TSTP Solution File: SEU320+1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SEU320+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n006.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 07:28:53 EDT 2022
% Result : Theorem 0.20s 0.38s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : SEU320+1 : TPTP v8.1.0. Released v3.3.0.
% 0.06/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.33 % Computer : n006.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
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Sat Sep 3 12:02:08 EDT 2022
% 0.14/0.33 % 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.
% 0.20/0.38 % SZS status Theorem
% 0.20/0.38 % SZS output start Proof
% 0.20/0.38 tff(open_subset_type, type, (
% 0.20/0.38 open_subset: ( $i * $i ) > $o)).
% 0.20/0.38 tff(tptp_fun_A_3_type, type, (
% 0.20/0.38 tptp_fun_A_3: $i)).
% 0.20/0.38 tff(subset_complement_type, type, (
% 0.20/0.38 subset_complement: ( $i * $i ) > $i)).
% 0.20/0.38 tff(tptp_fun_B_4_type, type, (
% 0.20/0.38 tptp_fun_B_4: $i)).
% 0.20/0.38 tff(the_carrier_type, type, (
% 0.20/0.38 the_carrier: $i > $i)).
% 0.20/0.38 tff(element_type, type, (
% 0.20/0.38 element: ( $i * $i ) > $o)).
% 0.20/0.38 tff(powerset_type, type, (
% 0.20/0.38 powerset: $i > $i)).
% 0.20/0.38 tff(closed_subset_type, type, (
% 0.20/0.38 closed_subset: ( $i * $i ) > $o)).
% 0.20/0.38 tff(top_str_type, type, (
% 0.20/0.38 top_str: $i > $o)).
% 0.20/0.38 tff(1,plain,
% 0.20/0.38 (((~(~top_str(A!3))) & (~((~element(B!4, powerset(the_carrier(A!3)))) | (open_subset(B!4, A!3) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3))))) <=> (top_str(A!3) & (~((~element(B!4, powerset(the_carrier(A!3)))) | (open_subset(B!4, A!3) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(2,plain,
% 0.20/0.38 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) <=> (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(3,plain,
% 0.20/0.38 ((~![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))) <=> (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A)))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(4,axiom,(~![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t30_tops_1')).
% 0.20/0.38 tff(5,plain,
% 0.20/0.38 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.38 tff(6,plain,
% 0.20/0.38 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.20/0.38 tff(7,plain,
% 0.20/0.38 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.38 tff(8,plain,
% 0.20/0.38 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.20/0.38 tff(9,plain,
% 0.20/0.38 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[8, 2])).
% 0.20/0.38 tff(10,plain,
% 0.20/0.38 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[9, 2])).
% 0.20/0.38 tff(11,plain,
% 0.20/0.38 (~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (open_subset(B, A) <=> closed_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.20/0.38 tff(12,plain,
% 0.20/0.38 (top_str(A!3) & (~((~element(B!4, powerset(the_carrier(A!3)))) | (open_subset(B!4, A!3) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[11, 1])).
% 0.20/0.38 tff(13,plain,
% 0.20/0.38 (~((~element(B!4, powerset(the_carrier(A!3)))) | (open_subset(B!4, A!3) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)))),
% 0.20/0.38 inference(and_elim,[status(thm)],[12])).
% 0.20/0.38 tff(14,plain,
% 0.20/0.38 (element(B!4, powerset(the_carrier(A!3)))),
% 0.20/0.38 inference(or_elim,[status(thm)],[13])).
% 0.20/0.38 tff(15,plain,
% 0.20/0.38 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(16,plain,
% 0.20/0.38 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[15])).
% 0.20/0.38 tff(17,plain,
% 0.20/0.38 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(18,plain,
% 0.20/0.38 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B)) <=> ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B)))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(19,plain,
% 0.20/0.38 (![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B)) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[18])).
% 0.20/0.38 tff(20,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','involutiveness_k3_subset_1')).
% 0.20/0.38 tff(21,plain,
% 0.20/0.38 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.20/0.38 tff(22,plain,
% 0.20/0.38 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.20/0.38 tff(23,plain,(
% 0.20/0.38 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.20/0.38 inference(skolemize,[status(sab)],[22])).
% 0.20/0.38 tff(24,plain,
% 0.20/0.38 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[23, 16])).
% 0.20/0.38 tff(25,plain,
% 0.20/0.38 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(B!4, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)) = B!4))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(B!4, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)) = B!4))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(26,plain,
% 0.20/0.38 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | ((~element(B!4, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)) = B!4))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(27,plain,
% 0.20/0.38 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))) | (~element(B!4, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)) = B!4)),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.20/0.38 tff(28,plain,
% 0.20/0.38 (subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)) = B!4),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[27, 24, 14])).
% 0.20/0.38 tff(29,plain,
% 0.20/0.38 (open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3) <=> open_subset(B!4, A!3)),
% 0.20/0.38 inference(monotonicity,[status(thm)],[28])).
% 0.20/0.38 tff(30,plain,
% 0.20/0.38 (open_subset(B!4, A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.20/0.38 inference(symmetry,[status(thm)],[29])).
% 0.20/0.38 tff(31,plain,
% 0.20/0.38 ((~open_subset(B!4, A!3)) <=> (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))),
% 0.20/0.38 inference(monotonicity,[status(thm)],[30])).
% 0.20/0.38 tff(32,assumption,(~closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)), introduced(assumption)).
% 0.20/0.38 tff(33,plain,
% 0.20/0.38 ((~(open_subset(B!4, A!3) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3))) <=> ((~open_subset(B!4, A!3)) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(34,plain,
% 0.20/0.38 (~(open_subset(B!4, A!3) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.20/0.38 inference(or_elim,[status(thm)],[13])).
% 0.20/0.38 tff(35,plain,
% 0.20/0.38 ((~open_subset(B!4, A!3)) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.20/0.38 tff(36,plain,
% 0.20/0.38 (open_subset(B!4, A!3) | closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) | (~((~open_subset(B!4, A!3)) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)))),
% 0.20/0.38 inference(tautology,[status(thm)],[])).
% 0.20/0.38 tff(37,plain,
% 0.20/0.38 (open_subset(B!4, A!3) | closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[36, 35])).
% 0.20/0.38 tff(38,plain,
% 0.20/0.38 (open_subset(B!4, A!3)),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[37, 32])).
% 0.20/0.38 tff(39,plain,
% 0.20/0.38 (open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[38, 30])).
% 0.20/0.38 tff(40,plain,
% 0.20/0.38 (top_str(A!3)),
% 0.20/0.38 inference(and_elim,[status(thm)],[12])).
% 0.20/0.38 tff(41,plain,
% 0.20/0.38 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(42,plain,
% 0.20/0.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[41])).
% 0.20/0.38 tff(43,plain,
% 0.20/0.38 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(44,plain,
% 0.20/0.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[43])).
% 0.20/0.38 tff(45,plain,
% 0.20/0.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(transitivity,[status(thm)],[44, 42])).
% 0.20/0.38 tff(46,plain,
% 0.20/0.38 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(47,plain,
% 0.20/0.38 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))) <=> ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(48,plain,
% 0.20/0.39 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[47])).
% 0.20/0.39 tff(49,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t29_tops_1')).
% 0.20/0.39 tff(50,plain,
% 0.20/0.39 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.20/0.39 tff(51,plain,
% 0.20/0.39 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[50, 46])).
% 0.20/0.39 tff(52,plain,(
% 0.20/0.39 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.39 inference(skolemize,[status(sab)],[51])).
% 0.20/0.39 tff(53,plain,
% 0.20/0.39 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[52, 45])).
% 0.20/0.39 tff(54,plain,
% 0.20/0.39 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(55,plain,
% 0.20/0.39 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | ((~top_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(56,plain,
% 0.20/0.39 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))) | (~top_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.20/0.39 tff(57,plain,
% 0.20/0.39 (![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3)))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[56, 53, 40])).
% 0.20/0.39 tff(58,plain,
% 0.20/0.39 (^[A: $i, B: $i] : refl(((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(59,plain,
% 0.20/0.39 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[58])).
% 0.20/0.39 tff(60,plain,
% 0.20/0.39 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(61,plain,
% 0.20/0.39 (^[A: $i, B: $i] : rewrite((element(B, powerset(A)) => element(subset_complement(A, B), powerset(A))) <=> ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(62,plain,
% 0.20/0.39 (![A: $i, B: $i] : (element(B, powerset(A)) => element(subset_complement(A, B), powerset(A))) <=> ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[61])).
% 0.20/0.39 tff(63,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => element(subset_complement(A, B), powerset(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k3_subset_1')).
% 0.20/0.39 tff(64,plain,
% 0.20/0.39 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[63, 62])).
% 0.20/0.39 tff(65,plain,
% 0.20/0.39 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[64, 60])).
% 0.20/0.39 tff(66,plain,(
% 0.20/0.39 ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[65])).
% 0.20/0.39 tff(67,plain,
% 0.20/0.39 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[66, 59])).
% 0.20/0.39 tff(68,plain,
% 0.20/0.39 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(B!4, powerset(the_carrier(A!3)))) | element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(B!4, powerset(the_carrier(A!3)))) | element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(69,plain,
% 0.20/0.39 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(B!4, powerset(the_carrier(A!3)))) | element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(70,plain,
% 0.20/0.39 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(B!4, powerset(the_carrier(A!3)))) | element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.20/0.39 tff(71,plain,
% 0.20/0.39 (element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3)))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[70, 67, 14])).
% 0.20/0.39 tff(72,plain,
% 0.20/0.39 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3)))) | ((~element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3)))) | (closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3)))) | (~element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3)))) | (closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(73,plain,
% 0.20/0.39 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3)))) | ((~element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3)))) | (closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(74,plain,
% 0.20/0.39 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3)))) | (~element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3)))) | (closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.20/0.39 tff(75,plain,
% 0.20/0.39 (closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[74, 71, 57])).
% 0.20/0.39 tff(76,plain,
% 0.20/0.39 ((~(closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))) | closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) | (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(77,plain,
% 0.20/0.39 (closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) | (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.20/0.39 tff(78,plain,
% 0.20/0.39 (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[77, 32])).
% 0.20/0.39 tff(79,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[78, 39])).
% 0.20/0.39 tff(80,plain,(closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(81,plain,
% 0.20/0.39 ((~open_subset(B!4, A!3)) | (~closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)) | (~((~open_subset(B!4, A!3)) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)))),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(82,plain,
% 0.20/0.39 ((~open_subset(B!4, A!3)) | (~closed_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[81, 35])).
% 0.20/0.39 tff(83,plain,
% 0.20/0.39 (~open_subset(B!4, A!3)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[82, 80])).
% 0.20/0.39 tff(84,plain,
% 0.20/0.39 (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[83, 31])).
% 0.20/0.39 tff(85,plain,
% 0.20/0.39 ((~(closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))) | (~closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)) | open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.20/0.39 inference(tautology,[status(thm)],[])).
% 0.20/0.39 tff(86,plain,
% 0.20/0.39 ((~closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)) | open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[85, 75])).
% 0.20/0.39 tff(87,plain,
% 0.20/0.39 (open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[86, 80])).
% 0.20/0.39 tff(88,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[87, 84])).
% 0.20/0.39 % SZS output end Proof
%------------------------------------------------------------------------------