TSTP Solution File: TOP011-1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : TOP011-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n028.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 : 600s
% DateTime : Thu Jul 21 21:33:50 EDT 2022
% Result : Timeout 300.05s 300.35s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : TOP011-1 : TPTP v8.1.0. Released v1.0.0.
% 0.11/0.12 % Command : tptp2X_and_run_prover9 %d %s
% 0.13/0.33 % Computer : n028.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 600
% 0.13/0.33 % DateTime : Sun May 29 06:49:18 EDT 2022
% 0.13/0.33 % CPUTime :
% 0.42/1.00 ============================== Prover9 ===============================
% 0.42/1.00 Prover9 (32) version 2009-11A, November 2009.
% 0.42/1.00 Process 17189 was started by sandbox on n028.cluster.edu,
% 0.42/1.00 Sun May 29 06:49:18 2022
% 0.42/1.00 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_17036_n028.cluster.edu".
% 0.42/1.00 ============================== end of head ===========================
% 0.42/1.00
% 0.42/1.00 ============================== INPUT =================================
% 0.42/1.00
% 0.42/1.00 % Reading from file /tmp/Prover9_17036_n028.cluster.edu
% 0.42/1.00
% 0.42/1.00 set(prolog_style_variables).
% 0.42/1.00 set(auto2).
% 0.42/1.00 % set(auto2) -> set(auto).
% 0.42/1.00 % set(auto) -> set(auto_inference).
% 0.42/1.00 % set(auto) -> set(auto_setup).
% 0.42/1.00 % set(auto_setup) -> set(predicate_elim).
% 0.42/1.00 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.42/1.00 % set(auto) -> set(auto_limits).
% 0.42/1.00 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.42/1.00 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.42/1.00 % set(auto) -> set(auto_denials).
% 0.42/1.00 % set(auto) -> set(auto_process).
% 0.42/1.00 % set(auto2) -> assign(new_constants, 1).
% 0.42/1.00 % set(auto2) -> assign(fold_denial_max, 3).
% 0.42/1.00 % set(auto2) -> assign(max_weight, "200.000").
% 0.42/1.00 % set(auto2) -> assign(max_hours, 1).
% 0.42/1.00 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.42/1.00 % set(auto2) -> assign(max_seconds, 0).
% 0.42/1.00 % set(auto2) -> assign(max_minutes, 5).
% 0.42/1.00 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.42/1.00 % set(auto2) -> set(sort_initial_sos).
% 0.42/1.00 % set(auto2) -> assign(sos_limit, -1).
% 0.42/1.00 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.42/1.00 % set(auto2) -> assign(max_megs, 400).
% 0.42/1.00 % set(auto2) -> assign(stats, some).
% 0.42/1.00 % set(auto2) -> clear(echo_input).
% 0.42/1.00 % set(auto2) -> set(quiet).
% 0.42/1.00 % set(auto2) -> clear(print_initial_clauses).
% 0.42/1.00 % set(auto2) -> clear(print_given).
% 0.42/1.00 assign(lrs_ticks,-1).
% 0.42/1.00 assign(sos_limit,10000).
% 0.42/1.00 assign(order,kbo).
% 0.42/1.00 set(lex_order_vars).
% 0.42/1.00 clear(print_given).
% 0.42/1.00
% 0.42/1.00 % formulas(sos). % not echoed (112 formulas)
% 0.42/1.00
% 0.42/1.00 ============================== end of input ==========================
% 0.42/1.00
% 0.42/1.00 % From the command line: assign(max_seconds, 300).
% 0.42/1.00
% 0.42/1.00 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.42/1.00
% 0.42/1.00 % Formulas that are not ordinary clauses:
% 0.42/1.00
% 0.42/1.00 ============================== end of process non-clausal formulas ===
% 0.42/1.00
% 0.42/1.00 ============================== PROCESS INITIAL CLAUSES ===============
% 0.42/1.00
% 0.42/1.00 ============================== PREDICATE ELIMINATION =================
% 0.42/1.00 1 topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f3(A,B),B) | subset_collections(f5(A,B),B) # label(topological_space_12) # label(axiom). [assumption].
% 0.42/1.00 2 -topological_space(A,B) | -subset_collections(C,B) | element_of_collection(union_of_members(C),B) # label(topological_space_11) # label(axiom). [assumption].
% 0.42/1.00 Derived: topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f3(A,B),B) | -topological_space(C,B) | element_of_collection(union_of_members(f5(A,B)),B). [resolve(1,f,2,b)].
% 0.42/1.00 3 topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f4(A,B),B) | subset_collections(f5(A,B),B) # label(topological_space_14) # label(axiom). [assumption].
% 0.42/1.00 Derived: topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f4(A,B),B) | -topological_space(C,B) | element_of_collection(union_of_members(f5(A,B)),B). [resolve(3,f,2,b)].
% 0.42/1.00 4 topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | -element_of_collection(intersection_of_sets(f3(A,B),f4(A,B)),B) | subset_collections(f5(A,B),B) # label(topological_space_16) # label(axiom). [assumption].
% 0.42/1.00 Derived: topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | -element_of_collection(intersection_of_sets(f3(A,B),f4(A,B)),B) | -topological_space(C,B) | element_of_collection(union_of_members(f5(A,B)),B). [resolve(4,f,2,b)].
% 0.42/1.00 5 -finer(A,B,C) | subset_collections(B,A) # label(finer_topology_26) # label(axiom). [assumption].
% 0.42/1.00 Derived: -finer(A,B,C) | -topological_space(D,A) | element_of_collection(union_of_members(B),A). [resolve(5,b,2,b)].
% 0.42/1.00 6 finer(A,B,C) | -topological_space(C,A) | -topological_space(C,B) | -subset_collections(B,A) # label(finer_topology_27) # label(axiom). [assumption].
% 0.42/1.00 Derived: finer(A,f5(B,A),C) | -topological_space(C,A) | -topological_space(C,f5(B,A)) | topological_space(B,A) | -equal_sets(union_of_members(A),B) | -element_of_collection(empty_set,A) | -element_of_collection(B,A) | element_of_collection(f3(B,A),A). [resolve(6,d,1,f)].
% 0.42/1.00 Derived: finer(A,f5(B,A),C) | -topological_space(C,A) | -topological_space(C,f5(B,A)) | topological_space(B,A) | -equal_sets(union_of_members(A),B) | -element_of_collection(empty_set,A) | -element_of_collection(B,A) | element_of_collection(f4(B,A),A). [resolve(6,d,3,f)].
% 0.42/1.00 Derived: finer(A,f5(B,A),C) | -topological_space(C,A) | -topological_space(C,f5(B,A)) | topological_space(B,A) | -equal_sets(union_of_members(A),B) | -element_of_collection(empty_set,A) | -element_of_collection(B,A) | -element_of_collection(intersection_of_sets(f3(B,A),f4(B,A)),A). [resolve(6,d,4,f)].
% 0.42/1.00 Derived: finer(A,B,C) | -topological_space(C,A) | -topological_space(C,B) | -finer(A,B,D). [resolve(6,d,5,b)].
% 0.42/1.00 7 -open_covering(A,B,C) | subset_collections(A,C) # label(open_covering_97) # label(axiom). [assumption].
% 0.42/1.00 Derived: -open_covering(A,B,C) | -topological_space(D,C) | element_of_collection(union_of_members(A),C). [resolve(7,b,2,b)].
% 0.42/1.00 Derived: -open_covering(A,B,C) | finer(C,A,D) | -topological_space(D,C) | -topological_space(D,A). [resolve(7,b,6,d)].
% 0.42/1.00 8 open_covering(A,B,C) | -topological_space(B,C) | -subset_collections(A,C) | -equal_sets(union_of_members(A),B) # label(open_covering_99) # label(axiom). [assumption].
% 0.42/1.00 Derived: open_covering(f5(A,B),C,B) | -topological_space(C,B) | -equal_sets(union_of_members(f5(A,B)),C) | topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f3(A,B),B). [resolve(8,c,1,f)].
% 0.42/1.00 Derived: open_covering(f5(A,B),C,B) | -topological_space(C,B) | -equal_sets(union_of_members(f5(A,B)),C) | topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f4(A,B),B). [resolve(8,c,3,f)].
% 0.42/1.00 Derived: open_covering(f5(A,B),C,B) | -topological_space(C,B) | -equal_sets(union_of_members(f5(A,B)),C) | topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | -element_of_collection(intersection_of_sets(f3(A,B),f4(A,B)),B). [resolve(8,c,4,f)].
% 0.42/1.00 Derived: open_covering(A,B,C) | -topological_space(B,C) | -equal_sets(union_of_members(A),B) | -finer(C,A,D). [resolve(8,c,5,b)].
% 0.42/1.00 Derived: open_covering(A,B,C) | -topological_space(B,C) | -equal_sets(union_of_members(A),B) | -open_covering(A,D,C). [resolve(8,c,7,b)].
% 0.42/1.00 9 -compact_space(A,B) | -open_covering(C,A,B) | subset_collections(f23(A,B,C),C) # label(compact_space_102) # label(axiom). [assumption].
% 0.42/1.00 Derived: -compact_space(A,B) | -open_covering(C,A,B) | -topological_space(D,C) | element_of_collection(union_of_members(f23(A,B,C)),C). [resolve(9,c,2,b)].
% 0.42/1.00 Derived: -compact_space(A,B) | -open_covering(C,A,B) | finer(C,f23(A,B,C),D) | -topological_space(D,C) | -topological_space(D,f23(A,B,C)). [resolve(9,c,6,d)].
% 0.42/1.00 Derived: -compact_space(A,B) | -open_covering(C,A,B) | open_covering(f23(A,B,C),D,C) | -topological_space(D,C) | -equal_sets(union_of_members(f23(A,B,C)),D). [resolve(9,c,8,c)].
% 0.42/1.00 10 compact_space(A,B) | -topological_space(A,B) | -finite(C) | -subset_collections(C,f24(A,B)) | -open_covering(C,A,B) # label(compact_space_105) # label(axiom). [assumption].
% 0.42/1.00 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | element_of_collection(f3(C,f24(A,B)),f24(A,B)). [resolve(10,d,1,f)].
% 0.42/1.00 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | element_of_collection(f4(C,f24(A,B)),f24(A,B)). [resolve(10,d,3,f)].
% 0.42/1.00 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | -element_of_collection(intersection_of_sets(f3(C,f24(A,B)),f4(C,f24(A,B))),f24(A,B)). [resolve(10,d,4,f)].
% 0.42/1.00 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(C) | -open_covering(C,A,B) | -finer(f24(A,B),C,D). [resolve(10,d,5,b)].
% 0.42/1.00 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(C) | -open_covering(C,A,B) | -open_covering(C,D,f24(A,B)). [resolve(10,d,7,b)].
% 0.42/1.00 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(f23(C,D,f24(A,B))) | -open_covering(f23(C,D,f24(A,B)),A,B) | -compact_space(C,D) | -open_covering(f24(A,B),C,D). [resolve(10,d,9,c)].
% 0.42/1.00 11 element_of_set(cu,top_of_basis(f)) | subset_collections(g,f) # label(problem_6_126) # label(negated_conjecture). [assumption].
% 0.42/1.00 Derived: element_of_set(cu,top_of_basis(f)) | -topological_space(A,f) | element_of_collection(union_of_members(g),f). [resolve(11,b,2,b)].
% 0.42/1.00 Derived: element_of_set(cu,top_of_basis(f)) | finer(f,g,A) | -topological_space(A,f) | -topological_space(A,g). [resolve(11,b,6,d)].
% 0.42/1.00 Derived: element_of_set(cu,top_of_basis(f)) | open_covering(g,A,f) | -topological_space(A,f) | -equal_sets(union_of_members(g),A). [resolve(11,b,8,c)].
% 0.42/1.00 12 -element_of_set(cu,top_of_basis(f)) | -subset_collections(A,f) | -equal_sets(cu,union_of_members(A)) # label(problem_6_128) # label(negated_conjecture). [assumption].
% 0.42/1.00 Derived: -element_of_set(cu,top_of_basis(f)) | -equal_sets(cu,union_of_members(f5(A,f))) | topological_space(A,f) | -equal_sets(union_of_members(f),A) | -element_of_collection(empty_set,f) | -element_of_collection(A,f) | element_of_collection(f3(A,f),f). [resolve(12,b,1,f)].
% 0.42/1.00 Derived: -element_of_set(cu,top_of_basis(f)) | -equal_sets(cu,union_of_members(f5(A,f))) | topological_space(A,f) | -equal_sets(union_of_members(f),A) | -element_of_collection(empty_set,f) | -element_of_collection(A,f) | element_of_collection(f4(A,f),f). [resolve(12,b,3,f)].
% 0.42/1.00 Derived: -element_of_set(cu,top_of_basis(f)) | -equal_sets(cu,union_of_members(f5(A,f))) | topological_space(A,f) | -equal_sets(union_of_members(f),A) | -element_of_collection(empty_set,f) | -element_of_collection(A,f) | -element_of_collection(intersection_of_sets(f3(A,f),f4(A,f)),f). [resolve(12,b,4,f)].
% 0.42/1.00 Derived: -element_of_set(cu,top_of_basis(f)) | -equal_sets(cu,union_of_members(A)) | -finer(f,A,B). [resolve(12,b,5,b)].
% 0.42/1.00 Derived: -element_of_set(cu,top_of_basis(f)) | -equal_sets(cu,union_of_members(A)) | -open_covering(A,B,f). [resolve(12,b,7,b)].
% 0.42/1.00 Derived: -element_of_set(cu,top_of_basis(f)) | -equal_sets(cu,union_of_members(f23(A,B,f))) | -compact_space(A,B) | -open_covering(f,A,B). [resolve(12,b,9,c)].
% 0.42/1.00 13 open(A,B,C) | -topological_space(B,C) | -element_of_collection(A,C) # label(open_set_20) # label(axiom). [assumption].
% 0.42/1.00 14 -open(A,B,C) | topological_space(B,C) # label(open_set_18) # label(axiom). [assumption].
% 0.42/1.00 15 -open(A,B,C) | element_of_collection(A,C) # label(open_set_19) # label(axiom). [assumption].
% 0.42/1.00 16 -closed(A,B,C) | open(relative_complement_sets(A,B),B,C) # label(closed_set_22) # label(axiom). [assumption].
% 0.42/1.00 Derived: -closed(A,B,C) | topological_space(B,C). [resolve(16,b,14,a)].
% 0.42/1.00 Derived: -closed(A,B,C) | element_of_collection(relative_complement_sets(A,B),C). [resolve(16,b,15,a)].
% 0.42/1.00 17 closed(A,B,C) | -topological_space(B,C) | -open(relative_complement_sets(A,B),B,C) # label(closed_set_23) # label(axiom). [assumption].
% 0.74/1.01 Derived: closed(A,B,C) | -topological_space(B,C) | -topological_space(B,C) | -element_of_collection(relative_complement_sets(A,B),C). [resolve(17,c,13,a)].
% 0.74/1.01 18 -element_of_set(A,interior(B,C,D)) | open(f13(B,C,D,A),C,D) # label(interior_51) # label(axiom). [assumption].
% 0.74/1.01 Derived: -element_of_set(A,interior(B,C,D)) | topological_space(C,D). [resolve(18,b,14,a)].
% 0.74/1.01 Derived: -element_of_set(A,interior(B,C,D)) | element_of_collection(f13(B,C,D,A),D). [resolve(18,b,15,a)].
% 0.74/1.01 19 element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,E) | -subset_sets(E,B) | -open(E,C,D) # label(interior_52) # label(axiom). [assumption].
% 0.74/1.01 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,E) | -subset_sets(E,B) | -topological_space(C,D) | -element_of_collection(E,D). [resolve(19,f,13,a)].
% 0.74/1.01 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,relative_complement_sets(E,C)) | -subset_sets(relative_complement_sets(E,C),B) | -closed(E,C,D). [resolve(19,f,16,b)].
% 0.74/1.01 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,f13(E,C,D,F)) | -subset_sets(f13(E,C,D,F),B) | -element_of_set(F,interior(E,C,D)). [resolve(19,f,18,b)].
% 0.74/1.01 20 -neighborhood(A,B,C,D) | open(A,C,D) # label(neighborhood_60) # label(axiom). [assumption].
% 0.74/1.01 Derived: -neighborhood(A,B,C,D) | topological_space(C,D). [resolve(20,b,14,a)].
% 0.74/1.01 Derived: -neighborhood(A,B,C,D) | element_of_collection(A,D). [resolve(20,b,15,a)].
% 0.74/1.01 Derived: -neighborhood(relative_complement_sets(A,B),C,B,D) | closed(A,B,D) | -topological_space(B,D). [resolve(20,b,17,c)].
% 0.74/1.01 Derived: -neighborhood(A,B,C,D) | element_of_set(E,interior(F,C,D)) | -topological_space(C,D) | -subset_sets(F,C) | -element_of_set(E,A) | -subset_sets(A,F). [resolve(20,b,19,f)].
% 0.74/1.01 21 neighborhood(A,B,C,D) | -topological_space(C,D) | -open(A,C,D) | -element_of_set(B,A) # label(neighborhood_62) # label(axiom). [assumption].
% 0.74/1.01 Derived: neighborhood(A,B,C,D) | -topological_space(C,D) | -element_of_set(B,A) | -topological_space(C,D) | -element_of_collection(A,D). [resolve(21,c,13,a)].
% 0.74/1.01 Derived: neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | -element_of_set(C,relative_complement_sets(A,B)) | -closed(A,B,D). [resolve(21,c,16,b)].
% 0.74/1.01 Derived: neighborhood(f13(A,B,C,D),E,B,C) | -topological_space(B,C) | -element_of_set(E,f13(A,B,C,D)) | -element_of_set(D,interior(A,B,C)). [resolve(21,c,18,b)].
% 0.74/1.01 Derived: neighborhood(A,B,C,D) | -topological_space(C,D) | -element_of_set(B,A) | -neighborhood(A,E,C,D). [resolve(21,c,20,b)].
% 0.74/1.01 22 element_of_set(A,closure(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | closed(f14(B,C,D,A),C,D) # label(closure_57) # label(axiom). [assumption].
% 0.74/1.01 23 -closed(A,B,C) | topological_space(B,C) # label(closed_set_21) # label(axiom). [assumption].
% 0.74/1.01 24 -element_of_set(A,closure(B,C,D)) | -subset_sets(B,E) | -closed(E,C,D) | element_of_set(A,E) # label(closure_55) # label(axiom). [assumption].
% 0.74/1.01 Derived: element_of_set(A,closure(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(E,closure(F,C,D)) | -subset_sets(F,f14(B,C,D,A)) | element_of_set(E,f14(B,C,D,A)). [resolve(22,d,24,c)].
% 0.74/1.01 25 -closed(A,B,C) | topological_space(B,C). [resolve(16,b,14,a)].
% 0.74/1.01 26 -closed(A,B,C) | element_of_collection(relative_complement_sets(A,B),C). [resolve(16,b,15,a)].
% 0.74/1.01 Derived: element_of_collection(relative_complement_sets(f14(A,B,C,D),B),C) | element_of_set(D,closure(A,B,C)) | -topological_space(B,C) | -subset_sets(A,B). [resolve(26,a,22,d)].
% 0.74/1.01 27 closed(A,B,C) | -topological_space(B,C) | -topological_space(B,C) | -element_of_collection(relative_complement_sets(A,B),C). [resolve(17,c,13,a)].
% 0.74/1.01 Derived: -topological_space(A,B) | -topological_space(A,B) | -element_of_collection(relative_complement_sets(C,A),B) | -element_of_set(D,closure(E,A,B)) | -subset_sets(E,C) | element_of_set(D,C). [resolve(27,a,24,c)].
% 0.74/1.01 28 element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,relative_complement_sets(E,C)) | -subset_sets(relative_complement_sets(E,C),B) | -closed(E,C,D). [resolve(19,f,16,b)].
% 0.74/1.01 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,relative_complement_sets(f14(E,C,D,F),C)) | -subset_sets(relative_complement_sets(f14(E,C,D,F),C),B) | element_of_set(F,closure(E,C,D)) | -topological_space(C,D) | -subset_sets(E,C). [resolve(28,f,22,d)].
% 0.74/1.01 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,relative_complement_sets(E,C)) | -subset_sets(relative_complement_sets(E,C),B) | -topological_space(C,D) | -topological_space(C,D) | -element_of_collection(relative_complement_sets(E,C),D). [resolve(28,f,27,a)].
% 0.74/1.01 29 -neighborhood(relative_complement_sets(A,B),C,B,D) | closed(A,B,D) | -topological_space(B,D). [resolve(20,b,17,c)].
% 0.74/1.01 Derived: -neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | -element_of_set(E,closure(F,B,D)) | -subset_sets(F,A) | element_of_set(E,A). [resolve(29,b,24,c)].
% 0.74/1.01 Derived: -neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | element_of_collection(relative_complement_sets(A,B),D). [resolve(29,b,26,a)].
% 0.74/1.01 30 neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | -element_of_set(C,relative_complement_sets(A,B)) | -closed(A,B,D). [resolve(21,c,16,b)].
% 0.74/1.01 Derived: neighborhood(relative_complement_sets(f14(A,B,C,D),B),E,B,C) | -topological_space(B,C) | -element_of_set(E,relative_complement_sets(f14(A,B,C,D),B)) | element_of_set(D,closure(A,B,C)) | -topological_space(B,C) | -subset_sets(A,B). [resolve(30,d,22,d)].
% 0.74/1.01 Derived: neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | -element_of_set(C,relative_complement_sets(A,B)) | -topological_space(B,D) | -topological_space(B,D) | -element_of_collection(relative_complement_sets(A,B),D). [resolve(30,d,27,a)].
% 0.74/1.01 31 basis(A,B) | -equal_sets(union_of_members(B),A) | element_of_set(f7(A,B),A) # label(basis_for_topology_32) # label(axiom). [assumption].
% 0.74/1.01 32 -basis(A,B) | equal_sets(union_of_members(B),A) # label(basis_for_topology_28) # label(axiom). [assumption].
% 0.74/1.01 33 -basis(A,B) | -element_of_set(C,A) | -element_of_collection(D,B) | -element_of_collection(E,B) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(A,B,C,D,E)) # label(basis_for_topology_29) # label(axiom). [assumption].
% 0.74/1.01 34 -basis(A,B) | -element_of_set(C,A) | -element_of_collection(D,B) | -element_of_collection(E,B) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(A,B,C,D,E),B) # label(basis_for_topology_30) # label(axiom). [assumption].
% 0.74/1.01 35 -basis(A,B) | -element_of_set(C,A) | -element_of_collection(D,B) | -element_of_collection(E,B) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(A,B,C,D,E),intersection_of_sets(D,E)) # label(basis_for_topology_31) # label(axiom). [assumption].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),B) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(B,A,C,D,E)). [resolve(31,a,33,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),B) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(B,A,C,D,E),A). [resolve(31,a,34,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),B) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(B,A,C,D,E),intersection_of_sets(D,E)). [resolve(31,a,35,a)].
% 0.74/1.01 36 basis(A,B) | -equal_sets(union_of_members(B),A) | element_of_collection(f8(A,B),B) # label(basis_for_topology_33) # label(axiom). [assumption].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f8(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(B,A,C,D,E)). [resolve(36,a,33,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f8(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(B,A,C,D,E),A). [resolve(36,a,34,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f8(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(B,A,C,D,E),intersection_of_sets(D,E)). [resolve(36,a,35,a)].
% 0.74/1.01 37 basis(A,B) | -equal_sets(union_of_members(B),A) | element_of_collection(f9(A,B),B) # label(basis_for_topology_34) # label(axiom). [assumption].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f9(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(B,A,C,D,E)). [resolve(37,a,33,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f9(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(B,A,C,D,E),A). [resolve(37,a,34,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f9(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(B,A,C,D,E),intersection_of_sets(D,E)). [resolve(37,a,35,a)].
% 0.74/1.01 38 basis(A,B) | -equal_sets(union_of_members(B),A) | element_of_set(f7(A,B),intersection_of_sets(f8(A,B),f9(A,B))) # label(basis_for_topology_35) # label(axiom). [assumption].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(B,A,C,D,E)). [resolve(38,a,33,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(B,A,C,D,E),A). [resolve(38,a,34,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(B,A,C,D,E),intersection_of_sets(D,E)). [resolve(38,a,35,a)].
% 0.74/1.01 39 basis(A,B) | -equal_sets(union_of_members(B),A) | -element_of_set(f7(A,B),C) | -element_of_collection(C,B) | -subset_sets(C,intersection_of_sets(f8(A,B),f9(A,B))) # label(basis_for_topology_36) # label(axiom). [assumption].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | -element_of_set(f7(B,A),C) | -element_of_collection(C,A) | -subset_sets(C,intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(D,B) | -element_of_collection(E,A) | -element_of_collection(F,A) | -element_of_set(D,intersection_of_sets(E,F)) | element_of_set(D,f6(B,A,D,E,F)). [resolve(39,a,33,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | -element_of_set(f7(B,A),C) | -element_of_collection(C,A) | -subset_sets(C,intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(D,B) | -element_of_collection(E,A) | -element_of_collection(F,A) | -element_of_set(D,intersection_of_sets(E,F)) | element_of_collection(f6(B,A,D,E,F),A). [resolve(39,a,34,a)].
% 0.74/1.01 Derived: -equal_sets(union_of_members(A),B) | -element_of_set(f7(B,A),C) | -element_of_collection(C,A) | -subset_sets(C,intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(D,B) | -element_of_collection(E,A) | -element_of_collection(F,A) | -element_of_set(D,intersection_of_sets(E,F)) | subset_sets(f6(B,A,D,E,F),intersection_of_sets(E,F)). [resolve(39,a,35,a)].
% 0.74/1.01 40 limit_point(A,B,C,D) | -topological_space(C,D) | -subset_sets(B,C) | neighborhood(f16(A,B,C,D),A,C,D) # label(limit_point_67) # label(axiom). [assumption].
% 0.74/1.02 41 -limit_point(A,B,C,D) | topological_space(C,D) # label(limit_point_63) # label(axiom). [assumption].
% 0.74/1.02 42 -limit_point(A,B,C,D) | subset_sets(B,C) # label(limit_point_64) # label(axiom). [assumption].
% 0.74/1.02 43 -limit_point(A,B,C,D) | -neighborhood(E,A,C,D) | element_of_set(f15(A,B,C,D,E),intersection_of_sets(E,B)) # label(limit_point_65) # label(axiom). [assumption].
% 0.74/1.02 44 -limit_point(A,B,C,D) | -neighborhood(E,A,C,D) | -eq_p(f15(A,B,C,D,E),A) # label(limit_point_66) # label(axiom). [assumption].
% 0.74/1.02 Derived: -topological_space(A,B) | -subset_sets(C,A) | neighborhood(f16(D,C,A,B),D,A,B) | -neighborhood(E,D,A,B) | element_of_set(f15(D,C,A,B,E),intersection_of_sets(E,C)). [resolve(40,a,43,a)].
% 0.74/1.02 Derived: -topological_space(A,B) | -subset_sets(C,A) | neighborhood(f16(D,C,A,B),D,A,B) | -neighborhood(E,D,A,B) | -eq_p(f15(D,C,A,B,E),D). [resolve(40,a,44,a)].
% 0.74/1.02 45 limit_point(A,B,C,D) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(E,intersection_of_sets(f16(A,B,C,D),B)) | eq_p(E,A) # label(limit_point_68) # label(axiom). [assumption].
% 0.74/1.02 Derived: -topological_space(A,B) | -subset_sets(C,A) | -element_of_set(D,intersection_of_sets(f16(E,C,A,B),C)) | eq_p(D,E) | -neighborhood(F,E,A,B) | element_of_set(f15(E,C,A,B,F),intersection_of_sets(F,C)). [resolve(45,a,43,a)].
% 0.74/1.02 Derived: -topological_space(A,B) | -subset_sets(C,A) | -element_of_set(D,intersection_of_sets(f16(E,C,A,B),C)) | eq_p(D,E) | -neighborhood(F,E,A,B) | -eq_p(f15(E,C,A,B,F),E). [resolve(45,a,44,a)].
% 0.74/1.02 46 hausdorff(A,B) | -topological_space(A,B) | element_of_set(f19(A,B),A) # label(hausdorff_77) # label(axiom). [assumption].
% 0.74/1.02 47 -hausdorff(A,B) | topological_space(A,B) # label(hausdorff_73) # label(axiom). [assumption].
% 0.74/1.02 48 -hausdorff(A,B) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f17(A,B,C,D),C,A,B) # label(hausdorff_74) # label(axiom). [assumption].
% 0.74/1.02 49 -hausdorff(A,B) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f18(A,B,C,D),D,A,B) # label(hausdorff_75) # label(axiom). [assumption].
% 0.74/1.02 50 -hausdorff(A,B) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | disjoint_s(f17(A,B,C,D),f18(A,B,C,D)) # label(hausdorff_76) # label(axiom). [assumption].
% 0.74/1.02 Derived: -topological_space(A,B) | element_of_set(f19(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f17(A,B,C,D),C,A,B). [resolve(46,a,48,a)].
% 0.74/1.02 Derived: -topological_space(A,B) | element_of_set(f19(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f18(A,B,C,D),D,A,B). [resolve(46,a,49,a)].
% 0.74/1.02 Derived: -topological_space(A,B) | element_of_set(f19(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | disjoint_s(f17(A,B,C,D),f18(A,B,C,D)). [resolve(46,a,50,a)].
% 0.74/1.02 51 hausdorff(A,B) | -topological_space(A,B) | element_of_set(f20(A,B),A) # label(hausdorff_78) # label(axiom). [assumption].
% 0.74/1.02 Derived: -topological_space(A,B) | element_of_set(f20(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f17(A,B,C,D),C,A,B). [resolve(51,a,48,a)].
% 0.74/1.02 Derived: -topological_space(A,B) | element_of_set(f20(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f18(A,B,C,D),D,A,B). [resolve(51,a,49,a)].
% 0.74/1.02 Derived: -topological_space(A,B) | element_of_set(f20(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | disjoint_s(f17(A,B,C,D),f18(A,B,C,D)). [resolve(51,a,50,a)].
% 0.74/1.02 52 hausdorff(A,B) | -topological_space(A,B) | -eq_p(f19(A,B),f20(A,B)) # label(hausdorff_79) # label(axiom). [assumption].
% 0.74/1.02 Derived: -topological_space(A,B) | -eq_p(f19(A,B),f20(A,B)) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f17(A,B,C,D),C,A,B). [resolve(52,a,48,a)].
% 0.74/1.02 Derived: -topological_space(A,B) | -eq_p(f19(A,B),f20(A,B)) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f18(A,B,C,D),D,A,B). [resolve(52,a,49,a)].
% 0.74/1.02 Derived: -topological_space(A,B) | -eq_p(f19(A,B),f20(A,B)) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | disjoint_s(f17(A,B,C,D),f18(A,B,C,D)). [resolve(52,a,50,a)].
% 0.74/1.03 53 hausdorff(A,B) | -topological_space(A,B) | -neighborhood(C,f19(A,B),A,B) | -neighborhood(D,f20(A,B),A,B) | -disjoint_s(C,D) # label(hausdorff_80) # label(axiom). [assumption].
% 0.74/1.03 Derived: -topological_space(A,B) | -neighborhood(C,f19(A,B),A,B) | -neighborhood(D,f20(A,B),A,B) | -disjoint_s(C,D) | -element_of_set(E,A) | -element_of_set(F,A) | eq_p(E,F) | neighborhood(f17(A,B,E,F),E,A,B). [resolve(53,a,48,a)].
% 0.74/1.03 Derived: -topological_space(A,B) | -neighborhood(C,f19(A,B),A,B) | -neighborhood(D,f20(A,B),A,B) | -disjoint_s(C,D) | -element_of_set(E,A) | -element_of_set(F,A) | eq_p(E,F) | neighborhood(f18(A,B,E,F),F,A,B). [resolve(53,a,49,a)].
% 0.74/1.03 Derived: -topological_space(A,B) | -neighborhood(C,f19(A,B),A,B) | -neighborhood(D,f20(A,B),A,B) | -disjoint_s(C,D) | -element_of_set(E,A) | -element_of_set(F,A) | eq_p(E,F) | disjoint_s(f17(A,B,E,F),f18(A,B,E,F)). [resolve(53,a,50,a)].
% 0.74/1.03 54 separation(A,B,C,D) | -topological_space(C,D) | equal_sets(A,empty_set) | equal_sets(B,empty_set) | -element_of_collection(A,D) | -element_of_collection(B,D) | -equal_sets(union_of_sets(A,B),C) | -disjoint_s(A,B) # label(separation_88) # label(axiom). [assumption].
% 0.74/1.03 55 -separation(A,B,C,D) | topological_space(C,D) # label(separation_81) # label(axiom). [assumption].
% 0.74/1.03 56 -separation(A,B,C,D) | -equal_sets(A,empty_set) # label(separation_82) # label(axiom). [assumption].
% 0.74/1.03 57 -separation(A,B,C,D) | -equal_sets(B,empty_set) # label(separation_83) # label(axiom). [assumption].
% 0.74/1.03 58 -separation(A,B,C,D) | element_of_collection(A,D) # label(separation_84) # label(axiom). [assumption].
% 0.74/1.03 59 -separation(A,B,C,D) | element_of_collection(B,D) # label(separation_85) # label(axiom). [assumption].
% 0.74/1.03 60 -separation(A,B,C,D) | equal_sets(union_of_sets(A,B),C) # label(separation_86) # label(axiom). [assumption].
% 0.74/1.03 61 -separation(A,B,C,D) | disjoint_s(A,B) # label(separation_87) # label(axiom). [assumption].
% 0.74/1.03 62 -connected_space(A,B) | -separation(C,D,A,B) # label(connected_space_90) # label(axiom). [assumption].
% 0.74/1.03 Derived: -connected_space(A,B) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(62,b,54,a)].
% 0.74/1.03 63 connected_space(A,B) | -topological_space(A,B) | separation(f21(A,B),f22(A,B),A,B) # label(connected_space_91) # label(axiom). [assumption].
% 0.74/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | -equal_sets(f21(A,B),empty_set). [resolve(63,c,56,a)].
% 0.74/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | -equal_sets(f22(A,B),empty_set). [resolve(63,c,57,a)].
% 0.74/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | element_of_collection(f21(A,B),B). [resolve(63,c,58,a)].
% 0.74/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | element_of_collection(f22(A,B),B). [resolve(63,c,59,a)].
% 0.74/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | equal_sets(union_of_sets(f21(A,B),f22(A,B)),A). [resolve(63,c,60,a)].
% 0.74/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | disjoint_s(f21(A,B),f22(A,B)). [resolve(63,c,61,a)].
% 0.74/1.03 64 -connected_set(A,B,C) | connected_space(A,subspace_topology(B,C,A)) # label(connected_set_94) # label(axiom). [assumption].
% 0.74/1.03 65 -connected_space(A,B) | topological_space(A,B) # label(connected_space_89) # label(axiom). [assumption].
% 0.74/1.03 Derived: -connected_set(A,B,C) | topological_space(A,subspace_topology(B,C,A)). [resolve(64,b,65,a)].
% 0.74/1.03 66 connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B) | -connected_space(A,subspace_topology(B,C,A)) # label(connected_set_95) # label(axiom). [assumption].
% 0.74/1.03 67 -connected_space(A,B) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(62,b,54,a)].
% 0.74/1.03 Derived: -topological_space(A,subspace_topology(B,C,A)) | equal_sets(D,empty_set) | equal_sets(E,empty_set) | -element_of_collection(D,subspace_topology(B,C,A)) | -element_of_collection(E,subspace_topology(B,C,A)) | -equal_sets(union_of_sets(D,E),A) | -disjoint_s(D,E) | -connected_set(A,B,C). [resolve(67,a,64,b)].
% 0.74/1.03 68 connected_space(A,B) | -topological_space(A,B) | -equal_sets(f21(A,B),empty_set). [resolve(63,c,56,a)].
% 0.74/1.03 Derived: -topological_space(A,subspace_topology(B,C,A)) | -equal_sets(f21(A,subspace_topology(B,C,A)),empty_set) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(68,a,66,d)].
% 0.74/1.03 Derived: -topological_space(A,B) | -equal_sets(f21(A,B),empty_set) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(68,a,67,a)].
% 0.74/1.03 69 connected_space(A,B) | -topological_space(A,B) | -equal_sets(f22(A,B),empty_set). [resolve(63,c,57,a)].
% 0.74/1.03 Derived: -topological_space(A,subspace_topology(B,C,A)) | -equal_sets(f22(A,subspace_topology(B,C,A)),empty_set) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(69,a,66,d)].
% 0.74/1.03 Derived: -topological_space(A,B) | -equal_sets(f22(A,B),empty_set) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(69,a,67,a)].
% 0.74/1.03 70 connected_space(A,B) | -topological_space(A,B) | element_of_collection(f21(A,B),B). [resolve(63,c,58,a)].
% 0.74/1.03 Derived: -topological_space(A,subspace_topology(B,C,A)) | element_of_collection(f21(A,subspace_topology(B,C,A)),subspace_topology(B,C,A)) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(70,a,66,d)].
% 0.74/1.03 Derived: -topological_space(A,B) | element_of_collection(f21(A,B),B) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(70,a,67,a)].
% 0.74/1.03 71 connected_space(A,B) | -topological_space(A,B) | element_of_collection(f22(A,B),B). [resolve(63,c,59,a)].
% 0.74/1.03 Derived: -topological_space(A,subspace_topology(B,C,A)) | element_of_collection(f22(A,subspace_topology(B,C,A)),subspace_topology(B,C,A)) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(71,a,66,d)].
% 0.74/1.03 Derived: -topological_space(A,B) | element_of_collection(f22(A,B),B) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(71,a,67,a)].
% 0.74/1.03 72 connected_space(A,B) | -topological_space(A,B) | equal_sets(union_of_sets(f21(A,B),f22(A,B)),A). [resolve(63,c,60,a)].
% 0.74/1.03 Derived: -topological_space(A,subspace_topology(B,C,A)) | equal_sets(union_of_sets(f21(A,subspace_topology(B,C,A)),f22(A,subspace_topology(B,C,A))),A) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(72,a,66,d)].
% 0.74/1.03 Derived: -topological_space(A,B) | equal_sets(union_of_sets(f21(A,B),f22(A,B)),A) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(72,a,67,a)].
% 0.74/1.03 73 connected_space(A,B) | -topological_space(A,B) | disjoint_s(f21(A,B),f22(A,B)). [resolve(63,c,61,a)].
% 0.74/1.03 Derived: -topological_space(A,subspace_topology(B,C,A)) | disjoint_s(f21(A,subspace_topology(B,C,A)),f22(A,subspace_topology(B,C,A))) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(73,a,66,d)].
% 0.74/1.03 Derived: -topological_space(A,B) | disjoint_s(f21(A,B),f22(A,B)) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(73,a,67,a)].
% 0.74/1.03 74 -topological_space(A,subspace_topology(B,C,A)) | -equal_sets(f21(A,subspace_topology(B,C,A)),empty_set) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(68,a,66,d)].
% 0.74/1.03 75 -connected_set(A,B,C) | topological_space(B,C) # label(connected_set_92) # label(axiom). [assumption].
% 0.74/1.04 76 -connected_set(A,B,C) | subset_sets(A,B) # label(connected_set_93) # label(axiom). [assumption].
% 0.74/1.04 77 -connected_set(A,B,C) | topological_space(A,subspace_topology(B,C,A)). [resolve(64,b,65,a)].
% 0.74/1.04 78 -topological_space(A,subspace_topology(B,C,A)) | equal_sets(D,empty_set) | equal_sets(E,empty_set) | -element_of_collection(D,subspace_topology(B,C,A)) | -element_of_collection(E,subspace_topology(B,C,A)) | -equal_sets(union_of_sets(D,E),A) | -disjoint_s(D,E) | -connected_set(A,B,C). [resolve(67,a,64,b)].
% 0.74/1.04 79 -topological_space(A,subspace_topology(B,C,A)) | -equal_sets(f22(A,subspace_topology(B,C,A)),empty_set) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(69,a,66,d)].
% 0.74/1.04 80 -topological_space(A,subspace_topology(B,C,A)) | element_of_collection(f21(A,subspace_topology(B,C,A)),subspace_topology(B,C,A)) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(70,a,66,d)].
% 0.74/1.04 81 -topological_space(A,subspace_topology(B,C,A)) | element_of_collection(f22(A,subspace_topology(B,C,A)),subspace_topology(B,C,A)) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(71,a,66,d)].
% 0.74/1.04 82 -topological_space(A,subspace_topology(B,C,A)) | equal_sets(union_of_sets(f21(A,subspace_topology(B,C,A)),f22(A,subspace_topology(B,C,A))),A) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(72,a,66,d)].
% 0.74/1.04 83 -topological_space(A,subspace_topology(B,C,A)) | disjoint_s(f21(A,subspace_topology(B,C,A)),f22(A,subspace_topology(B,C,A))) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(73,a,66,d)].
% 0.74/1.04 84 compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | element_of_collection(f3(C,f24(A,B)),f24(A,B)). [resolve(10,d,1,f)].
% 0.74/1.04 85 -compact_space(A,B) | -open_covering(C,A,B) | finite(f23(A,B,C)) # label(compact_space_101) # label(axiom). [assumption].
% 0.74/1.04 86 compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | element_of_collection(f4(C,f24(A,B)),f24(A,B)). [resolve(10,d,3,f)].
% 0.74/1.04 87 compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | -element_of_collection(intersection_of_sets(f3(C,f24(A,B)),f4(C,f24(A,B))),f24(A,B)). [resolve(10,d,4,f)].
% 0.74/1.04 88 compact_space(A,B) | -topological_space(A,B) | -finite(C) | -open_covering(C,A,B) | -finer(f24(A,B),C,D). [resolve(10,d,5,b)].
% 0.74/1.04 Derived: compact_space(A,B) | -topological_space(A,B) | -open_covering(f23(C,D,E),A,B) | -finer(f24(A,B),f23(C,D,E),F) | -compact_space(C,D) | -open_covering(E,C,D). [resolve(88,c,85,c)].
% 0.74/1.04 89 compact_space(A,B) | -topological_space(A,B) | -finite(C) | -open_covering(C,A,B) | -open_covering(C,D,f24(A,B)). [resolve(10,d,7,b)].
% 0.74/1.04 Derived: compact_space(A,B) | -topological_space(A,B) | -open_covering(f23(C,D,E),A,B) | -open_covering(f23(C,D,E),F,f24(A,B)) | -compact_space(C,D) | -open_covering(E,C,D). [resolve(89,c,85,c)].
% 0.74/1.04 90 compact_space(A,B) | -topological_space(A,B) | -finite(f23(C,D,f24(A,B))) | -open_covering(f23(C,D,f24(A,B)),A,B) | -compact_space(C,D) | -open_covering(f24(A,B),C,D). [resolve(10,d,9,c)].
% 0.74/1.04 Derived: compact_space(A,B) | -topological_space(A,B) | -open_covering(f23(C,D,f24(A,B)),A,B) | -compact_space(C,D) | -open_covering(f24(A,B),C,D) | -compact_space(C,D) | -open_covering(f24(A,B),C,D). [resolve(90,c,85,c)].
% 0.74/1.04 91 compact_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B) | -compact_space(A,subspace_topology(B,C,A)) # label(compact_set_109) # label(axiom). [assumption].
% 220.50/220.82 92 -compact_set(A,B,C) | topological_space(B,C) # label(compact_set_106) # label(axiom). [assumption].
% 220.50/220.82 93 -compact_set(A,B,C) | subset_sets(A,B) # label(compact_set_107) # label(axiom). [assumption].
% 220.50/220.82 94 -compact_set(A,B,C) | compact_space(A,subspace_topology(B,C,A)) # label(compact_set_108) # label(axiom). [assumption].
% 220.50/220.82
% 220.50/220.82 ============================== end predicate elimination =============
% 220.50/220.82
% 220.50/220.82 Auto_denials: (non-Horn, no changes).
% 220.50/220.82
% 220.50/220.82 Term ordering decisions:
% 220.50/220.82 Function symbol KB weights: f=1. empty_set=1. cu=1. g=1. intersection_of_sets=1. f5=1. relative_complement_sets=1. f3=1. f4=1. f19=1. f20=1. f7=1. f8=1. f9=1. union_of_sets=1. f24=1. f21=1. f22=1. f1=1. f11=1. f2=1. union_of_members=1. top_of_basis=1. intersection_of_members=1. closure=1. interior=1. f23=1. subspace_topology=1. boundary=1. f10=1. f14=1. f17=1. f18=1. f13=1. f16=1. f12=1. f6=1. f15=1.
% 220.50/220.82
% 220.50/220.82 ============================== end of process initial clauses ========
% 220.50/220.82
% 220.50/220.82 ============================== CLAUSES FOR SEARCH ====================
% 220.50/220.82
% 220.50/220.82 ============================== end of clauses for search =============
% 220.50/220.82
% 220.50/220.82 ============================== SEARCH ================================
% 220.50/220.82
% 220.50/220.82 % Starting search at 0.09 seconds.
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=60.000, iters=3386
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=57.000, iters=3366
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=53.000, iters=3393
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=50.000, iters=3352
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=41.000, iters=3349
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=38.000, iters=3335
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=35.000, iters=3336
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=33.000, iters=3345
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=30.000, iters=3345
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=29.000, iters=3337
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=27.000, iters=3337
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=25.000, iters=3365
% 220.50/220.82
% 220.50/220.82 Low Water (keep): wt=24.000, iters=3333
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27516, wt=199.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27590, wt=196.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27136, wt=194.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27272, wt=193.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27271, wt=192.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27134, wt=191.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27270, wt=189.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27132, wt=188.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27130, wt=187.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27128, wt=184.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27020, wt=182.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27448, wt=180.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27018, wt=179.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27522, wt=177.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27016, wt=176.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27014, wt=175.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27596, wt=174.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27210, wt=173.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27012, wt=172.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27103, wt=171.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27208, wt=170.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=26901, wt=169.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27102, wt=168.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27101, wt=167.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=26900, wt=166.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27080, wt=165.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27154, wt=164.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=26872, wt=163.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27078, wt=162.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27152, wt=161.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27215, wt=160.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27034, wt=159.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27447, wt=158.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27213, wt=157.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27167, wt=156.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27521, wt=155.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27211, wt=154.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27439, wt=153.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27595, wt=152.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=26759, wt=151.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27512, wt=150.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27164, wt=149.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27081, wt=148.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27586, wt=147.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27163, wt=146.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27079, wt=145.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27178, wt=144.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27177, wt=143.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27077, wt=142.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27436, wt=141.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27176, wt=140.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=27509, wt=138.000
% 220.50/220.82
% 220.50/220.82 Low Water (displace): id=26952, wt=137.000
% 220.50/220.82
% 220.50/220.82 LCputime limit exceeded (core dumped)
%------------------------------------------------------------------------------