%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU323+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 : Sat Jun 21 05:27:36 AM UTC 2025
% Result : Theorem 0.14s 0.40s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.14 % Problem : SEU323+1 : TPTP v9.0.0. Released v3.3.0.
% 0.07/0.14 % Command : run_E %s %d THM
% 0.14/0.36 % Computer : n015.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Fri Jun 20 06:44:55 EDT 2025
% 0.14/0.36 % CPUTime :
% 0.14/0.40 % SZS status Theorem
% 0.14/0.40 % SZS output start Proof
% 0.14/0.40 tff(open_subset_type, type, (
% 0.14/0.40 open_subset: ( $i * $i ) > $o)).
% 0.14/0.40 tff(tptp_fun_A_5_type, type, (
% 0.14/0.40 tptp_fun_A_5: $i)).
% 0.14/0.40 tff(subset_complement_type, type, (
% 0.14/0.40 subset_complement: ( $i * $i ) > $i)).
% 0.14/0.40 tff(topstr_closure_type, type, (
% 0.14/0.40 topstr_closure: ( $i * $i ) > $i)).
% 0.14/0.40 tff(tptp_fun_B_6_type, type, (
% 0.14/0.40 tptp_fun_B_6: $i)).
% 0.14/0.40 tff(the_carrier_type, type, (
% 0.14/0.40 the_carrier: $i > $i)).
% 0.14/0.40 tff(interior_type, type, (
% 0.14/0.40 interior: ( $i * $i ) > $i)).
% 0.14/0.40 tff(element_type, type, (
% 0.14/0.40 element: ( $i * $i ) > $o)).
% 0.14/0.40 tff(powerset_type, type, (
% 0.14/0.40 powerset: $i > $i)).
% 0.14/0.40 tff(top_str_type, type, (
% 0.14/0.40 top_str: $i > $o)).
% 0.14/0.40 tff(topological_space_type, type, (
% 0.14/0.40 topological_space: $i > $o)).
% 0.14/0.40 tff(closed_subset_type, type, (
% 0.14/0.40 closed_subset: ( $i * $i ) > $o)).
% 0.14/0.40 tff(1,plain,
% 0.14/0.40 (((topological_space(A!5) & top_str(A!5)) & (~((~element(B!6, powerset(the_carrier(A!5)))) | open_subset(interior(A!5, B!6), A!5)))) <=> (topological_space(A!5) & top_str(A!5) & (~((~element(B!6, powerset(the_carrier(A!5)))) | open_subset(interior(A!5, B!6), A!5))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(2,plain,
% 0.14/0.40 ((~![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | open_subset(interior(A, B), A)))) <=> (~![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | open_subset(interior(A, B), A))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(3,plain,
% 0.14/0.40 ((~![A: $i] : ((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(the_carrier(A))) => open_subset(interior(A, B), A)))) <=> (~![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | open_subset(interior(A, B), A))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(4,axiom,(~![A: $i] : ((topological_space(A) & top_str(A)) => ![B: $i] : (element(B, powerset(the_carrier(A))) => open_subset(interior(A, B), A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t51_tops_1')).
% 0.14/0.40 tff(5,plain,
% 0.14/0.40 (~![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | open_subset(interior(A, B), A)))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.14/0.40 tff(6,plain,
% 0.14/0.40 (~![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | open_subset(interior(A, B), A)))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.14/0.40 tff(7,plain,
% 0.14/0.40 (~![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | open_subset(interior(A, B), A)))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.14/0.40 tff(8,plain,
% 0.14/0.40 (~![A: $i] : ((~(topological_space(A) & top_str(A))) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | open_subset(interior(A, B), A)))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.14/0.40 tff(9,plain,
% 0.14/0.40 (topological_space(A!5) & top_str(A!5) & (~((~element(B!6, powerset(the_carrier(A!5)))) | open_subset(interior(A!5, B!6), A!5)))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[8, 1])).
% 0.14/0.40 tff(10,plain,
% 0.14/0.40 (top_str(A!5)),
% 0.14/0.40 inference(and_elim,[status(thm)],[9])).
% 0.14/0.40 tff(11,plain,
% 0.14/0.40 (^[A: $i] : refl(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(12,plain,
% 0.14/0.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[11])).
% 0.14/0.40 tff(13,plain,
% 0.14/0.40 (^[A: $i] : rewrite(((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(14,plain,
% 0.14/0.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[13])).
% 0.14/0.40 tff(15,plain,
% 0.14/0.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))),
% 0.14/0.40 inference(transitivity,[status(thm)],[14, 12])).
% 0.14/0.40 tff(16,plain,
% 0.14/0.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(17,plain,
% 0.14/0.40 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))) <=> ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))), (![B: $i] : (element(B, powerset(the_carrier(A))) => (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))) <=> ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> (top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))))), rewrite((top_str(A) => ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))), ((top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(18,plain,
% 0.14/0.40 (![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B)))))) <=> ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[17])).
% 0.14/0.40 tff(19,axiom,(![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_tops_1')).
% 0.14/0.40 tff(20,plain,
% 0.14/0.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[19, 18])).
% 0.14/0.40 tff(21,plain,
% 0.14/0.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[20, 16])).
% 0.14/0.40 tff(22,plain,(
% 0.14/0.40 ![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))),
% 0.14/0.40 inference(skolemize,[status(sab)],[21])).
% 0.14/0.40 tff(23,plain,
% 0.14/0.40 (![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[22, 15])).
% 0.14/0.40 tff(24,plain,
% 0.14/0.40 (((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))) | ((~top_str(A!5)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | (interior(A!5, B) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B))))))) <=> ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))) | (~top_str(A!5)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | (interior(A!5, B) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B))))))),
% 0.14/0.41 inference(rewrite,[status(thm)],[])).
% 0.14/0.41 tff(25,plain,
% 0.14/0.41 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))) | ((~top_str(A!5)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | (interior(A!5, B) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B))))))),
% 0.14/0.41 inference(quant_inst,[status(thm)],[])).
% 0.14/0.41 tff(26,plain,
% 0.14/0.41 ((~![A: $i] : ((~top_str(A)) | ![B: $i] : ((~element(B, powerset(the_carrier(A)))) | (interior(A, B) = subset_complement(the_carrier(A), topstr_closure(A, subset_complement(the_carrier(A), B))))))) | (~top_str(A!5)) | ![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | (interior(A!5, B) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B)))))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.14/0.41 tff(27,plain,
% 0.14/0.41 (![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | (interior(A!5, B) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B)))))),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[26, 23, 10])).
% 0.14/0.41 tff(28,plain,
% 0.14/0.41 (~((~element(B!6, powerset(the_carrier(A!5)))) | open_subset(interior(A!5, B!6), A!5))),
% 0.14/0.41 inference(and_elim,[status(thm)],[9])).
% 0.14/0.41 tff(29,plain,
% 0.14/0.41 (element(B!6, powerset(the_carrier(A!5)))),
% 0.14/0.41 inference(or_elim,[status(thm)],[28])).
% 0.14/0.41 tff(30,plain,
% 0.14/0.41 (((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | (interior(A!5, B) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B)))))) | ((~element(B!6, powerset(the_carrier(A!5)))) | (interior(A!5, B!6) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)))))) <=> ((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | (interior(A!5, B) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B)))))) | (~element(B!6, powerset(the_carrier(A!5)))) | (interior(A!5, B!6) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)))))),
% 0.14/0.41 inference(rewrite,[status(thm)],[])).
% 0.14/0.41 tff(31,plain,
% 0.14/0.41 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | (interior(A!5, B) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B)))))) | ((~element(B!6, powerset(the_carrier(A!5)))) | (interior(A!5, B!6) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)))))),
% 0.14/0.41 inference(quant_inst,[status(thm)],[])).
% 0.14/0.41 tff(32,plain,
% 0.14/0.41 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!5)))) | (interior(A!5, B) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B)))))) | (~element(B!6, powerset(the_carrier(A!5)))) | (interior(A!5, B!6) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.14/0.41 tff(33,plain,
% 0.14/0.41 (interior(A!5, B!6) = subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)))),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[32, 29, 27])).
% 0.14/0.41 tff(34,plain,
% 0.14/0.41 (subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))) = interior(A!5, B!6)),
% 0.14/0.41 inference(symmetry,[status(thm)],[33])).
% 0.14/0.41 tff(35,plain,
% 0.14/0.41 (open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) <=> open_subset(interior(A!5, B!6), A!5)),
% 0.14/0.41 inference(monotonicity,[status(thm)],[34])).
% 0.14/0.41 tff(36,plain,
% 0.14/0.41 (open_subset(interior(A!5, B!6), A!5) <=> open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5)),
% 0.14/0.41 inference(symmetry,[status(thm)],[35])).
% 0.14/0.41 tff(37,plain,
% 0.14/0.41 ((~open_subset(interior(A!5, B!6), A!5)) <=> (~open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5))),
% 0.14/0.41 inference(monotonicity,[status(thm)],[36])).
% 0.14/0.41 tff(38,plain,
% 0.14/0.41 (~open_subset(interior(A!5, B!6), A!5)),
% 0.14/0.41 inference(or_elim,[status(thm)],[28])).
% 0.14/0.41 tff(39,plain,
% 0.14/0.41 (~open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5)),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.14/0.41 tff(40,plain,
% 0.14/0.41 (^[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))))),
% 0.14/0.41 inference(bind,[status(th)],[])).
% 0.14/0.41 tff(41,plain,
% 0.14/0.41 (![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)))),
% 0.14/0.41 inference(quant_intro,[status(thm)],[40])).
% 0.14/0.41 tff(42,plain,
% 0.14/0.41 (![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)))),
% 0.14/0.41 inference(rewrite,[status(thm)],[])).
% 0.14/0.41 tff(43,plain,
% 0.14/0.41 (^[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))))),
% 0.14/0.41 inference(bind,[status(th)],[])).
% 0.14/0.41 tff(44,plain,
% 0.14/0.41 (![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)))),
% 0.14/0.41 inference(quant_intro,[status(thm)],[43])).
% 0.14/0.41 tff(45,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => element(subset_complement(A, B), powerset(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k3_subset_1')).
% 0.14/0.41 tff(46,plain,
% 0.14/0.41 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.14/0.41 tff(47,plain,
% 0.14/0.41 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.14/0.41 tff(48,plain,(
% 0.14/0.41 ![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.14/0.41 inference(skolemize,[status(sab)],[47])).
% 0.14/0.41 tff(49,plain,
% 0.14/0.41 (![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[48, 41])).
% 0.14/0.41 tff(50,plain,
% 0.14/0.41 (((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(B!6, powerset(the_carrier(A!5)))) | element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5))))) <=> ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(B!6, powerset(the_carrier(A!5)))) | element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5))))),
% 0.14/0.41 inference(rewrite,[status(thm)],[])).
% 0.14/0.41 tff(51,plain,
% 0.14/0.41 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | ((~element(B!6, powerset(the_carrier(A!5)))) | element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5))))),
% 0.14/0.41 inference(quant_inst,[status(thm)],[])).
% 0.14/0.41 tff(52,plain,
% 0.14/0.41 ((~![A: $i, B: $i] : ((~element(B, powerset(A))) | element(subset_complement(A, B), powerset(A)))) | (~element(B!6, powerset(the_carrier(A!5)))) | element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.14/0.41 tff(53,plain,
% 0.14/0.41 (element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[52, 49, 29])).
% 0.14/0.41 tff(54,plain,
% 0.14/0.41 (topological_space(A!5)),
% 0.14/0.41 inference(and_elim,[status(thm)],[9])).
% 0.14/0.41 tff(55,plain,
% 0.14/0.41 (^[A: $i, B: $i] : refl((closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A))) <=> (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A))))),
% 0.14/0.41 inference(bind,[status(th)],[])).
% 0.14/0.41 tff(56,plain,
% 0.14/0.41 (![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A))) <=> ![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))),
% 0.14/0.41 inference(quant_intro,[status(thm)],[55])).
% 0.14/0.41 tff(57,plain,
% 0.14/0.41 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A)))) <=> (~((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A))))), ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) <=> (~(~((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A))))))), rewrite((~(~((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A))))) <=> ((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))), ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) <=> ((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A))))), (((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A)) <=> (((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A))) | closed_subset(topstr_closure(A, B), A)))), rewrite((((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A))) | closed_subset(topstr_closure(A, B), A)) <=> (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))), (((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A)) <=> (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))))),
% 0.14/0.41 inference(bind,[status(th)],[])).
% 0.14/0.41 tff(58,plain,
% 0.14/0.41 (![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A)) <=> ![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))),
% 0.14/0.41 inference(quant_intro,[status(thm)],[57])).
% 0.14/0.41 tff(59,plain,
% 0.14/0.41 (![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A)) <=> ![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A))),
% 0.14/0.41 inference(rewrite,[status(thm)],[])).
% 0.14/0.41 tff(60,plain,
% 0.14/0.41 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((topological_space(A) & top_str(A)) & element(B, powerset(the_carrier(A)))) <=> (topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))), ((((topological_space(A) & top_str(A)) & element(B, powerset(the_carrier(A)))) => closed_subset(topstr_closure(A, B), A)) <=> ((topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A)))) => closed_subset(topstr_closure(A, B), A)))), rewrite(((topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A)))) => closed_subset(topstr_closure(A, B), A)) <=> ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A))), ((((topological_space(A) & top_str(A)) & element(B, powerset(the_carrier(A)))) => closed_subset(topstr_closure(A, B), A)) <=> ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A))))),
% 0.14/0.41 inference(bind,[status(th)],[])).
% 0.14/0.41 tff(61,plain,
% 0.14/0.41 (![A: $i, B: $i] : (((topological_space(A) & top_str(A)) & element(B, powerset(the_carrier(A)))) => closed_subset(topstr_closure(A, B), A)) <=> ![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A))),
% 0.14/0.41 inference(quant_intro,[status(thm)],[60])).
% 0.14/0.41 tff(62,axiom,(![A: $i, B: $i] : (((topological_space(A) & top_str(A)) & element(B, powerset(the_carrier(A)))) => closed_subset(topstr_closure(A, B), A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','fc2_tops_1')).
% 0.14/0.41 tff(63,plain,
% 0.14/0.41 (![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.14/0.41 tff(64,plain,
% 0.14/0.41 (![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[63, 59])).
% 0.14/0.41 tff(65,plain,(
% 0.14/0.41 ![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & element(B, powerset(the_carrier(A))))) | closed_subset(topstr_closure(A, B), A))),
% 0.14/0.41 inference(skolemize,[status(sab)],[64])).
% 0.14/0.41 tff(66,plain,
% 0.14/0.41 (![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[65, 58])).
% 0.14/0.41 tff(67,plain,
% 0.14/0.41 (![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[66, 56])).
% 0.14/0.41 tff(68,plain,
% 0.14/0.41 (((~![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))) | ((~topological_space(A!5)) | (~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5))) <=> ((~![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(69,plain,
% 0.22/0.41 ((closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | (~topological_space(A!5)) | (~top_str(A!5))) <=> ((~topological_space(A!5)) | (~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(70,plain,
% 0.22/0.41 (((~![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))) | (closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | (~topological_space(A!5)) | (~top_str(A!5)))) <=> ((~![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))) | ((~topological_space(A!5)) | (~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)))),
% 0.22/0.41 inference(monotonicity,[status(thm)],[69])).
% 0.22/0.41 tff(71,plain,
% 0.22/0.41 (((~![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))) | (closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | (~topological_space(A!5)) | (~top_str(A!5)))) <=> ((~![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5))),
% 0.22/0.41 inference(transitivity,[status(thm)],[70, 68])).
% 0.22/0.41 tff(72,plain,
% 0.22/0.41 ((~![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))) | (closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | (~topological_space(A!5)) | (~top_str(A!5)))),
% 0.22/0.41 inference(quant_inst,[status(thm)],[])).
% 0.22/0.41 tff(73,plain,
% 0.22/0.41 ((~![A: $i, B: $i] : (closed_subset(topstr_closure(A, B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.22/0.41 tff(74,plain,
% 0.22/0.41 (closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[73, 67, 54, 10, 53])).
% 0.22/0.41 tff(75,plain,
% 0.22/0.41 (^[A: $i, B: $i] : refl((element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A))) <=> (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(76,plain,
% 0.22/0.41 (![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A))) <=> ![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[75])).
% 0.22/0.41 tff(77,plain,
% 0.22/0.41 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((top_str(A) & element(B, powerset(the_carrier(A)))) <=> (~((~top_str(A)) | (~element(B, powerset(the_carrier(A))))))), ((~(top_str(A) & element(B, powerset(the_carrier(A))))) <=> (~(~((~top_str(A)) | (~element(B, powerset(the_carrier(A))))))))), rewrite((~(~((~top_str(A)) | (~element(B, powerset(the_carrier(A))))))) <=> ((~top_str(A)) | (~element(B, powerset(the_carrier(A)))))), ((~(top_str(A) & element(B, powerset(the_carrier(A))))) <=> ((~top_str(A)) | (~element(B, powerset(the_carrier(A))))))), (((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> (((~top_str(A)) | (~element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))))), rewrite((((~top_str(A)) | (~element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))), (((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(78,plain,
% 0.22/0.41 (![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> ![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[77])).
% 0.22/0.41 tff(79,plain,
% 0.22/0.41 (![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> ![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(80,plain,
% 0.22/0.41 (^[A: $i, B: $i] : rewrite(((top_str(A) & element(B, powerset(the_carrier(A)))) => element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A)))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(81,plain,
% 0.22/0.41 (![A: $i, B: $i] : ((top_str(A) & element(B, powerset(the_carrier(A)))) => element(topstr_closure(A, B), powerset(the_carrier(A)))) <=> ![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[80])).
% 0.22/0.41 tff(82,axiom,(![A: $i, B: $i] : ((top_str(A) & element(B, powerset(the_carrier(A)))) => element(topstr_closure(A, B), powerset(the_carrier(A))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','dt_k6_pre_topc')).
% 0.22/0.41 tff(83,plain,
% 0.22/0.41 (![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.22/0.41 tff(84,plain,
% 0.22/0.41 (![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[83, 79])).
% 0.22/0.41 tff(85,plain,(
% 0.22/0.41 ![A: $i, B: $i] : ((~(top_str(A) & element(B, powerset(the_carrier(A))))) | element(topstr_closure(A, B), powerset(the_carrier(A))))),
% 0.22/0.41 inference(skolemize,[status(sab)],[84])).
% 0.22/0.41 tff(86,plain,
% 0.22/0.41 (![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[85, 78])).
% 0.22/0.41 tff(87,plain,
% 0.22/0.41 (![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[86, 76])).
% 0.22/0.41 tff(88,plain,
% 0.22/0.41 (((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | ((~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5))))) <=> ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(89,plain,
% 0.22/0.42 ((element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5))) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | (~top_str(A!5))) <=> ((~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(90,plain,
% 0.22/0.42 (((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5))) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | (~top_str(A!5)))) <=> ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | ((~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))))),
% 0.22/0.42 inference(monotonicity,[status(thm)],[89])).
% 0.22/0.42 tff(91,plain,
% 0.22/0.42 (((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5))) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | (~top_str(A!5)))) <=> ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5))))),
% 0.22/0.42 inference(transitivity,[status(thm)],[90, 88])).
% 0.22/0.42 tff(92,plain,
% 0.22/0.42 ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5))) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | (~top_str(A!5)))),
% 0.22/0.42 inference(quant_inst,[status(thm)],[])).
% 0.22/0.42 tff(93,plain,
% 0.22/0.42 ((~![A: $i, B: $i] : (element(topstr_closure(A, B), powerset(the_carrier(A))) | (~element(B, powerset(the_carrier(A)))) | (~top_str(A)))) | (~top_str(A!5)) | (~element(subset_complement(the_carrier(A!5), B!6), powerset(the_carrier(A!5)))) | element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.22/0.42 tff(94,plain,
% 0.22/0.42 (element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[93, 87, 10, 53])).
% 0.22/0.42 tff(95,plain,
% 0.22/0.42 (^[A: $i, B: $i] : refl((open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A))) <=> (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(96,plain,
% 0.22/0.42 (![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A))) <=> ![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[95])).
% 0.22/0.42 tff(97,plain,
% 0.22/0.42 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A)))) <=> (~((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A))))), ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) <=> (~(~((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A))))))), rewrite((~(~((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A))))) <=> ((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))), ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) <=> ((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A))))), (((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A)) <=> (((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A))) | open_subset(subset_complement(the_carrier(A), B), A)))), rewrite((((~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A))) | open_subset(subset_complement(the_carrier(A), B), A)) <=> (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))), (((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A)) <=> (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(98,plain,
% 0.22/0.42 (![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A)) <=> ![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[97])).
% 0.22/0.42 tff(99,plain,
% 0.22/0.42 (![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A)) <=> ![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(100,plain,
% 0.22/0.42 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite(((topological_space(A) & top_str(A)) & closed_subset(B, A)) <=> (topological_space(A) & top_str(A) & closed_subset(B, A))), ((((topological_space(A) & top_str(A)) & closed_subset(B, A)) & element(B, powerset(the_carrier(A)))) <=> ((topological_space(A) & top_str(A) & closed_subset(B, A)) & element(B, powerset(the_carrier(A)))))), rewrite(((topological_space(A) & top_str(A) & closed_subset(B, A)) & element(B, powerset(the_carrier(A)))) <=> (topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))), ((((topological_space(A) & top_str(A)) & closed_subset(B, A)) & element(B, powerset(the_carrier(A)))) <=> (topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A)))))), (((((topological_space(A) & top_str(A)) & closed_subset(B, A)) & element(B, powerset(the_carrier(A)))) => open_subset(subset_complement(the_carrier(A), B), A)) <=> ((topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A)))) => open_subset(subset_complement(the_carrier(A), B), A)))), rewrite(((topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A)))) => open_subset(subset_complement(the_carrier(A), B), A)) <=> ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A))), (((((topological_space(A) & top_str(A)) & closed_subset(B, A)) & element(B, powerset(the_carrier(A)))) => open_subset(subset_complement(the_carrier(A), B), A)) <=> ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(101,plain,
% 0.22/0.42 (![A: $i, B: $i] : ((((topological_space(A) & top_str(A)) & closed_subset(B, A)) & element(B, powerset(the_carrier(A)))) => open_subset(subset_complement(the_carrier(A), B), A)) <=> ![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[100])).
% 0.22/0.42 tff(102,axiom,(![A: $i, B: $i] : ((((topological_space(A) & top_str(A)) & closed_subset(B, A)) & element(B, powerset(the_carrier(A)))) => open_subset(subset_complement(the_carrier(A), B), A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','fc3_tops_1')).
% 0.22/0.42 tff(103,plain,
% 0.22/0.42 (![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.22/0.42 tff(104,plain,
% 0.22/0.42 (![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[103, 99])).
% 0.22/0.42 tff(105,plain,(
% 0.22/0.42 ![A: $i, B: $i] : ((~(topological_space(A) & top_str(A) & closed_subset(B, A) & element(B, powerset(the_carrier(A))))) | open_subset(subset_complement(the_carrier(A), B), A))),
% 0.22/0.42 inference(skolemize,[status(sab)],[104])).
% 0.22/0.42 tff(106,plain,
% 0.22/0.42 (![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[105, 98])).
% 0.22/0.42 tff(107,plain,
% 0.22/0.42 (![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[106, 96])).
% 0.22/0.42 tff(108,plain,
% 0.22/0.42 (((~![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))) | ((~topological_space(A!5)) | (~top_str(A!5)) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)))) <=> ((~![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(109,plain,
% 0.22/0.42 ((open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5))) <=> ((~topological_space(A!5)) | (~top_str(A!5)) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(110,plain,
% 0.22/0.42 (((~![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))) | (open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)))) <=> ((~![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))) | ((~topological_space(A!5)) | (~top_str(A!5)) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5))))),
% 0.22/0.42 inference(monotonicity,[status(thm)],[109])).
% 0.22/0.42 tff(111,plain,
% 0.22/0.42 (((~![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))) | (open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)))) <=> ((~![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)))),
% 0.22/0.42 inference(transitivity,[status(thm)],[110, 108])).
% 0.22/0.42 tff(112,plain,
% 0.22/0.42 ((~![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))) | (open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5)))),
% 0.22/0.42 inference(quant_inst,[status(thm)],[])).
% 0.22/0.42 tff(113,plain,
% 0.22/0.42 ((~![A: $i, B: $i] : (open_subset(subset_complement(the_carrier(A), B), A) | (~element(B, powerset(the_carrier(A)))) | (~topological_space(A)) | (~top_str(A)) | (~closed_subset(B, A)))) | (~topological_space(A!5)) | (~top_str(A!5)) | (~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[112, 111])).
% 0.22/0.43 tff(114,plain,
% 0.22/0.43 ((~element(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), powerset(the_carrier(A!5)))) | open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5) | (~closed_subset(topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6)), A!5))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[113, 107, 54, 10])).
% 0.22/0.43 tff(115,plain,
% 0.22/0.43 (open_subset(subset_complement(the_carrier(A!5), topstr_closure(A!5, subset_complement(the_carrier(A!5), B!6))), A!5)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[114, 94, 74])).
% 0.22/0.43 tff(116,plain,
% 0.22/0.43 ($false),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[115, 39])).
% 0.22/0.43 % SZS output end Proof
% 0.22/0.43 % E exiting
%------------------------------------------------------------------------------