TSTP Solution File: SEU319+1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SEU319+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 07:28:53 EDT 2022
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SEU319+1 : TPTP v8.1.0. Released v3.3.0.
% 0.12/0.14 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.35 % Computer : n002.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat Sep 3 12:14:32 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.20/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.20/0.35 Usage: tptp [options] [-file:]file
% 0.20/0.35 -h, -? prints this message.
% 0.20/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.20/0.35 -m, -model generate model.
% 0.20/0.35 -p, -proof generate proof.
% 0.20/0.35 -c, -core generate unsat core of named formulas.
% 0.20/0.35 -st, -statistics display statistics.
% 0.20/0.35 -t:timeout set timeout (in second).
% 0.20/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.20/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.20/0.35 -<param>:<value> configuration parameter and value.
% 0.20/0.35 -o:<output-file> file to place output in.
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 tff(open_subset_type, type, (
% 0.20/0.40 open_subset: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_A_3_type, type, (
% 0.20/0.40 tptp_fun_A_3: $i)).
% 0.20/0.40 tff(subset_difference_type, type, (
% 0.20/0.40 subset_difference: ( $i * $i * $i ) > $i)).
% 0.20/0.40 tff(tptp_fun_B_4_type, type, (
% 0.20/0.40 tptp_fun_B_4: $i)).
% 0.20/0.40 tff(cast_as_carrier_subset_type, type, (
% 0.20/0.40 cast_as_carrier_subset: $i > $i)).
% 0.20/0.40 tff(the_carrier_type, type, (
% 0.20/0.40 the_carrier: $i > $i)).
% 0.20/0.40 tff(subset_complement_type, type, (
% 0.20/0.40 subset_complement: ( $i * $i ) > $i)).
% 0.20/0.40 tff(element_type, type, (
% 0.20/0.40 element: ( $i * $i ) > $o)).
% 0.20/0.40 tff(powerset_type, type, (
% 0.20/0.40 powerset: $i > $i)).
% 0.20/0.40 tff(one_sorted_str_type, type, (
% 0.20/0.40 one_sorted_str: $i > $o)).
% 0.20/0.40 tff(top_str_type, type, (
% 0.20/0.40 top_str: $i > $o)).
% 0.20/0.40 tff(closed_subset_type, type, (
% 0.20/0.40 closed_subset: ( $i * $i ) > $o)).
% 0.20/0.40 tff(1,plain,
% 0.20/0.40 (((~(~top_str(A!3))) & (~((~element(B!4, powerset(the_carrier(A!3)))) | (closed_subset(B!4, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3))))) <=> (top_str(A!3) & (~((~element(B!4, powerset(the_carrier(A!3)))) | (closed_subset(B!4, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(2,plain,
% 0.20/0.40 ((~![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)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(3,plain,
% 0.20/0.40 ((~![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)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(4,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/sandbox2/benchmark/theBenchmark.p','t29_tops_1')).
% 0.20/0.40 tff(5,plain,
% 0.20/0.40 (~![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))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.40 tff(6,plain,
% 0.20/0.40 (~![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))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.20/0.40 tff(7,plain,
% 0.20/0.40 (~![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))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.40 tff(8,plain,
% 0.20/0.40 (~![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))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.20/0.40 tff(9,plain,
% 0.20/0.40 (~![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))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[8, 2])).
% 0.20/0.40 tff(10,plain,
% 0.20/0.40 (~![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))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[9, 2])).
% 0.20/0.40 tff(11,plain,
% 0.20/0.40 (~![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))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.20/0.40 tff(12,plain,
% 0.20/0.40 (top_str(A!3) & (~((~element(B!4, powerset(the_carrier(A!3)))) | (closed_subset(B!4, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[11, 1])).
% 0.20/0.40 tff(13,plain,
% 0.20/0.40 (top_str(A!3)),
% 0.20/0.40 inference(and_elim,[status(thm)],[12])).
% 0.20/0.40 tff(14,plain,
% 0.20/0.40 (^[A: $i] : refl((one_sorted_str(A) | (~top_str(A))) <=> (one_sorted_str(A) | (~top_str(A))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(15,plain,
% 0.20/0.40 (![A: $i] : (one_sorted_str(A) | (~top_str(A))) <=> ![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[14])).
% 0.20/0.40 tff(16,plain,
% 0.20/0.40 (![A: $i] : (one_sorted_str(A) | (~top_str(A))) <=> ![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(17,plain,
% 0.20/0.40 (^[A: $i] : rewrite((top_str(A) => one_sorted_str(A)) <=> (one_sorted_str(A) | (~top_str(A))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(18,plain,
% 0.20/0.40 (![A: $i] : (top_str(A) => one_sorted_str(A)) <=> ![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[17])).
% 0.20/0.40 tff(19,axiom,(![A: $i] : (top_str(A) => one_sorted_str(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_l1_pre_topc')).
% 0.20/0.40 tff(20,plain,
% 0.20/0.40 (![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[19, 18])).
% 0.20/0.40 tff(21,plain,
% 0.20/0.40 (![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[20, 16])).
% 0.20/0.40 tff(22,plain,(
% 0.20/0.40 ![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[21])).
% 0.20/0.40 tff(23,plain,
% 0.20/0.40 (![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[22, 15])).
% 0.20/0.40 tff(24,plain,
% 0.20/0.40 (((~![A: $i] : (one_sorted_str(A) | (~top_str(A)))) | (one_sorted_str(A!3) | (~top_str(A!3)))) <=> ((~![A: $i] : (one_sorted_str(A) | (~top_str(A)))) | one_sorted_str(A!3) | (~top_str(A!3)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(25,plain,
% 0.20/0.40 ((~![A: $i] : (one_sorted_str(A) | (~top_str(A)))) | (one_sorted_str(A!3) | (~top_str(A!3)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(26,plain,
% 0.20/0.40 ((~![A: $i] : (one_sorted_str(A) | (~top_str(A)))) | one_sorted_str(A!3) | (~top_str(A!3))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.20/0.40 tff(27,plain,
% 0.20/0.40 (one_sorted_str(A!3)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[26, 23, 13])).
% 0.20/0.40 tff(28,plain,
% 0.20/0.40 (^[A: $i] : refl(((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(29,plain,
% 0.20/0.40 (![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> ![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[28])).
% 0.20/0.40 tff(30,plain,
% 0.20/0.40 (^[A: $i] : rewrite(((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(31,plain,
% 0.20/0.40 (![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> ![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[30])).
% 0.20/0.40 tff(32,plain,
% 0.20/0.40 (![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> ![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))),
% 0.20/0.40 inference(transitivity,[status(thm)],[31, 29])).
% 0.20/0.40 tff(33,plain,
% 0.20/0.40 (![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> ![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(34,plain,
% 0.20/0.40 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))) <=> ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))), ((one_sorted_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> (one_sorted_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))))), rewrite((one_sorted_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))), ((one_sorted_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(35,plain,
% 0.20/0.40 (![A: $i] : (one_sorted_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B)))) <=> ![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[34])).
% 0.20/0.40 tff(36,axiom,(![A: $i] : (one_sorted_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t17_pre_topc')).
% 0.20/0.40 tff(37,plain,
% 0.20/0.40 (![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.20/0.40 tff(38,plain,
% 0.20/0.40 (![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.20/0.40 tff(39,plain,(
% 0.20/0.40 ![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[38])).
% 0.20/0.40 tff(40,plain,
% 0.20/0.40 (![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.20/0.40 tff(41,plain,
% 0.20/0.40 (((~![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))) | ((~one_sorted_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B))))) <=> ((~![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))) | (~one_sorted_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(42,plain,
% 0.20/0.40 ((~![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))) | ((~one_sorted_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(43,plain,
% 0.20/0.40 ((~![A: $i] : ((~one_sorted_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (subset_complement(the_carrier(A), B) = subset_difference(the_carrier(A), cast_as_carrier_subset(A), B))))) | (~one_sorted_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.20/0.40 tff(44,plain,
% 0.20/0.40 (![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B)))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[43, 40, 27])).
% 0.20/0.40 tff(45,plain,
% 0.20/0.40 (~((~element(B!4, powerset(the_carrier(A!3)))) | (closed_subset(B!4, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3)))),
% 0.20/0.40 inference(and_elim,[status(thm)],[12])).
% 0.20/0.40 tff(46,plain,
% 0.20/0.40 (element(B!4, powerset(the_carrier(A!3)))),
% 0.20/0.40 inference(or_elim,[status(thm)],[45])).
% 0.20/0.40 tff(47,plain,
% 0.20/0.40 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B)))) | ((~element(B!4, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B!4) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4)))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B)))) | (~element(B!4, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B!4) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(48,plain,
% 0.20/0.40 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B)))) | ((~element(B!4, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B!4) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(49,plain,
% 0.20/0.40 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B)))) | (~element(B!4, powerset(the_carrier(A!3)))) | (subset_complement(the_carrier(A!3), B!4) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.20/0.41 tff(50,plain,
% 0.20/0.41 (subset_complement(the_carrier(A!3), B!4) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[49, 46, 44])).
% 0.20/0.41 tff(51,plain,
% 0.20/0.41 (open_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.20/0.41 inference(monotonicity,[status(thm)],[50])).
% 0.20/0.41 tff(52,plain,
% 0.20/0.41 ((~open_subset(subset_complement(the_carrier(A!3), B!4), A!3)) <=> (~open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[51])).
% 0.20/0.41 tff(53,assumption,(~open_subset(subset_complement(the_carrier(A!3), B!4), A!3)), introduced(assumption)).
% 0.20/0.41 tff(54,plain,
% 0.20/0.41 (~open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.20/0.41 tff(55,plain,
% 0.20/0.41 ((~(closed_subset(B!4, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3))) <=> ((~closed_subset(B!4, A!3)) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(56,plain,
% 0.20/0.41 (~(closed_subset(B!4, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.20/0.41 inference(or_elim,[status(thm)],[45])).
% 0.20/0.41 tff(57,plain,
% 0.20/0.41 ((~closed_subset(B!4, A!3)) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.20/0.41 tff(58,plain,
% 0.20/0.41 (closed_subset(B!4, A!3) | open_subset(subset_complement(the_carrier(A!3), B!4), A!3) | (~((~closed_subset(B!4, A!3)) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3)))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(59,plain,
% 0.20/0.41 (closed_subset(B!4, A!3) | open_subset(subset_complement(the_carrier(A!3), B!4), A!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.20/0.41 tff(60,plain,
% 0.20/0.41 (closed_subset(B!4, A!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[59, 53])).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(62,plain,
% 0.20/0.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[61])).
% 0.20/0.41 tff(63,plain,
% 0.20/0.41 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(64,plain,
% 0.20/0.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[63])).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[64, 62])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(67,plain,
% 0.20/0.41 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))) <=> ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(68,plain,
% 0.20/0.41 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A)))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[67])).
% 0.20/0.41 tff(69,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d6_pre_topc')).
% 0.20/0.41 tff(70,plain,
% 0.20/0.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.20/0.41 tff(71,plain,
% 0.20/0.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[70, 66])).
% 0.20/0.41 tff(72,plain,(
% 0.20/0.41 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[71])).
% 0.20/0.41 tff(73,plain,
% 0.20/0.41 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[72, 65])).
% 0.20/0.41 tff(74,plain,
% 0.20/0.41 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))) | ((~top_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(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_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))) | (~top_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B), A!3))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(75,plain,
% 0.20/0.41 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))) | ((~top_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B), A!3))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(76,plain,
% 0.20/0.41 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (closed_subset(B, A) <=> open_subset(subset_difference(the_carrier(A), cast_as_carrier_subset(A), B), A))))) | (~top_str(A!3)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B), A!3)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.20/0.41 tff(77,plain,
% 0.20/0.41 (![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B), A!3)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[76, 73, 13])).
% 0.20/0.41 tff(78,plain,
% 0.20/0.41 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B), A!3)))) | ((~element(B!4, powerset(the_carrier(A!3)))) | (closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B), A!3)))) | (~element(B!4, powerset(the_carrier(A!3)))) | (closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(79,plain,
% 0.20/0.41 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B), A!3)))) | ((~element(B!4, powerset(the_carrier(A!3)))) | (closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(80,plain,
% 0.20/0.41 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B), A!3)))) | (~element(B!4, powerset(the_carrier(A!3)))) | (closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.20/0.41 tff(81,plain,
% 0.20/0.41 (closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[80, 46, 77])).
% 0.20/0.41 tff(82,plain,
% 0.20/0.41 ((~(closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3))) | (~closed_subset(B!4, A!3)) | open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(83,plain,
% 0.20/0.41 ((~closed_subset(B!4, A!3)) | open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[82, 81])).
% 0.20/0.41 tff(84,plain,
% 0.20/0.41 (open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[83, 60])).
% 0.20/0.41 tff(85,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[84, 54])).
% 0.20/0.41 tff(86,plain,(open_subset(subset_complement(the_carrier(A!3), B!4), A!3)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(87,plain,
% 0.20/0.41 (open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[86, 51])).
% 0.20/0.41 tff(88,plain,
% 0.20/0.41 ((~closed_subset(B!4, A!3)) | (~open_subset(subset_complement(the_carrier(A!3), B!4), A!3)) | (~((~closed_subset(B!4, A!3)) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3)))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(89,plain,
% 0.20/0.41 ((~closed_subset(B!4, A!3)) | (~open_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[88, 57])).
% 0.20/0.41 tff(90,plain,
% 0.20/0.41 (~closed_subset(B!4, A!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[89, 86])).
% 0.20/0.41 tff(91,plain,
% 0.20/0.41 ((~(closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3))) | closed_subset(B!4, A!3) | (~open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(92,plain,
% 0.20/0.41 (closed_subset(B!4, A!3) | (~open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[91, 81])).
% 0.20/0.41 tff(93,plain,
% 0.20/0.41 (~open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[92, 90])).
% 0.20/0.41 tff(94,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[93, 87])).
% 0.20/0.41 % SZS output end Proof
%------------------------------------------------------------------------------