TSTP Solution File: SEU327+2 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SEU327+2 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n013.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 20 07:28:57 EDT 2022
% Result : Theorem 2.22s 1.73s
% Output : Proof 2.33s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SEU327+2 : TPTP v8.1.0. Released v3.3.0.
% 0.06/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.33 % Computer : n013.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sat Sep 3 12:04:47 EDT 2022
% 0.14/0.34 % CPUTime :
% 0.14/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.34 Usage: tptp [options] [-file:]file
% 0.14/0.34 -h, -? prints this message.
% 0.14/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.34 -m, -model generate model.
% 0.14/0.34 -p, -proof generate proof.
% 0.14/0.34 -c, -core generate unsat core of named formulas.
% 0.14/0.34 -st, -statistics display statistics.
% 0.14/0.34 -t:timeout set timeout (in second).
% 0.14/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.34 -<param>:<value> configuration parameter and value.
% 0.14/0.34 -o:<output-file> file to place output in.
% 2.22/1.73 % SZS status Theorem
% 2.22/1.73 % SZS output start Proof
% 2.22/1.73 tff(subset_complement_type, type, (
% 2.22/1.73 subset_complement: ( $i * $i ) > $i)).
% 2.22/1.73 tff(union_of_subsets_type, type, (
% 2.22/1.73 union_of_subsets: ( $i * $i ) > $i)).
% 2.22/1.73 tff(tptp_fun_B_301_type, type, (
% 2.22/1.73 tptp_fun_B_301: $i)).
% 2.22/1.73 tff(tptp_fun_A_302_type, type, (
% 2.22/1.73 tptp_fun_A_302: $i)).
% 2.22/1.73 tff(meet_of_subsets_type, type, (
% 2.22/1.73 meet_of_subsets: ( $i * $i ) > $i)).
% 2.22/1.73 tff(complements_of_subsets_type, type, (
% 2.22/1.73 complements_of_subsets: ( $i * $i ) > $i)).
% 2.22/1.73 tff(element_type, type, (
% 2.22/1.73 element: ( $i * $i ) > $o)).
% 2.22/1.73 tff(powerset_type, type, (
% 2.22/1.73 powerset: $i > $i)).
% 2.22/1.73 tff(empty_set_type, type, (
% 2.22/1.73 empty_set: $i)).
% 2.22/1.73 tff(subset_difference_type, type, (
% 2.22/1.73 subset_difference: ( $i * $i * $i ) > $i)).
% 2.22/1.73 tff(cast_to_subset_type, type, (
% 2.22/1.73 cast_to_subset: $i > $i)).
% 2.22/1.73 tff(set_difference_type, type, (
% 2.22/1.73 set_difference: ( $i * $i ) > $i)).
% 2.22/1.73 tff(1,plain,
% 2.22/1.73 ((~((B!301 = empty_set) | (meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = subset_complement(A!302, union_of_subsets(A!302, B!301))) | (~element(B!301, powerset(powerset(A!302)))))) <=> (~((B!301 = empty_set) | (meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = subset_complement(A!302, union_of_subsets(A!302, B!301))) | (~element(B!301, powerset(powerset(A!302))))))),
% 2.22/1.73 inference(rewrite,[status(thm)],[])).
% 2.22/1.73 tff(2,plain,
% 2.22/1.73 ((~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) <=> (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 2.22/1.73 inference(rewrite,[status(thm)],[])).
% 2.22/1.73 tff(3,plain,
% 2.22/1.73 ((~![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] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 2.22/1.73 inference(rewrite,[status(thm)],[])).
% 2.22/1.73 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/sandbox/benchmark/theBenchmark.p','t11_tops_2')).
% 2.22/1.73 tff(5,plain,
% 2.22/1.73 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.73 inference(modus_ponens,[status(thm)],[4, 3])).
% 2.22/1.73 tff(6,plain,
% 2.22/1.73 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.73 inference(modus_ponens,[status(thm)],[5, 2])).
% 2.22/1.73 tff(7,plain,
% 2.22/1.73 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.73 inference(modus_ponens,[status(thm)],[6, 2])).
% 2.22/1.73 tff(8,plain,
% 2.22/1.73 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.73 inference(modus_ponens,[status(thm)],[7, 2])).
% 2.22/1.73 tff(9,plain,
% 2.22/1.73 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.73 inference(modus_ponens,[status(thm)],[8, 2])).
% 2.22/1.73 tff(10,plain,
% 2.22/1.73 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.73 inference(modus_ponens,[status(thm)],[9, 2])).
% 2.22/1.73 tff(11,plain,
% 2.22/1.73 (~![A: $i, B: $i] : ((B = empty_set) | (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[10, 2])).
% 2.22/1.74 tff(12,plain,(
% 2.22/1.74 ~((B!301 = empty_set) | (meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = subset_complement(A!302, union_of_subsets(A!302, B!301))) | (~element(B!301, powerset(powerset(A!302)))))),
% 2.22/1.74 inference(skolemize,[status(sab)],[11])).
% 2.22/1.74 tff(13,plain,
% 2.22/1.74 (~((B!301 = empty_set) | (meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = subset_complement(A!302, union_of_subsets(A!302, B!301))) | (~element(B!301, powerset(powerset(A!302)))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[12, 1])).
% 2.22/1.74 tff(14,plain,
% 2.22/1.74 (element(B!301, powerset(powerset(A!302)))),
% 2.22/1.74 inference(or_elim,[status(thm)],[13])).
% 2.22/1.74 tff(15,plain,
% 2.22/1.74 (^[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)))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(16,plain,
% 2.22/1.74 (![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))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[15])).
% 2.22/1.74 tff(17,plain,
% 2.22/1.74 (![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))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(18,plain,
% 2.22/1.74 (^[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)))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(19,plain,
% 2.22/1.74 (![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))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[18])).
% 2.22/1.74 tff(20,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')).
% 2.22/1.74 tff(21,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[20, 19])).
% 2.22/1.74 tff(22,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[21, 17])).
% 2.22/1.74 tff(23,plain,(
% 2.22/1.74 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 2.22/1.74 inference(skolemize,[status(sab)],[22])).
% 2.22/1.74 tff(24,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[23, 16])).
% 2.22/1.74 tff(25,plain,
% 2.22/1.74 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!301, powerset(powerset(A!302)))) | (complements_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = B!301))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!301, powerset(powerset(A!302)))) | (complements_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = B!301))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(26,plain,
% 2.22/1.74 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | ((~element(B!301, powerset(powerset(A!302)))) | (complements_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = B!301))),
% 2.22/1.74 inference(quant_inst,[status(thm)],[])).
% 2.22/1.74 tff(27,plain,
% 2.22/1.74 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (complements_of_subsets(A, complements_of_subsets(A, B)) = B))) | (~element(B!301, powerset(powerset(A!302)))) | (complements_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = B!301)),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[26, 25])).
% 2.22/1.74 tff(28,plain,
% 2.22/1.74 (complements_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = B!301),
% 2.22/1.74 inference(unit_resolution,[status(thm)],[27, 24, 14])).
% 2.22/1.74 tff(29,plain,
% 2.22/1.74 (union_of_subsets(A!302, complements_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = union_of_subsets(A!302, B!301)),
% 2.22/1.74 inference(monotonicity,[status(thm)],[28])).
% 2.22/1.74 tff(30,plain,
% 2.22/1.74 (^[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)))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(31,plain,
% 2.22/1.74 (![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))))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[30])).
% 2.22/1.74 tff(32,plain,
% 2.22/1.74 (![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))))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(33,plain,
% 2.22/1.74 (^[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)))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(34,plain,
% 2.22/1.74 (![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))))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[33])).
% 2.22/1.74 tff(35,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')).
% 2.22/1.74 tff(36,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[35, 34])).
% 2.22/1.74 tff(37,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[36, 32])).
% 2.22/1.74 tff(38,plain,(
% 2.22/1.74 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 2.22/1.74 inference(skolemize,[status(sab)],[37])).
% 2.22/1.74 tff(39,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[38, 31])).
% 2.22/1.74 tff(40,plain,
% 2.22/1.74 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!301, powerset(powerset(A!302)))) | element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!301, powerset(powerset(A!302)))) | element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302))))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(41,plain,
% 2.22/1.74 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | ((~element(B!301, powerset(powerset(A!302)))) | element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302))))),
% 2.22/1.74 inference(quant_inst,[status(thm)],[])).
% 2.22/1.74 tff(42,plain,
% 2.22/1.74 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(complements_of_subsets(A, B), powerset(powerset(A))))) | (~element(B!301, powerset(powerset(A!302)))) | element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302)))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[41, 40])).
% 2.22/1.74 tff(43,plain,
% 2.22/1.74 (element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302)))),
% 2.22/1.74 inference(unit_resolution,[status(thm)],[42, 39, 14])).
% 2.22/1.74 tff(44,plain,
% 2.22/1.74 (^[A: $i, B: $i] : refl(((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))))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(45,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[44])).
% 2.22/1.74 tff(46,plain,
% 2.22/1.74 (^[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) | (~(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))))) <=> ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(47,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[46])).
% 2.22/1.74 tff(48,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(49,plain,
% 2.22/1.74 (^[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))))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(50,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[49])).
% 2.22/1.74 tff(51,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')).
% 2.22/1.74 tff(52,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[51, 50])).
% 2.22/1.74 tff(53,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[52, 48])).
% 2.22/1.74 tff(54,plain,(
% 2.22/1.74 ![A: $i, B: $i] : ((~((~(B = empty_set)) & (complements_of_subsets(A, B) = empty_set))) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.74 inference(skolemize,[status(sab)],[53])).
% 2.22/1.74 tff(55,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[54, 47])).
% 2.22/1.74 tff(56,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[55, 45])).
% 2.22/1.74 tff(57,plain,
% 2.22/1.74 (~(B!301 = empty_set)),
% 2.22/1.74 inference(or_elim,[status(thm)],[13])).
% 2.22/1.74 tff(58,plain,
% 2.22/1.74 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!301 = empty_set) | (~element(B!301, powerset(powerset(A!302)))) | (~(complements_of_subsets(A!302, B!301) = empty_set)))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (B!301 = empty_set) | (~element(B!301, powerset(powerset(A!302)))) | (~(complements_of_subsets(A!302, B!301) = empty_set)))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(59,plain,
% 2.22/1.74 (((B!301 = empty_set) | (~(complements_of_subsets(A!302, B!301) = empty_set)) | (~element(B!301, powerset(powerset(A!302))))) <=> ((B!301 = empty_set) | (~element(B!301, powerset(powerset(A!302)))) | (~(complements_of_subsets(A!302, B!301) = empty_set)))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(60,plain,
% 2.22/1.74 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!301 = empty_set) | (~(complements_of_subsets(A!302, B!301) = empty_set)) | (~element(B!301, powerset(powerset(A!302)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!301 = empty_set) | (~element(B!301, powerset(powerset(A!302)))) | (~(complements_of_subsets(A!302, B!301) = empty_set))))),
% 2.22/1.74 inference(monotonicity,[status(thm)],[59])).
% 2.22/1.74 tff(61,plain,
% 2.22/1.74 (((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!301 = empty_set) | (~(complements_of_subsets(A!302, B!301) = empty_set)) | (~element(B!301, powerset(powerset(A!302)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (B!301 = empty_set) | (~element(B!301, powerset(powerset(A!302)))) | (~(complements_of_subsets(A!302, B!301) = empty_set)))),
% 2.22/1.74 inference(transitivity,[status(thm)],[60, 58])).
% 2.22/1.74 tff(62,plain,
% 2.22/1.74 ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | ((B!301 = empty_set) | (~(complements_of_subsets(A!302, B!301) = empty_set)) | (~element(B!301, powerset(powerset(A!302)))))),
% 2.22/1.74 inference(quant_inst,[status(thm)],[])).
% 2.22/1.74 tff(63,plain,
% 2.22/1.74 ((~![A: $i, B: $i] : ((B = empty_set) | (~(complements_of_subsets(A, B) = empty_set)) | (~element(B, powerset(powerset(A)))))) | (B!301 = empty_set) | (~element(B!301, powerset(powerset(A!302)))) | (~(complements_of_subsets(A!302, B!301) = empty_set))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[62, 61])).
% 2.22/1.74 tff(64,plain,
% 2.22/1.74 (~(complements_of_subsets(A!302, B!301) = empty_set)),
% 2.22/1.74 inference(unit_resolution,[status(thm)],[63, 57, 14, 56])).
% 2.22/1.74 tff(65,plain,
% 2.22/1.74 (^[A: $i, B: $i] : refl(((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))))) <=> ((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))))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(66,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[65])).
% 2.22/1.74 tff(67,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(68,plain,
% 2.22/1.74 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = empty_set)) => (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B)))) <=> ((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))) => ((~(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))) => ((B = empty_set) | (union_of_subsets(A, complements_of_subsets(A, B)) = subset_difference(A, cast_to_subset(A), meet_of_subsets(A, B))))))), rewrite((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))))) <=> ((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)))))), ((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))))) <=> ((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)))))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(69,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[68])).
% 2.22/1.74 tff(70,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')).
% 2.22/1.74 tff(71,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[70, 69])).
% 2.22/1.74 tff(72,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[71, 67])).
% 2.22/1.74 tff(73,plain,(
% 2.22/1.74 ![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)))))),
% 2.22/1.74 inference(skolemize,[status(sab)],[72])).
% 2.22/1.74 tff(74,plain,
% 2.22/1.74 (![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)))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[73, 66])).
% 2.22/1.74 tff(75,plain,
% 2.22/1.74 (((~![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)))))) | ((complements_of_subsets(A!302, B!301) = empty_set) | (union_of_subsets(A!302, complements_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) | (~element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302)))))) <=> ((~![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)))))) | (complements_of_subsets(A!302, B!301) = empty_set) | (union_of_subsets(A!302, complements_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) | (~element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302)))))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(76,plain,
% 2.22/1.74 ((~![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)))))) | ((complements_of_subsets(A!302, B!301) = empty_set) | (union_of_subsets(A!302, complements_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) | (~element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302)))))),
% 2.22/1.74 inference(quant_inst,[status(thm)],[])).
% 2.22/1.74 tff(77,plain,
% 2.22/1.74 ((~![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)))))) | (complements_of_subsets(A!302, B!301) = empty_set) | (union_of_subsets(A!302, complements_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) | (~element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[76, 75])).
% 2.22/1.74 tff(78,plain,
% 2.22/1.74 (union_of_subsets(A!302, complements_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))),
% 2.22/1.74 inference(unit_resolution,[status(thm)],[77, 74, 64, 43])).
% 2.22/1.74 tff(79,plain,
% 2.22/1.74 (subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = union_of_subsets(A!302, complements_of_subsets(A!302, complements_of_subsets(A!302, B!301)))),
% 2.22/1.74 inference(symmetry,[status(thm)],[78])).
% 2.22/1.74 tff(80,plain,
% 2.22/1.74 (^[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))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(81,plain,
% 2.22/1.74 (![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)))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[80])).
% 2.22/1.74 tff(82,plain,
% 2.22/1.74 (![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)))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(83,plain,
% 2.22/1.74 (^[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))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(84,plain,
% 2.22/1.74 (![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)))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[83])).
% 2.22/1.74 tff(85,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')).
% 2.22/1.74 tff(86,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[85, 84])).
% 2.22/1.74 tff(87,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[86, 82])).
% 2.22/1.74 tff(88,plain,(
% 2.22/1.74 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 2.22/1.74 inference(skolemize,[status(sab)],[87])).
% 2.22/1.74 tff(89,plain,
% 2.22/1.74 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[88, 81])).
% 2.22/1.74 tff(90,plain,
% 2.22/1.74 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | ((~element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302)))) | element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | (~element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302)))) | element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302)))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(91,plain,
% 2.22/1.74 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | ((~element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302)))) | element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302)))),
% 2.22/1.74 inference(quant_inst,[status(thm)],[])).
% 2.22/1.74 tff(92,plain,
% 2.22/1.74 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(meet_of_subsets(A, B), powerset(A)))) | (~element(complements_of_subsets(A!302, B!301), powerset(powerset(A!302)))) | element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[91, 90])).
% 2.22/1.74 tff(93,plain,
% 2.22/1.74 (element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))),
% 2.22/1.74 inference(unit_resolution,[status(thm)],[92, 89, 43])).
% 2.22/1.74 tff(94,plain,
% 2.22/1.74 (^[A: $i] : refl(element(cast_to_subset(A), powerset(A)) <=> element(cast_to_subset(A), powerset(A)))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(95,plain,
% 2.22/1.74 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[94])).
% 2.22/1.74 tff(96,plain,
% 2.22/1.74 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(97,axiom,(![A: $i] : element(cast_to_subset(A), powerset(A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k2_subset_1')).
% 2.22/1.74 tff(98,plain,
% 2.22/1.74 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[97, 96])).
% 2.22/1.74 tff(99,plain,(
% 2.22/1.74 ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 2.22/1.74 inference(skolemize,[status(sab)],[98])).
% 2.22/1.74 tff(100,plain,
% 2.22/1.74 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[99, 95])).
% 2.22/1.74 tff(101,plain,
% 2.22/1.74 ((~![A: $i] : element(cast_to_subset(A), powerset(A))) | element(cast_to_subset(A!302), powerset(A!302))),
% 2.22/1.74 inference(quant_inst,[status(thm)],[])).
% 2.22/1.74 tff(102,plain,
% 2.22/1.74 (element(cast_to_subset(A!302), powerset(A!302))),
% 2.22/1.74 inference(unit_resolution,[status(thm)],[101, 100])).
% 2.22/1.74 tff(103,plain,
% 2.22/1.74 (^[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)))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(104,plain,
% 2.22/1.74 (![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))))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[103])).
% 2.22/1.74 tff(105,plain,
% 2.22/1.74 (^[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))))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(106,plain,
% 2.22/1.74 (![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))))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[105])).
% 2.22/1.74 tff(107,plain,
% 2.22/1.74 (![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)))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(108,plain,
% 2.22/1.74 (^[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))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(109,plain,
% 2.22/1.74 (![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)))),
% 2.22/1.74 inference(quant_intro,[status(thm)],[108])).
% 2.22/1.74 tff(110,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')).
% 2.22/1.74 tff(111,plain,
% 2.22/1.74 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[110, 109])).
% 2.22/1.74 tff(112,plain,
% 2.22/1.74 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[111, 107])).
% 2.22/1.74 tff(113,plain,(
% 2.22/1.74 ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 2.22/1.74 inference(skolemize,[status(sab)],[112])).
% 2.22/1.74 tff(114,plain,
% 2.22/1.74 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[113, 106])).
% 2.22/1.74 tff(115,plain,
% 2.22/1.74 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[114, 104])).
% 2.22/1.74 tff(116,plain,
% 2.22/1.74 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(cast_to_subset(A!302), powerset(A!302))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(cast_to_subset(A!302), powerset(A!302))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(117,plain,
% 2.22/1.74 (((subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (~element(cast_to_subset(A!302), powerset(A!302)))) <=> ((~element(cast_to_subset(A!302), powerset(A!302))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))))),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(118,plain,
% 2.22/1.74 (((~![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!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (~element(cast_to_subset(A!302), powerset(A!302))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | ((~element(cast_to_subset(A!302), powerset(A!302))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))))))),
% 2.22/1.74 inference(monotonicity,[status(thm)],[117])).
% 2.22/1.74 tff(119,plain,
% 2.22/1.74 (((~![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!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (~element(cast_to_subset(A!302), powerset(A!302))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(cast_to_subset(A!302), powerset(A!302))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))))),
% 2.22/1.74 inference(transitivity,[status(thm)],[118, 116])).
% 2.22/1.74 tff(120,plain,
% 2.22/1.74 ((~![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!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (~element(cast_to_subset(A!302), powerset(A!302))))),
% 2.22/1.74 inference(quant_inst,[status(thm)],[])).
% 2.22/1.74 tff(121,plain,
% 2.22/1.74 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(C, powerset(A))) | (~element(B, powerset(A))))) | (~element(cast_to_subset(A!302), powerset(A!302))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))))),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[120, 119])).
% 2.22/1.74 tff(122,plain,
% 2.22/1.74 (subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))),
% 2.22/1.74 inference(unit_resolution,[status(thm)],[121, 115, 102, 93])).
% 2.22/1.74 tff(123,plain,
% 2.22/1.74 (set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = subset_difference(A!302, cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))),
% 2.22/1.74 inference(symmetry,[status(thm)],[122])).
% 2.22/1.74 tff(124,plain,
% 2.22/1.74 (^[A: $i] : refl((cast_to_subset(A) = A) <=> (cast_to_subset(A) = A))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(125,plain,
% 2.22/1.74 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 2.22/1.74 inference(quant_intro,[status(thm)],[124])).
% 2.22/1.74 tff(126,plain,
% 2.22/1.74 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 2.22/1.74 inference(rewrite,[status(thm)],[])).
% 2.22/1.74 tff(127,axiom,(![A: $i] : (cast_to_subset(A) = A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d4_subset_1')).
% 2.22/1.74 tff(128,plain,
% 2.22/1.74 (![A: $i] : (cast_to_subset(A) = A)),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[127, 126])).
% 2.22/1.74 tff(129,plain,(
% 2.22/1.74 ![A: $i] : (cast_to_subset(A) = A)),
% 2.22/1.74 inference(skolemize,[status(sab)],[128])).
% 2.22/1.74 tff(130,plain,
% 2.22/1.74 (![A: $i] : (cast_to_subset(A) = A)),
% 2.22/1.74 inference(modus_ponens,[status(thm)],[129, 125])).
% 2.22/1.74 tff(131,plain,
% 2.22/1.74 ((~![A: $i] : (cast_to_subset(A) = A)) | (cast_to_subset(A!302) = A!302)),
% 2.22/1.74 inference(quant_inst,[status(thm)],[])).
% 2.22/1.74 tff(132,plain,
% 2.22/1.74 (cast_to_subset(A!302) = A!302),
% 2.22/1.74 inference(unit_resolution,[status(thm)],[131, 130])).
% 2.22/1.74 tff(133,plain,
% 2.22/1.74 (A!302 = cast_to_subset(A!302)),
% 2.22/1.74 inference(symmetry,[status(thm)],[132])).
% 2.22/1.74 tff(134,plain,
% 2.22/1.74 (set_difference(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(cast_to_subset(A!302), meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))),
% 2.22/1.74 inference(monotonicity,[status(thm)],[133])).
% 2.22/1.74 tff(135,plain,
% 2.22/1.74 (^[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))))),
% 2.22/1.74 inference(bind,[status(th)],[])).
% 2.22/1.74 tff(136,plain,
% 2.22/1.74 (![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)))),
% 2.22/1.75 inference(quant_intro,[status(thm)],[135])).
% 2.22/1.75 tff(137,plain,
% 2.22/1.75 (![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)))),
% 2.22/1.75 inference(rewrite,[status(thm)],[])).
% 2.22/1.75 tff(138,plain,
% 2.22/1.75 (^[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))))),
% 2.22/1.75 inference(bind,[status(th)],[])).
% 2.22/1.75 tff(139,plain,
% 2.22/1.75 (![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)))),
% 2.22/1.75 inference(quant_intro,[status(thm)],[138])).
% 2.22/1.75 tff(140,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')).
% 2.22/1.75 tff(141,plain,
% 2.22/1.75 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 2.22/1.75 inference(modus_ponens,[status(thm)],[140, 139])).
% 2.22/1.75 tff(142,plain,
% 2.22/1.75 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 2.22/1.75 inference(modus_ponens,[status(thm)],[141, 137])).
% 2.22/1.75 tff(143,plain,(
% 2.22/1.75 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 2.22/1.75 inference(skolemize,[status(sab)],[142])).
% 2.22/1.75 tff(144,plain,
% 2.22/1.75 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 2.22/1.75 inference(modus_ponens,[status(thm)],[143, 136])).
% 2.22/1.75 tff(145,plain,
% 2.22/1.75 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))))),
% 2.22/1.75 inference(rewrite,[status(thm)],[])).
% 2.22/1.75 tff(146,plain,
% 2.22/1.75 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))))),
% 2.22/1.75 inference(quant_inst,[status(thm)],[])).
% 2.22/1.75 tff(147,plain,
% 2.22/1.75 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))))),
% 2.22/1.75 inference(modus_ponens,[status(thm)],[146, 145])).
% 2.22/1.75 tff(148,plain,
% 2.22/1.75 (subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = set_difference(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))),
% 2.22/1.75 inference(unit_resolution,[status(thm)],[147, 144, 93])).
% 2.22/1.75 tff(149,plain,
% 2.22/1.75 (subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))) = union_of_subsets(A!302, B!301)),
% 2.22/1.75 inference(transitivity,[status(thm)],[148, 134, 123, 79, 29])).
% 2.22/1.75 tff(150,plain,
% 2.22/1.75 (subset_complement(A!302, subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) = subset_complement(A!302, union_of_subsets(A!302, B!301))),
% 2.22/1.75 inference(monotonicity,[status(thm)],[149])).
% 2.22/1.75 tff(151,plain,
% 2.22/1.75 (^[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)))),
% 2.22/1.75 inference(bind,[status(th)],[])).
% 2.22/1.75 tff(152,plain,
% 2.22/1.75 (![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))),
% 2.22/1.75 inference(quant_intro,[status(thm)],[151])).
% 2.22/1.75 tff(153,plain,
% 2.22/1.75 (![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))),
% 2.22/1.75 inference(rewrite,[status(thm)],[])).
% 2.22/1.75 tff(154,plain,
% 2.22/1.75 (^[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)))),
% 2.22/1.75 inference(bind,[status(th)],[])).
% 2.22/1.75 tff(155,plain,
% 2.22/1.75 (![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))),
% 2.22/1.75 inference(quant_intro,[status(thm)],[154])).
% 2.22/1.75 tff(156,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')).
% 2.22/1.75 tff(157,plain,
% 2.22/1.75 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 2.22/1.75 inference(modus_ponens,[status(thm)],[156, 155])).
% 2.22/1.75 tff(158,plain,
% 2.22/1.75 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 2.22/1.75 inference(modus_ponens,[status(thm)],[157, 153])).
% 2.22/1.75 tff(159,plain,(
% 2.22/1.75 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 2.22/1.75 inference(skolemize,[status(sab)],[158])).
% 2.22/1.75 tff(160,plain,
% 2.22/1.75 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 2.22/1.75 inference(modus_ponens,[status(thm)],[159, 152])).
% 2.22/1.75 tff(161,plain,
% 2.22/1.75 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_complement(A!302, subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) = meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_complement(A!302, subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) = meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))))),
% 2.22/1.75 inference(rewrite,[status(thm)],[])).
% 2.22/1.75 tff(162,plain,
% 2.22/1.75 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_complement(A!302, subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) = meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))))),
% 2.22/1.75 inference(quant_inst,[status(thm)],[])).
% 2.22/1.75 tff(163,plain,
% 2.22/1.75 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)), powerset(A!302))) | (subset_complement(A!302, subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) = meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))),
% 2.22/1.75 inference(modus_ponens,[status(thm)],[162, 161])).
% 2.22/1.75 tff(164,plain,
% 2.22/1.75 (subset_complement(A!302, subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)))) = meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))),
% 2.33/1.77 inference(unit_resolution,[status(thm)],[163, 160, 93])).
% 2.33/1.77 tff(165,plain,
% 2.33/1.77 (meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = subset_complement(A!302, subset_complement(A!302, meet_of_subsets(A!302, complements_of_subsets(A!302, B!301))))),
% 2.33/1.77 inference(symmetry,[status(thm)],[164])).
% 2.33/1.77 tff(166,plain,
% 2.33/1.77 (meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = subset_complement(A!302, union_of_subsets(A!302, B!301))),
% 2.33/1.77 inference(transitivity,[status(thm)],[165, 150])).
% 2.33/1.77 tff(167,plain,
% 2.33/1.77 (~(meet_of_subsets(A!302, complements_of_subsets(A!302, B!301)) = subset_complement(A!302, union_of_subsets(A!302, B!301)))),
% 2.33/1.77 inference(or_elim,[status(thm)],[13])).
% 2.33/1.77 tff(168,plain,
% 2.33/1.77 ($false),
% 2.33/1.77 inference(unit_resolution,[status(thm)],[167, 166])).
% 2.33/1.77 % SZS output end Proof
%------------------------------------------------------------------------------