%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU320+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.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 05:27:35 AM UTC 2025
% Result : Theorem 0.14s 0.40s
% Output : Proof 0.14s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.14 % Problem : SEU320+1 : TPTP v9.0.0. Released v3.3.0.
% 0.13/0.14 % Command : run_E %s %d THM
% 0.14/0.36 % Computer : n011.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 06:44:42 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_3_type, type, (
% 0.14/0.40 tptp_fun_A_3: $i)).
% 0.14/0.40 tff(subset_complement_type, type, (
% 0.14/0.40 subset_complement: ( $i * $i ) > $i)).
% 0.14/0.40 tff(tptp_fun_B_4_type, type, (
% 0.14/0.40 tptp_fun_B_4: $i)).
% 0.14/0.40 tff(the_carrier_type, type, (
% 0.14/0.40 the_carrier: $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(closed_subset_type, type, (
% 0.14/0.40 closed_subset: ( $i * $i ) > $o)).
% 0.14/0.40 tff(top_str_type, type, (
% 0.14/0.40 top_str: $i > $o)).
% 0.14/0.40 tff(1,plain,
% 0.14/0.40 ((~![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)))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(2,plain,
% 0.14/0.40 ((~![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)))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(3,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/sandbox2/benchmark/theBenchmark.p','t30_tops_1')).
% 0.14/0.40 tff(4,plain,
% 0.14/0.40 (~![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))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[3, 2])).
% 0.14/0.40 tff(5,plain,
% 0.14/0.40 (~![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))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[4, 1])).
% 0.14/0.40 tff(6,plain,
% 0.14/0.40 (~![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))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[5, 1])).
% 0.14/0.40 tff(7,plain,
% 0.14/0.40 (~![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))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[6, 1])).
% 0.14/0.40 tff(8,plain,
% 0.14/0.40 (~((~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)))),
% 0.14/0.40 inference(and_elim,[status(thm)],[7])).
% 0.14/0.40 tff(9,plain,
% 0.14/0.40 (element(B!4, powerset(the_carrier(A!3)))),
% 0.14/0.40 inference(or_elim,[status(thm)],[8])).
% 0.14/0.40 tff(10,plain,
% 0.14/0.40 (^[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)))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(11,plain,
% 0.14/0.40 (![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))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[10])).
% 0.14/0.40 tff(12,plain,
% 0.14/0.40 (![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))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(13,plain,
% 0.14/0.40 (^[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)))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(14,plain,
% 0.14/0.40 (![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))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[13])).
% 0.14/0.40 tff(15,axiom,(![A: $i, B: $i] : (element(B, powerset(A)) => (subset_complement(A, subset_complement(A, B)) = B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','involutiveness_k3_subset_1')).
% 0.14/0.40 tff(16,plain,
% 0.14/0.40 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.14/0.40 tff(17,plain,
% 0.14/0.40 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.14/0.40 tff(18,plain,(
% 0.14/0.40 ![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.14/0.40 inference(skolemize,[status(sab)],[17])).
% 0.14/0.40 tff(19,plain,
% 0.14/0.40 (![A: $i, B: $i] : ((~element(B, powerset(A))) | (subset_complement(A, subset_complement(A, B)) = B))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.14/0.40 tff(20,plain,
% 0.14/0.40 (((~![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))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(21,plain,
% 0.14/0.40 ((~![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))),
% 0.14/0.40 inference(quant_inst,[status(thm)],[])).
% 0.14/0.40 tff(22,plain,
% 0.14/0.40 ((~![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)),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.14/0.40 tff(23,plain,
% 0.14/0.40 (subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)) = B!4),
% 0.14/0.40 inference(unit_resolution,[status(thm)],[22, 19, 9])).
% 0.14/0.40 tff(24,plain,
% 0.14/0.40 (open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3) <=> open_subset(B!4, A!3)),
% 0.14/0.40 inference(monotonicity,[status(thm)],[23])).
% 0.14/0.40 tff(25,plain,
% 0.14/0.40 (open_subset(B!4, A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.14/0.40 inference(symmetry,[status(thm)],[24])).
% 0.14/0.40 tff(26,plain,
% 0.14/0.40 ((~open_subset(B!4, A!3)) <=> (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))),
% 0.14/0.40 inference(monotonicity,[status(thm)],[25])).
% 0.14/0.40 tff(27,assumption,(~closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)), introduced(assumption)).
% 0.14/0.40 tff(28,plain,
% 0.14/0.40 (~(open_subset(B!4, A!3) <=> closed_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.14/0.40 inference(or_elim,[status(thm)],[8])).
% 0.14/0.40 tff(29,plain,
% 0.14/0.40 ((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)),
% 0.14/0.40 inference(tautology,[status(thm)],[])).
% 0.14/0.40 tff(30,plain,
% 0.14/0.40 (open_subset(B!4, A!3) | closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)),
% 0.14/0.40 inference(unit_resolution,[status(thm)],[29, 28])).
% 0.14/0.40 tff(31,plain,
% 0.14/0.40 (open_subset(B!4, A!3)),
% 0.14/0.40 inference(unit_resolution,[status(thm)],[30, 27])).
% 0.14/0.40 tff(32,plain,
% 0.14/0.40 (open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[31, 25])).
% 0.14/0.40 tff(33,plain,
% 0.14/0.40 (top_str(A!3)),
% 0.14/0.40 inference(and_elim,[status(thm)],[7])).
% 0.14/0.40 tff(34,plain,
% 0.14/0.40 (^[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)))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(35,plain,
% 0.14/0.40 (![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))))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[34])).
% 0.14/0.40 tff(36,plain,
% 0.14/0.40 (^[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)))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(37,plain,
% 0.14/0.40 (![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))))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[36])).
% 0.14/0.40 tff(38,plain,
% 0.14/0.40 (![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))))),
% 0.14/0.40 inference(transitivity,[status(thm)],[37, 35])).
% 0.14/0.40 tff(39,plain,
% 0.14/0.40 (![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))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(40,plain,
% 0.14/0.40 (^[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))))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(41,plain,
% 0.14/0.40 (![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))))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[40])).
% 0.14/0.40 tff(42,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')).
% 0.14/0.40 tff(43,plain,
% 0.14/0.40 (![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))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.14/0.40 tff(44,plain,
% 0.14/0.40 (![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))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[43, 39])).
% 0.14/0.40 tff(45,plain,(
% 0.14/0.40 ![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))))),
% 0.14/0.40 inference(skolemize,[status(sab)],[44])).
% 0.14/0.40 tff(46,plain,
% 0.14/0.40 (![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))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[45, 38])).
% 0.14/0.40 tff(47,plain,
% 0.14/0.40 (((~![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))))) | ((~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))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(48,plain,
% 0.14/0.40 ((~![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))))) | ((~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))))),
% 0.14/0.40 inference(quant_inst,[status(thm)],[])).
% 0.14/0.40 tff(49,plain,
% 0.14/0.40 ((~![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))))) | (~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)))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.14/0.40 tff(50,plain,
% 0.14/0.40 (![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)))),
% 0.14/0.40 inference(unit_resolution,[status(thm)],[49, 46, 33])).
% 0.14/0.40 tff(51,plain,
% 0.14/0.40 (^[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))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(52,plain,
% 0.14/0.40 (![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)))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[51])).
% 0.14/0.40 tff(53,plain,
% 0.14/0.40 (![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)))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(54,plain,
% 0.14/0.40 (^[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))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(55,plain,
% 0.14/0.40 (![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)],[54])).
% 0.14/0.41 tff(56,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(57,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)],[56, 55])).
% 0.14/0.41 tff(58,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)],[57, 53])).
% 0.14/0.41 tff(59,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)],[58])).
% 0.14/0.41 tff(60,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)],[59, 52])).
% 0.14/0.41 tff(61,plain,
% 0.14/0.41 (((~![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))))),
% 0.14/0.41 inference(rewrite,[status(thm)],[])).
% 0.14/0.41 tff(62,plain,
% 0.14/0.41 ((~![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))))),
% 0.14/0.41 inference(quant_inst,[status(thm)],[])).
% 0.14/0.41 tff(63,plain,
% 0.14/0.41 ((~![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)))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.14/0.41 tff(64,plain,
% 0.14/0.41 (element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3)))),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[63, 60, 9])).
% 0.14/0.41 tff(65,plain,
% 0.14/0.41 (((~![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)))),
% 0.14/0.41 inference(rewrite,[status(thm)],[])).
% 0.14/0.41 tff(66,plain,
% 0.14/0.41 ((~![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)))),
% 0.14/0.41 inference(quant_inst,[status(thm)],[])).
% 0.14/0.41 tff(67,plain,
% 0.14/0.41 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B), A!3)))) | (~element(subset_complement(the_carrier(A!3), B!4), powerset(the_carrier(A!3)))) | (closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[66, 65])).
% 0.14/0.41 tff(68,plain,
% 0.14/0.41 (closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[67, 64, 50])).
% 0.14/0.41 tff(69,plain,
% 0.14/0.41 ((~(closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) <=> open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))) | closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) | (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))),
% 0.14/0.41 inference(tautology,[status(thm)],[])).
% 0.14/0.41 tff(70,plain,
% 0.14/0.41 (closed_subset(subset_complement(the_carrier(A!3), B!4), A!3) | (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3))),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[69, 68])).
% 0.14/0.41 tff(71,plain,
% 0.14/0.41 (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[70, 27])).
% 0.14/0.41 tff(72,plain,
% 0.14/0.41 ($false),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[71, 32])).
% 0.14/0.41 tff(73,plain,(closed_subset(subset_complement(the_carrier(A!3), B!4), A!3)), inference(lemma,lemma(discharge,[]))).
% 0.14/0.41 tff(74,plain,
% 0.14/0.41 ((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))),
% 0.14/0.41 inference(tautology,[status(thm)],[])).
% 0.14/0.41 tff(75,plain,
% 0.14/0.41 ((~open_subset(B!4, A!3)) | (~closed_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[74, 28])).
% 0.14/0.41 tff(76,plain,
% 0.14/0.41 (~open_subset(B!4, A!3)),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[75, 73])).
% 0.14/0.41 tff(77,plain,
% 0.14/0.41 (~open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[76, 26])).
% 0.14/0.41 tff(78,plain,
% 0.14/0.41 ((~(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)),
% 0.14/0.41 inference(tautology,[status(thm)],[])).
% 0.14/0.41 tff(79,plain,
% 0.14/0.41 ((~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)),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[78, 68])).
% 0.14/0.41 tff(80,plain,
% 0.14/0.41 (open_subset(subset_complement(the_carrier(A!3), subset_complement(the_carrier(A!3), B!4)), A!3)),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[79, 73])).
% 0.14/0.41 tff(81,plain,
% 0.14/0.41 ($false),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[80, 77])).
% 0.14/0.41 % SZS output end Proof
% 0.14/0.41 % E exiting
%------------------------------------------------------------------------------