[83])).
% 0.62/0.70 tff(198,plain,
% 0.62/0.70 (element(cast_to_subset(A!29), powerset(A!29)) <=> element(A!29, powerset(A!29))),
% 0.62/0.70 inference(symmetry,[status(thm)],[197])).
% 0.62/0.70 tff(199,plain,
% 0.62/0.70 (element(A!29, powerset(A!29))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[9, 198])).
% 0.62/0.70 tff(200,plain,
% 0.62/0.70 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))))),
% 0.62/0.70 inference(rewrite,[status(thm)],[])).
% 0.62/0.70 tff(201,plain,
% 0.62/0.70 (((subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (~element(A!29, powerset(A!29)))) <=> ((~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))))),
% 0.62/0.70 inference(rewrite,[status(thm)],[])).
% 0.62/0.70 tff(202,plain,
% 0.62/0.70 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (~element(A!29, powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29)))))),
% 0.62/0.70 inference(monotonicity,[status(thm)],[201])).
% 0.62/0.70 tff(203,plain,
% 0.62/0.70 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (~element(A!29, powerset(A!29))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))))),
% 0.62/0.70 inference(transitivity,[status(thm)],[202, 200])).
% 0.62/0.70 tff(204,plain,
% 0.62/0.70 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (~element(A!29, powerset(A!29))))),
% 0.62/0.70 inference(quant_inst,[status(thm)],[])).
% 0.62/0.70 tff(205,plain,
% 0.62/0.70 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29)))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[204, 203])).
% 0.62/0.70 tff(206,plain,
% 0.62/0.70 ((~element(A!29, powerset(A!29))) | (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29)))),
% 0.62/0.70 inference(unit_resolution,[status(thm)],[205, 50])).
% 0.62/0.70 tff(207,plain,
% 0.62/0.70 (subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))),
% 0.62/0.70 inference(unit_resolution,[status(thm)],[206, 199, 196])).
% 0.62/0.70 tff(208,plain,
% 0.62/0.70 (set_difference(A!29, union(complements_of_subsets(A!29, B!28))) = subset_difference(A!29, A!29, union(complements_of_subsets(A!29, B!28)))),
% 0.62/0.70 inference(symmetry,[status(thm)],[207])).
% 0.62/0.70 tff(209,plain,
% 0.62/0.70 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))))),
% 0.62/0.70 inference(rewrite,[status(thm)],[])).
% 0.62/0.70 tff(210,plain,
% 0.62/0.70 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))))),
% 0.62/0.70 inference(quant_inst,[status(thm)],[])).
% 0.62/0.70 tff(211,plain,
% 0.62/0.70 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28))))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[210, 209])).
% 0.62/0.70 tff(212,plain,
% 0.62/0.70 ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28))))),
% 0.62/0.70 inference(unit_resolution,[status(thm)],[211, 97])).
% 0.62/0.70 tff(213,plain,
% 0.62/0.70 (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_difference(A!29, union(complements_of_subsets(A!29, B!28)))),
% 0.62/0.70 inference(unit_resolution,[status(thm)],[212, 196])).
% 0.62/0.70 tff(214,plain,
% 0.62/0.70 (subset_complement(A!29, union(complements_of_subsets(A!29, B!28))) = set_meet(B!28)),
% 0.62/0.70 inference(transitivity,[status(thm)],[213, 208, 179, 163, 117, 72])).
% 0.62/0.70 tff(215,plain,
% 0.62/0.70 (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = subset_complement(A!29, set_meet(B!28))),
% 0.62/0.70 inference(monotonicity,[status(thm)],[214])).
% 0.62/0.70 tff(216,plain,
% 0.62/0.70 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)))),
% 0.62/0.70 inference(bind,[status(th)],[])).
% 0.62/0.70 tff(217,plain,
% 0.62/0.70 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.62/0.70 inference(quant_intro,[status(thm)],[216])).
% 0.62/0.70 tff(218,plain,
% 0.62/0.70 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.62/0.70 inference(rewrite,[status(thm)],[])).
% 0.62/0.70 tff(219,plain,
% 0.62/0.70 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B)) <=> ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)))),
% 0.62/0.70 inference(bind,[status(th)],[])).
% 0.62/0.70 tff(220,plain,
% 0.62/0.70 (![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.62/0.70 inference(quant_intro,[status(thm)],[219])).
% 0.62/0.70 tff(221,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','involutiveness_k3_subset_1')).
% 0.62/0.70 tff(222,plain,
% 0.62/0.70 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[221, 220])).
% 0.62/0.70 tff(223,plain,
% 0.62/0.70 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[222, 218])).
% 0.62/0.70 tff(224,plain,(
% 0.62/0.70 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.62/0.70 inference(skolemize,[status(sab)],[223])).
% 0.62/0.70 tff(225,plain,
% 0.62/0.70 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[224, 217])).
% 0.62/0.70 tff(226,plain,
% 0.62/0.70 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28))))),
% 0.62/0.70 inference(rewrite,[status(thm)],[])).
% 0.62/0.70 tff(227,plain,
% 0.62/0.70 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28))))),
% 0.62/0.70 inference(quant_inst,[status(thm)],[])).
% 0.62/0.70 tff(228,plain,
% 0.62/0.70 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28)))),
% 0.62/0.70 inference(modus_ponens,[status(thm)],[227, 226])).
% 0.62/0.70 tff(229,plain,
% 0.62/0.70 ((~element(union(complements_of_subsets(A!29, B!28)), powerset(A!29))) | (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28)))),
% 0.62/0.70 inference(unit_resolution,[status(thm)],[228, 225])).
% 0.62/0.70 tff(230,plain,
% 0.62/0.70 (subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28)))) = union(complements_of_subsets(A!29, B!28))),
% 0.62/0.70 inference(unit_resolution,[status(thm)],[229, 196])).
% 0.62/0.70 tff(231,plain,
% 0.62/0.70 (union(complements_of_subsets(A!29, B!28)) = subset_complement(A!29, subset_complement(A!29, union(complements_of_subsets(A!29, B!28))))),
% 0.62/0.70 inference(symmetry,[status(thm)],[230])).
% 0.62/0.70 tff(232,plain,
% 0.62/0.70 (union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28))),
% 0.62/0.71 inference(transitivity,[status(thm)],[177, 231, 215, 102, 84, 58])).
% 0.62/0.71 tff(233,plain,
% 0.62/0.71 (~(union_of_subsets(A!29, complements_of_subsets(A!29, B!28)) = subset_difference(A!29, cast_to_subset(A!29), meet_of_subsets(A!29, B!28)))),
% 0.62/0.71 inference(or_elim,[status(thm)],[22])).
% 0.62/0.71 tff(234,plain,
% 0.62/0.71 ($false),
% 0.62/0.71 inference(unit_resolution,[status(thm)],[233, 232])).
% 0.62/0.71 % SZS output end Proof
%------------------------------------------------------------------------------