%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU319+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.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 0.12s 0.37s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SEU319+1 : TPTP v9.0.0. Released v3.3.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n008.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 06:44:33 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.37 % SZS status Theorem
% 0.12/0.37 % SZS output start Proof
% 0.12/0.37 tff(open_subset_type, type, (
% 0.12/0.37 open_subset: ( $i * $i ) > $o)).
% 0.12/0.37 tff(tptp_fun_A_3_type, type, (
% 0.12/0.37 tptp_fun_A_3: $i)).
% 0.12/0.37 tff(subset_difference_type, type, (
% 0.12/0.37 subset_difference: ( $i * $i * $i ) > $i)).
% 0.12/0.37 tff(tptp_fun_B_4_type, type, (
% 0.12/0.37 tptp_fun_B_4: $i)).
% 0.12/0.37 tff(cast_as_carrier_subset_type, type, (
% 0.12/0.37 cast_as_carrier_subset: $i > $i)).
% 0.12/0.37 tff(the_carrier_type, type, (
% 0.12/0.37 the_carrier: $i > $i)).
% 0.12/0.37 tff(subset_complement_type, type, (
% 0.12/0.37 subset_complement: ( $i * $i ) > $i)).
% 0.12/0.37 tff(element_type, type, (
% 0.12/0.37 element: ( $i * $i ) > $o)).
% 0.12/0.37 tff(powerset_type, type, (
% 0.12/0.37 powerset: $i > $i)).
% 0.12/0.37 tff(one_sorted_str_type, type, (
% 0.12/0.37 one_sorted_str: $i > $o)).
% 0.12/0.37 tff(top_str_type, type, (
% 0.12/0.37 top_str: $i > $o)).
% 0.12/0.37 tff(closed_subset_type, type, (
% 0.12/0.37 closed_subset: ( $i * $i ) > $o)).
% 0.12/0.37 tff(1,plain,
% 0.12/0.37 ((~![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)))))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(2,plain,
% 0.12/0.37 ((~![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)))))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(3,axiom,(~![A: $i] : (top_str(A) => ![B: $i] : (element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_complement(the_carrier(A), B), A))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t29_tops_1')).
% 0.12/0.37 tff(4,plain,
% 0.12/0.37 (~![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))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[3, 2])).
% 0.12/0.37 tff(5,plain,
% 0.12/0.37 (~![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))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[4, 1])).
% 0.12/0.37 tff(6,plain,
% 0.12/0.37 (~![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))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[5, 1])).
% 0.12/0.37 tff(7,plain,
% 0.12/0.37 (~![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))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[6, 1])).
% 0.12/0.37 tff(8,plain,
% 0.12/0.37 (top_str(A!3)),
% 0.12/0.37 inference(and_elim,[status(thm)],[7])).
% 0.12/0.37 tff(9,plain,
% 0.12/0.37 (^[A: $i] : refl((one_sorted_str(A) | (~top_str(A))) <=> (one_sorted_str(A) | (~top_str(A))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(10,plain,
% 0.12/0.37 (![A: $i] : (one_sorted_str(A) | (~top_str(A))) <=> ![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[9])).
% 0.12/0.37 tff(11,plain,
% 0.12/0.37 (![A: $i] : (one_sorted_str(A) | (~top_str(A))) <=> ![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(12,plain,
% 0.12/0.37 (^[A: $i] : rewrite((top_str(A) => one_sorted_str(A)) <=> (one_sorted_str(A) | (~top_str(A))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(13,plain,
% 0.12/0.37 (![A: $i] : (top_str(A) => one_sorted_str(A)) <=> ![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[12])).
% 0.12/0.37 tff(14,axiom,(![A: $i] : (top_str(A) => one_sorted_str(A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_l1_pre_topc')).
% 0.12/0.37 tff(15,plain,
% 0.12/0.37 (![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.12/0.37 tff(16,plain,
% 0.12/0.37 (![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.12/0.37 tff(17,plain,(
% 0.12/0.37 ![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.12/0.38 inference(skolemize,[status(sab)],[16])).
% 0.12/0.38 tff(18,plain,
% 0.12/0.38 (![A: $i] : (one_sorted_str(A) | (~top_str(A)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[17, 10])).
% 0.12/0.38 tff(19,plain,
% 0.12/0.38 (((~![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)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(20,plain,
% 0.12/0.38 ((~![A: $i] : (one_sorted_str(A) | (~top_str(A)))) | (one_sorted_str(A!3) | (~top_str(A!3)))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(21,plain,
% 0.12/0.38 ((~![A: $i] : (one_sorted_str(A) | (~top_str(A)))) | one_sorted_str(A!3) | (~top_str(A!3))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.12/0.38 tff(22,plain,
% 0.12/0.38 (one_sorted_str(A!3)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[21, 18, 8])).
% 0.12/0.38 tff(23,plain,
% 0.12/0.38 (^[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)))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(24,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[23])).
% 0.12/0.38 tff(25,plain,
% 0.12/0.38 (^[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)))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(26,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[25])).
% 0.12/0.38 tff(27,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[26, 24])).
% 0.12/0.38 tff(28,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(29,plain,
% 0.12/0.38 (^[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))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(30,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[29])).
% 0.12/0.38 tff(31,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/sandbox/benchmark/theBenchmark.p','t17_pre_topc')).
% 0.12/0.38 tff(32,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.12/0.38 tff(33,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.12/0.38 tff(34,plain,(
% 0.12/0.38 ![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))))),
% 0.12/0.38 inference(skolemize,[status(sab)],[33])).
% 0.12/0.38 tff(35,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[34, 27])).
% 0.12/0.38 tff(36,plain,
% 0.12/0.38 (((~![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))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(37,plain,
% 0.12/0.38 ((~![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))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(38,plain,
% 0.12/0.38 ((~![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)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.12/0.38 tff(39,plain,
% 0.12/0.38 (![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)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[38, 35, 22])).
% 0.12/0.38 tff(40,plain,
% 0.12/0.38 (~((~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)))),
% 0.12/0.38 inference(and_elim,[status(thm)],[7])).
% 0.12/0.38 tff(41,plain,
% 0.12/0.38 (element(B!4, powerset(the_carrier(A!3)))),
% 0.12/0.38 inference(or_elim,[status(thm)],[40])).
% 0.12/0.38 tff(42,plain,
% 0.12/0.38 (((~![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)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(43,plain,
% 0.12/0.38 ((~![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)))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(44,plain,
% 0.12/0.38 ((~![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))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.12/0.38 tff(45,plain,
% 0.12/0.38 (subset_complement(the_carrier(A!3), B!4) = subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[44, 41, 39])).
% 0.12/0.38 tff(46,plain,
% 0.12/0.38 (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)),
% 0.12/0.38 inference(monotonicity,[status(thm)],[45])).
% 0.12/0.38 tff(47,plain,
% 0.12/0.38 ((~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))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[46])).
% 0.12/0.38 tff(48,assumption,(~open_subset(subset_complement(the_carrier(A!3), B!4), A!3)), introduced(assumption)).
% 0.12/0.38 tff(49,plain,
% 0.12/0.38 (~open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.12/0.38 tff(50,plain,
% 0.12/0.38 (~(closed_subset(B!4, A!3) <=> open_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.12/0.38 inference(or_elim,[status(thm)],[40])).
% 0.12/0.38 tff(51,plain,
% 0.12/0.38 ((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)),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(52,plain,
% 0.12/0.38 (closed_subset(B!4, A!3) | open_subset(subset_complement(the_carrier(A!3), B!4), A!3)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[51, 50])).
% 0.12/0.38 tff(53,plain,
% 0.12/0.38 (closed_subset(B!4, A!3)),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[52, 48])).
% 0.12/0.38 tff(54,plain,
% 0.12/0.38 (^[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)))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(55,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[54])).
% 0.12/0.38 tff(56,plain,
% 0.12/0.38 (^[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)))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(57,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[56])).
% 0.12/0.38 tff(58,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[57, 55])).
% 0.12/0.38 tff(59,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(60,plain,
% 0.12/0.38 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : rewrite((element(B, powerset(the_carrier(A))) => (closed_subset(B, A) <=> open_subset(subset_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))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(61,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[60])).
% 0.12/0.38 tff(62,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/sandbox/benchmark/theBenchmark.p','d6_pre_topc')).
% 0.12/0.38 tff(63,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.12/0.38 tff(64,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[63, 59])).
% 0.12/0.38 tff(65,plain,(
% 0.12/0.38 ![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))))),
% 0.12/0.38 inference(skolemize,[status(sab)],[64])).
% 0.12/0.38 tff(66,plain,
% 0.12/0.38 (![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))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[65, 58])).
% 0.12/0.38 tff(67,plain,
% 0.12/0.38 (((~![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))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(68,plain,
% 0.12/0.38 ((~![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))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(69,plain,
% 0.12/0.38 ((~![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)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.12/0.39 tff(70,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[69, 66, 8])).
% 0.12/0.39 tff(71,plain,
% 0.12/0.39 (((~![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)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(72,plain,
% 0.12/0.39 ((~![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)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(73,plain,
% 0.12/0.39 ((~![B: $i] : ((~element(B, powerset(the_carrier(A!3)))) | (closed_subset(B, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B), A!3)))) | (~element(B!4, powerset(the_carrier(A!3)))) | (closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.12/0.39 tff(74,plain,
% 0.12/0.39 (closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[73, 41, 70])).
% 0.12/0.39 tff(75,plain,
% 0.12/0.39 ((~(closed_subset(B!4, A!3) <=> open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3))) | (~closed_subset(B!4, A!3)) | open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(76,plain,
% 0.12/0.39 ((~closed_subset(B!4, A!3)) | open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[75, 74])).
% 0.12/0.39 tff(77,plain,
% 0.12/0.39 (open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[76, 53])).
% 0.12/0.39 tff(78,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[77, 49])).
% 0.12/0.39 tff(79,plain,(open_subset(subset_complement(the_carrier(A!3), B!4), A!3)), inference(lemma,lemma(discharge,[]))).
% 0.12/0.39 tff(80,plain,
% 0.12/0.39 (open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[79, 46])).
% 0.12/0.39 tff(81,plain,
% 0.12/0.39 ((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))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(82,plain,
% 0.12/0.39 ((~closed_subset(B!4, A!3)) | (~open_subset(subset_complement(the_carrier(A!3), B!4), A!3))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[81, 50])).
% 0.12/0.39 tff(83,plain,
% 0.12/0.39 (~closed_subset(B!4, A!3)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[82, 79])).
% 0.12/0.39 tff(84,plain,
% 0.12/0.39 ((~(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))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(85,plain,
% 0.12/0.39 (closed_subset(B!4, A!3) | (~open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[84, 74])).
% 0.12/0.39 tff(86,plain,
% 0.12/0.39 (~open_subset(subset_difference(the_carrier(A!3), cast_as_carrier_subset(A!3), B!4), A!3)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[85, 83])).
% 0.12/0.39 tff(87,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[86, 80])).
% 0.12/0.39 % SZS output end Proof
% 0.12/0.39 % E exiting
%------------------------------------------------------------------------------