TSTP Solution File: SEU327+1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SEU327+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n002.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 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SEU327+1 : TPTP v8.1.0. Released v3.3.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n002.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sat Sep 3 12:19:47 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 0.19/0.41 % SZS status Theorem
% 0.19/0.41 % SZS output start Proof
% 0.19/0.41 tff(element_type, type, (
% 0.19/0.41 element: ( $i * $i ) > $o)).
% 0.19/0.41 tff(powerset_type, type, (
% 0.19/0.41 powerset: $i > $i)).
% 0.19/0.41 tff(tptp_fun_A_5_type, type, (
% 0.19/0.41 tptp_fun_A_5: $i)).
% 0.19/0.41 tff(union_type, type, (
% 0.19/0.41 union: $i > $i)).
% 0.19/0.41 tff(tptp_fun_B_4_type, type, (
% 0.19/0.41 tptp_fun_B_4: $i)).
% 0.19/0.41 tff(union_of_subsets_type, type, (
% 0.19/0.41 union_of_subsets: ( $i * $i ) > $i)).
% 0.19/0.41 tff(empty_set_type, type, (
% 0.19/0.41 empty_set: $i)).
% 0.19/0.41 tff(subset_complement_type, type, (
% 0.19/0.41 subset_complement: ( $i * $i ) > $i)).
% 0.19/0.41 tff(meet_of_subsets_type, type, (
% 0.19/0.41 meet_of_subsets: ( $i * $i ) > $i)).
% 0.19/0.41 tff(complements_of_subsets_type, type, (
% 0.19/0.41 complements_of_subsets: ( $i * $i ) > $i)).
% 0.19/0.41 tff(set_difference_type, type, (
% 0.19/0.41 set_difference: ( $i * $i ) > $i)).
% 0.19/0.41 tff(cast_to_subset_type, type, (
% 0.19/0.41 cast_to_subset: $i > $i)).
% 0.19/0.41 tff(subset_difference_type, type, (
% 0.19/0.41 subset_difference: ( $i * $i * $i ) > $i)).
% 0.19/0.41 tff(1,plain,
% 0.19/0.41 ((~((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))))) <=> (~((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(2,plain,
% 0.19/0.41 ((~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))) <=> (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(3,plain,
% 0.19/0.41 ((~![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B)))))) <=> (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A))))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 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')).
% 0.19/0.42 tff(5,plain,
% 0.19/0.42 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.42 tff(6,plain,
% 0.19/0.42 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.19/0.42 tff(7,plain,
% 0.19/0.42 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.42 tff(8,plain,
% 0.19/0.42 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.19/0.42 tff(9,plain,
% 0.19/0.42 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[8, 2])).
% 0.19/0.42 tff(10,plain,
% 0.19/0.42 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[9, 2])).
% 0.19/0.42 tff(11,plain,
% 0.19/0.42 (~![A: $i, B: $i] : ((meet_of_subsets(A, complements_of_subsets(A, B)) = subset_complement(A, union_of_subsets(A, B))) | (B = empty_set) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.19/0.42 tff(12,plain,(
% 0.19/0.42 ~((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))))),
% 0.19/0.42 inference(skolemize,[status(sab)],[11])).
% 0.19/0.42 tff(13,plain,
% 0.19/0.42 (~((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[12, 1])).
% 0.19/0.42 tff(14,plain,
% 0.19/0.42 (element(B!4, powerset(powerset(A!5)))),
% 0.19/0.42 inference(or_elim,[status(thm)],[13])).
% 0.19/0.42 tff(15,plain,
% 0.19/0.42 (^[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))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(16,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[15])).
% 0.19/0.42 tff(17,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(18,plain,
% 0.19/0.42 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => (union_of_subsets(A, B) = union(B))) <=> ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(19,plain,
% 0.19/0.42 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => (union_of_subsets(A, B) = union(B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[18])).
% 0.19/0.42 tff(20,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')).
% 0.19/0.42 tff(21,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.19/0.42 tff(22,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.19/0.42 tff(23,plain,(
% 0.19/0.42 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[22])).
% 0.19/0.42 tff(24,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[23, 16])).
% 0.19/0.42 tff(25,plain,
% 0.19/0.42 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | ((~element(B!4, powerset(powerset(A!5)))) | (union_of_subsets(A!5, B!4) = union(B!4)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | (~element(B!4, powerset(powerset(A!5)))) | (union_of_subsets(A!5, B!4) = union(B!4)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(26,plain,
% 0.19/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | ((~element(B!4, powerset(powerset(A!5)))) | (union_of_subsets(A!5, B!4) = union(B!4)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(27,plain,
% 0.19/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | (union_of_subsets(A, B) = union(B)))) | (~element(B!4, powerset(powerset(A!5)))) | (union_of_subsets(A!5, B!4) = union(B!4))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.19/0.42 tff(28,plain,
% 0.19/0.42 (union_of_subsets(A!5, B!4) = union(B!4)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[27, 24, 14])).
% 0.19/0.42 tff(29,plain,
% 0.19/0.42 (union(B!4) = union_of_subsets(A!5, B!4)),
% 0.19/0.42 inference(symmetry,[status(thm)],[28])).
% 0.19/0.42 tff(30,plain,
% 0.19/0.42 (element(union(B!4), powerset(A!5)) <=> element(union_of_subsets(A!5, B!4), powerset(A!5))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[29])).
% 0.19/0.42 tff(31,plain,
% 0.19/0.42 (element(union_of_subsets(A!5, B!4), powerset(A!5)) <=> element(union(B!4), powerset(A!5))),
% 0.19/0.42 inference(symmetry,[status(thm)],[30])).
% 0.19/0.42 tff(32,plain,
% 0.19/0.42 (^[A: $i, B: $i] : refl(((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(33,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[32])).
% 0.19/0.42 tff(34,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(35,plain,
% 0.19/0.42 (^[A: $i, B: $i] : rewrite((element(B, powerset(powerset(A))) => element(union_of_subsets(A, B), powerset(A))) <=> ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(36,plain,
% 0.19/0.42 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => element(union_of_subsets(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[35])).
% 0.19/0.42 tff(37,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')).
% 0.19/0.42 tff(38,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.19/0.42 tff(39,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[38, 34])).
% 0.19/0.42 tff(40,plain,(
% 0.19/0.42 ![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[39])).
% 0.19/0.42 tff(41,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[40, 33])).
% 0.19/0.42 tff(42,plain,
% 0.19/0.42 (((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | ((~element(B!4, powerset(powerset(A!5)))) | element(union_of_subsets(A!5, B!4), powerset(A!5)))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | (~element(B!4, powerset(powerset(A!5)))) | element(union_of_subsets(A!5, B!4), powerset(A!5)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(43,plain,
% 0.19/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | ((~element(B!4, powerset(powerset(A!5)))) | element(union_of_subsets(A!5, B!4), powerset(A!5)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(44,plain,
% 0.19/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(powerset(A)))) | element(union_of_subsets(A, B), powerset(A)))) | (~element(B!4, powerset(powerset(A!5)))) | element(union_of_subsets(A!5, B!4), powerset(A!5))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.19/0.42 tff(45,plain,
% 0.19/0.42 (element(union_of_subsets(A!5, B!4), powerset(A!5))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[44, 41, 14])).
% 0.19/0.42 tff(46,plain,
% 0.19/0.42 (element(union(B!4), powerset(A!5))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[45, 31])).
% 0.19/0.42 tff(47,plain,
% 0.19/0.42 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))) <=> ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(48,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[47])).
% 0.19/0.42 tff(49,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(50,plain,
% 0.19/0.42 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => (subset_complement(A, B) = set_difference(A, B))) <=> ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(51,plain,
% 0.19/0.42 (![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, B) = set_difference(A, B))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[50])).
% 0.19/0.42 tff(52,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')).
% 0.19/0.42 tff(53,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.19/0.42 tff(54,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.19/0.42 tff(55,plain,(
% 0.19/0.42 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[54])).
% 0.19/0.42 tff(56,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[55, 48])).
% 0.19/0.42 tff(57,plain,
% 0.19/0.42 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(union_of_subsets(A!5, B!4), powerset(A!5))) | (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(union_of_subsets(A!5, B!4), powerset(A!5))) | (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(58,plain,
% 0.19/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | ((~element(union_of_subsets(A!5, B!4), powerset(A!5))) | (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(59,plain,
% 0.19/0.42 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, B) = set_difference(A, B)))) | (~element(union_of_subsets(A!5, B!4), powerset(A!5))) | (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.19/0.42 tff(60,plain,
% 0.19/0.42 (subset_complement(A!5, union_of_subsets(A!5, B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[59, 56, 45])).
% 0.19/0.42 tff(61,plain,
% 0.19/0.42 (set_difference(A!5, union_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))),
% 0.19/0.42 inference(symmetry,[status(thm)],[60])).
% 0.19/0.42 tff(62,plain,
% 0.19/0.42 (^[A: $i] : refl((cast_to_subset(A) = A) <=> (cast_to_subset(A) = A))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(63,plain,
% 0.19/0.42 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 0.19/0.42 inference(quant_intro,[status(thm)],[62])).
% 0.19/0.42 tff(64,plain,
% 0.19/0.42 (![A: $i] : (cast_to_subset(A) = A) <=> ![A: $i] : (cast_to_subset(A) = A)),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(65,axiom,(![A: $i] : (cast_to_subset(A) = A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d4_subset_1')).
% 0.19/0.42 tff(66,plain,
% 0.19/0.42 (![A: $i] : (cast_to_subset(A) = A)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.19/0.42 tff(67,plain,(
% 0.19/0.42 ![A: $i] : (cast_to_subset(A) = A)),
% 0.19/0.42 inference(skolemize,[status(sab)],[66])).
% 0.19/0.42 tff(68,plain,
% 0.19/0.42 (![A: $i] : (cast_to_subset(A) = A)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[67, 63])).
% 0.19/0.42 tff(69,plain,
% 0.19/0.42 ((~![A: $i] : (cast_to_subset(A) = A)) | (cast_to_subset(A!5) = A!5)),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(70,plain,
% 0.19/0.42 (cast_to_subset(A!5) = A!5),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[69, 68])).
% 0.19/0.42 tff(71,plain,
% 0.19/0.42 (set_difference(cast_to_subset(A!5), union(B!4)) = set_difference(A!5, union_of_subsets(A!5, B!4))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[70, 29])).
% 0.19/0.42 tff(72,plain,
% 0.19/0.42 (set_difference(cast_to_subset(A!5), union(B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))),
% 0.19/0.42 inference(transitivity,[status(thm)],[71, 61])).
% 0.19/0.42 tff(73,plain,
% 0.19/0.42 (^[A: $i, B: $i] : refl(((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(74,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[73])).
% 0.19/0.42 tff(75,plain,
% 0.19/0.42 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(76,plain,
% 0.19/0.42 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B)))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))), ((element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> (element(B, powerset(powerset(A))) => ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))))), rewrite((element(B, powerset(powerset(A))) => ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))), ((element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(77,plain,
% 0.19/0.42 (![A: $i, B: $i] : (element(B, powerset(powerset(A))) => ((~(B = empty_set)) => (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))))) <=> ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[76])).
% 0.19/0.42 tff(78,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')).
% 0.19/0.43 tff(79,plain,
% 0.19/0.43 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.19/0.43 tff(80,plain,
% 0.19/0.43 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[79, 75])).
% 0.19/0.43 tff(81,plain,(
% 0.19/0.43 ![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.43 inference(skolemize,[status(sab)],[80])).
% 0.19/0.43 tff(82,plain,
% 0.19/0.43 (![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[81, 74])).
% 0.19/0.43 tff(83,plain,
% 0.19/0.43 (~(B!4 = empty_set)),
% 0.19/0.43 inference(or_elim,[status(thm)],[13])).
% 0.19/0.43 tff(84,plain,
% 0.19/0.43 (((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(85,plain,
% 0.19/0.43 (((B!4 = empty_set) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))) | (~element(B!4, powerset(powerset(A!5))))) <=> ((B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(86,plain,
% 0.19/0.43 (((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))) | (~element(B!4, powerset(powerset(A!5)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[85])).
% 0.19/0.43 tff(87,plain,
% 0.19/0.43 (((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))) | (~element(B!4, powerset(powerset(A!5)))))) <=> ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))))),
% 0.19/0.43 inference(transitivity,[status(thm)],[86, 84])).
% 0.19/0.43 tff(88,plain,
% 0.19/0.43 ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | ((B!4 = empty_set) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))) | (~element(B!4, powerset(powerset(A!5)))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(89,plain,
% 0.19/0.43 ((~![A: $i, B: $i] : ((B = empty_set) | (subset_difference(A, cast_to_subset(A), union_of_subsets(A, B)) = meet_of_subsets(A, complements_of_subsets(A, B))) | (~element(B, powerset(powerset(A)))))) | (B!4 = empty_set) | (~element(B!4, powerset(powerset(A!5)))) | (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.19/0.43 tff(90,plain,
% 0.19/0.43 (subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[89, 83, 14, 82])).
% 0.19/0.43 tff(91,plain,
% 0.19/0.43 (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = subset_difference(A!5, cast_to_subset(A!5), union_of_subsets(A!5, B!4))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[29])).
% 0.19/0.43 tff(92,plain,
% 0.19/0.43 (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = meet_of_subsets(A!5, complements_of_subsets(A!5, B!4))),
% 0.19/0.43 inference(transitivity,[status(thm)],[91, 90])).
% 0.19/0.43 tff(93,plain,
% 0.19/0.43 ((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) <=> (meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4)))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[92, 72])).
% 0.19/0.43 tff(94,plain,
% 0.19/0.43 ((meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4))) <=> (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))),
% 0.19/0.43 inference(symmetry,[status(thm)],[93])).
% 0.19/0.43 tff(95,plain,
% 0.19/0.43 ((~(meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4)))) <=> (~(subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[94])).
% 0.19/0.43 tff(96,plain,
% 0.19/0.43 (~(meet_of_subsets(A!5, complements_of_subsets(A!5, B!4)) = subset_complement(A!5, union_of_subsets(A!5, B!4)))),
% 0.19/0.43 inference(or_elim,[status(thm)],[13])).
% 0.19/0.43 tff(97,plain,
% 0.19/0.43 (~(subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.19/0.43 tff(98,plain,
% 0.19/0.43 (^[A: $i] : refl(element(cast_to_subset(A), powerset(A)) <=> element(cast_to_subset(A), powerset(A)))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(99,plain,
% 0.19/0.43 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[98])).
% 0.19/0.43 tff(100,plain,
% 0.19/0.43 (![A: $i] : element(cast_to_subset(A), powerset(A)) <=> ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(101,axiom,(![A: $i] : element(cast_to_subset(A), powerset(A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k2_subset_1')).
% 0.19/0.43 tff(102,plain,
% 0.19/0.43 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.19/0.43 tff(103,plain,(
% 0.19/0.43 ![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.19/0.43 inference(skolemize,[status(sab)],[102])).
% 0.19/0.43 tff(104,plain,
% 0.19/0.43 (![A: $i] : element(cast_to_subset(A), powerset(A))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[103, 99])).
% 0.19/0.43 tff(105,plain,
% 0.19/0.43 ((~![A: $i] : element(cast_to_subset(A), powerset(A))) | element(cast_to_subset(A!5), powerset(A!5))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(106,plain,
% 0.19/0.43 (element(cast_to_subset(A!5), powerset(A!5))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[105, 104])).
% 0.19/0.43 tff(107,plain,
% 0.19/0.43 (^[A: $i, B: $i, C: $i] : refl(((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A)))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(108,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A)))) <=> ![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[107])).
% 0.19/0.43 tff(109,plain,
% 0.19/0.43 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((element(B, powerset(A)) & element(C, powerset(A))) <=> (~((~element(B, powerset(A))) | (~element(C, powerset(A)))))), ((~(element(B, powerset(A)) & element(C, powerset(A)))) <=> (~(~((~element(B, powerset(A))) | (~element(C, powerset(A)))))))), rewrite((~(~((~element(B, powerset(A))) | (~element(C, powerset(A)))))) <=> ((~element(B, powerset(A))) | (~element(C, powerset(A))))), ((~(element(B, powerset(A)) & element(C, powerset(A)))) <=> ((~element(B, powerset(A))) | (~element(C, powerset(A)))))), (((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> (((~element(B, powerset(A))) | (~element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))))), rewrite((((~element(B, powerset(A))) | (~element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))), (((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(110,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[109])).
% 0.19/0.43 tff(111,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))) <=> ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(112,plain,
% 0.19/0.43 (^[A: $i, B: $i, C: $i] : rewrite(((element(B, powerset(A)) & element(C, powerset(A))) => (subset_difference(A, B, C) = set_difference(B, C))) <=> ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(113,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((element(B, powerset(A)) & element(C, powerset(A))) => (subset_difference(A, B, C) = set_difference(B, C))) <=> ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[112])).
% 0.19/0.43 tff(114,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')).
% 0.19/0.43 tff(115,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[114, 113])).
% 0.19/0.43 tff(116,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[115, 111])).
% 0.19/0.43 tff(117,plain,(
% 0.19/0.43 ![A: $i, B: $i, C: $i] : ((~(element(B, powerset(A)) & element(C, powerset(A)))) | (subset_difference(A, B, C) = set_difference(B, C)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[116])).
% 0.19/0.43 tff(118,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[117, 110])).
% 0.19/0.43 tff(119,plain,
% 0.19/0.43 (![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[118, 108])).
% 0.19/0.43 tff(120,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(121,plain,
% 0.19/0.43 (((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5)))) <=> ((~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(122,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[121])).
% 0.19/0.43 tff(123,plain,
% 0.19/0.43 (((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))))) <=> ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))))),
% 0.19/0.43 inference(transitivity,[status(thm)],[122, 120])).
% 0.19/0.43 tff(124,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | ((subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(125,plain,
% 0.19/0.43 ((~![A: $i, B: $i, C: $i] : ((subset_difference(A, B, C) = set_difference(B, C)) | (~element(B, powerset(A))) | (~element(C, powerset(A))))) | (~element(cast_to_subset(A!5), powerset(A!5))) | (~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.19/0.44 tff(126,plain,
% 0.19/0.44 ((~element(union(B!4), powerset(A!5))) | (subset_difference(A!5, cast_to_subset(A!5), union(B!4)) = set_difference(cast_to_subset(A!5), union(B!4)))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[125, 119, 106])).
% 0.19/0.44 tff(127,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[126, 97, 46])).
% 0.19/0.44 % SZS output end Proof
%------------------------------------------------------------------------------