TSTP Solution File: TOP009-1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : TOP009-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n026.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 300.10s 300.43s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : TOP009-1 : TPTP v8.1.0. Released v1.0.0.
% 0.10/0.13 % Command : tptp2X_and_run_prover9 %d %s
% 0.12/0.32 % Computer : n026.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 600
% 0.12/0.32 % DateTime : Sun May 29 06:38:54 EDT 2022
% 0.12/0.32 % CPUTime :
% 0.68/1.00 ============================== Prover9 ===============================
% 0.68/1.00 Prover9 (32) version 2009-11A, November 2009.
% 0.68/1.00 Process 23837 was started by sandbox2 on n026.cluster.edu,
% 0.68/1.00 Sun May 29 06:38:55 2022
% 0.68/1.00 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_23491_n026.cluster.edu".
% 0.68/1.00 ============================== end of head ===========================
% 0.68/1.00
% 0.68/1.00 ============================== INPUT =================================
% 0.68/1.00
% 0.68/1.00 % Reading from file /tmp/Prover9_23491_n026.cluster.edu
% 0.68/1.00
% 0.68/1.00 set(prolog_style_variables).
% 0.68/1.00 set(auto2).
% 0.68/1.00 % set(auto2) -> set(auto).
% 0.68/1.00 % set(auto) -> set(auto_inference).
% 0.68/1.00 % set(auto) -> set(auto_setup).
% 0.68/1.00 % set(auto_setup) -> set(predicate_elim).
% 0.68/1.00 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.68/1.00 % set(auto) -> set(auto_limits).
% 0.68/1.00 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.68/1.00 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.68/1.00 % set(auto) -> set(auto_denials).
% 0.68/1.00 % set(auto) -> set(auto_process).
% 0.68/1.00 % set(auto2) -> assign(new_constants, 1).
% 0.68/1.00 % set(auto2) -> assign(fold_denial_max, 3).
% 0.68/1.00 % set(auto2) -> assign(max_weight, "200.000").
% 0.68/1.00 % set(auto2) -> assign(max_hours, 1).
% 0.68/1.00 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.68/1.00 % set(auto2) -> assign(max_seconds, 0).
% 0.68/1.00 % set(auto2) -> assign(max_minutes, 5).
% 0.68/1.00 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.68/1.00 % set(auto2) -> set(sort_initial_sos).
% 0.68/1.00 % set(auto2) -> assign(sos_limit, -1).
% 0.68/1.00 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.68/1.00 % set(auto2) -> assign(max_megs, 400).
% 0.68/1.00 % set(auto2) -> assign(stats, some).
% 0.68/1.00 % set(auto2) -> clear(echo_input).
% 0.68/1.00 % set(auto2) -> set(quiet).
% 0.68/1.00 % set(auto2) -> clear(print_initial_clauses).
% 0.68/1.00 % set(auto2) -> clear(print_given).
% 0.68/1.00 assign(lrs_ticks,-1).
% 0.68/1.00 assign(sos_limit,10000).
% 0.68/1.00 assign(order,kbo).
% 0.68/1.00 set(lex_order_vars).
% 0.68/1.00 clear(print_given).
% 0.68/1.00
% 0.68/1.00 % formulas(sos). % not echoed (112 formulas)
% 0.68/1.00
% 0.68/1.00 ============================== end of input ==========================
% 0.68/1.00
% 0.68/1.00 % From the command line: assign(max_seconds, 300).
% 0.68/1.00
% 0.68/1.00 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.68/1.00
% 0.68/1.00 % Formulas that are not ordinary clauses:
% 0.68/1.00
% 0.68/1.00 ============================== end of process non-clausal formulas ===
% 0.68/1.00
% 0.68/1.00 ============================== PROCESS INITIAL CLAUSES ===============
% 0.68/1.00
% 0.68/1.00 ============================== PREDICATE ELIMINATION =================
% 0.68/1.00 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].
% 0.68/1.00 2 -topological_space(A,B) | -subset_collections(C,B) | element_of_collection(union_of_members(C),B) # label(topological_space_11) # label(axiom). [assumption].
% 0.68/1.00 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)].
% 0.68/1.00 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].
% 0.68/1.00 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)].
% 0.68/1.00 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].
% 0.68/1.00 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)].
% 0.72/1.01 5 -finer(A,B,C) | subset_collections(B,A) # label(finer_topology_26) # label(axiom). [assumption].
% 0.72/1.01 Derived: -finer(A,B,C) | -topological_space(D,A) | element_of_collection(union_of_members(B),A). [resolve(5,b,2,b)].
% 0.72/1.01 6 finer(A,B,C) | -topological_space(C,A) | -topological_space(C,B) | -subset_collections(B,A) # label(finer_topology_27) # label(axiom). [assumption].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 Derived: finer(A,B,C) | -topological_space(C,A) | -topological_space(C,B) | -finer(A,B,D). [resolve(6,d,5,b)].
% 0.72/1.01 7 -open_covering(A,B,C) | subset_collections(A,C) # label(open_covering_97) # label(axiom). [assumption].
% 0.72/1.01 Derived: -open_covering(A,B,C) | -topological_space(D,C) | element_of_collection(union_of_members(A),C). [resolve(7,b,2,b)].
% 0.72/1.01 Derived: -open_covering(A,B,C) | finer(C,A,D) | -topological_space(D,C) | -topological_space(D,A). [resolve(7,b,6,d)].
% 0.72/1.01 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].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 9 -compact_space(A,B) | -open_covering(C,A,B) | subset_collections(f23(A,B,C),C) # label(compact_space_102) # label(axiom). [assumption].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 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)].
% 0.72/1.01 11 open(A,B,C) | -topological_space(B,C) | -element_of_collection(A,C) # label(open_set_20) # label(axiom). [assumption].
% 0.72/1.01 12 -open(A,B,C) | topological_space(B,C) # label(open_set_18) # label(axiom). [assumption].
% 0.72/1.01 13 -open(A,B,C) | element_of_collection(A,C) # label(open_set_19) # label(axiom). [assumption].
% 0.72/1.01 14 -closed(A,B,C) | open(relative_complement_sets(A,B),B,C) # label(closed_set_22) # label(axiom). [assumption].
% 0.72/1.01 Derived: -closed(A,B,C) | topological_space(B,C). [resolve(14,b,12,a)].
% 0.72/1.01 Derived: -closed(A,B,C) | element_of_collection(relative_complement_sets(A,B),C). [resolve(14,b,13,a)].
% 0.72/1.01 15 closed(A,B,C) | -topological_space(B,C) | -open(relative_complement_sets(A,B),B,C) # label(closed_set_23) # label(axiom). [assumption].
% 0.72/1.01 Derived: closed(A,B,C) | -topological_space(B,C) | -topological_space(B,C) | -element_of_collection(relative_complement_sets(A,B),C). [resolve(15,c,11,a)].
% 0.72/1.01 16 -element_of_set(A,interior(B,C,D)) | open(f13(B,C,D,A),C,D) # label(interior_51) # label(axiom). [assumption].
% 0.72/1.01 Derived: -element_of_set(A,interior(B,C,D)) | topological_space(C,D). [resolve(16,b,12,a)].
% 0.72/1.01 Derived: -element_of_set(A,interior(B,C,D)) | element_of_collection(f13(B,C,D,A),D). [resolve(16,b,13,a)].
% 0.72/1.01 17 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].
% 0.72/1.01 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(17,f,11,a)].
% 0.72/1.01 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(17,f,14,b)].
% 0.72/1.01 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(17,f,16,b)].
% 0.72/1.01 18 -neighborhood(A,B,C,D) | open(A,C,D) # label(neighborhood_60) # label(axiom). [assumption].
% 0.72/1.01 Derived: -neighborhood(A,B,C,D) | topological_space(C,D). [resolve(18,b,12,a)].
% 0.72/1.01 Derived: -neighborhood(A,B,C,D) | element_of_collection(A,D). [resolve(18,b,13,a)].
% 0.72/1.01 Derived: -neighborhood(relative_complement_sets(A,B),C,B,D) | closed(A,B,D) | -topological_space(B,D). [resolve(18,b,15,c)].
% 0.72/1.01 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(18,b,17,f)].
% 0.72/1.01 19 neighborhood(A,B,C,D) | -topological_space(C,D) | -open(A,C,D) | -element_of_set(B,A) # label(neighborhood_62) # label(axiom). [assumption].
% 0.72/1.01 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(19,c,11,a)].
% 0.72/1.01 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(19,c,14,b)].
% 0.72/1.01 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(19,c,16,b)].
% 0.72/1.01 Derived: neighborhood(A,B,C,D) | -topological_space(C,D) | -element_of_set(B,A) | -neighborhood(A,E,C,D). [resolve(19,c,18,b)].
% 0.72/1.01 20 open(cy,cx,ct) # label(problem_4_120) # label(negated_conjecture). [assumption].
% 0.72/1.01 Derived: topological_space(cx,ct). [resolve(20,a,12,a)].
% 0.72/1.01 Derived: element_of_collection(cy,ct). [resolve(20,a,13,a)].
% 0.72/1.01 Derived: element_of_set(A,interior(B,cx,ct)) | -topological_space(cx,ct) | -subset_sets(B,cx) | -element_of_set(A,cy) | -subset_sets(cy,B). [resolve(20,a,17,f)].
% 0.72/1.01 Derived: neighborhood(cy,A,cx,ct) | -topological_space(cx,ct) | -element_of_set(A,cy). [resolve(20,a,19,c)].
% 0.72/1.01 21 open(a,cy,subspace_topology(cx,ct,cy)) # label(problem_4_121) # label(negated_conjecture). [assumption].
% 0.72/1.01 Derived: topological_space(cy,subspace_topology(cx,ct,cy)). [resolve(21,a,12,a)].
% 0.72/1.01 Derived: element_of_collection(a,subspace_topology(cx,ct,cy)). [resolve(21,a,13,a)].
% 0.72/1.01 Derived: element_of_set(A,interior(B,cy,subspace_topology(cx,ct,cy))) | -topological_space(cy,subspace_topology(cx,ct,cy)) | -subset_sets(B,cy) | -element_of_set(A,a) | -subset_sets(a,B). [resolve(21,a,17,f)].
% 0.72/1.01 Derived: neighborhood(a,A,cy,subspace_topology(cx,ct,cy)) | -topological_space(cy,subspace_topology(cx,ct,cy)) | -element_of_set(A,a). [resolve(21,a,19,c)].
% 0.72/1.01 22 -open(a,cx,ct) # label(problem_4_122) # label(negated_conjecture). [assumption].
% 0.72/1.01 Derived: -topological_space(cx,ct) | -element_of_collection(a,ct). [resolve(22,a,11,a)].
% 0.72/1.01 Derived: -neighborhood(a,A,cx,ct). [resolve(22,a,18,b)].
% 0.72/1.01 23 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].
% 0.72/1.01 24 -closed(A,B,C) | topological_space(B,C) # label(closed_set_21) # label(axiom). [assumption].
% 0.72/1.01 25 -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].
% 0.72/1.01 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(23,d,25,c)].
% 0.72/1.01 26 -closed(A,B,C) | topological_space(B,C). [resolve(14,b,12,a)].
% 0.72/1.01 27 -closed(A,B,C) | element_of_collection(relative_complement_sets(A,B),C). [resolve(14,b,13,a)].
% 0.72/1.01 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(27,a,23,d)].
% 0.72/1.01 28 closed(A,B,C) | -topological_space(B,C) | -topological_space(B,C) | -element_of_collection(relative_complement_sets(A,B),C). [resolve(15,c,11,a)].
% 0.72/1.01 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(28,a,25,c)].
% 0.72/1.01 29 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(17,f,14,b)].
% 0.72/1.01 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(29,f,23,d)].
% 0.72/1.01 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(29,f,28,a)].
% 0.72/1.01 30 -neighborhood(relative_complement_sets(A,B),C,B,D) | closed(A,B,D) | -topological_space(B,D). [resolve(18,b,15,c)].
% 0.72/1.01 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(30,b,25,c)].
% 0.72/1.01 Derived: -neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | element_of_collection(relative_complement_sets(A,B),D). [resolve(30,b,27,a)].
% 0.72/1.01 31 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(19,c,14,b)].
% 0.72/1.01 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(31,d,23,d)].
% 0.72/1.01 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(31,d,28,a)].
% 0.72/1.01 32 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].
% 0.72/1.01 33 -basis(A,B) | equal_sets(union_of_members(B),A) # label(basis_for_topology_28) # label(axiom). [assumption].
% 0.72/1.01 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_set(C,f6(A,B,C,D,E)) # label(basis_for_topology_29) # label(axiom). [assumption].
% 0.72/1.01 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)) | element_of_collection(f6(A,B,C,D,E),B) # label(basis_for_topology_30) # label(axiom). [assumption].
% 0.72/1.01 36 -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].
% 0.72/1.01 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(32,a,34,a)].
% 0.72/1.01 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(32,a,35,a)].
% 0.72/1.01 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(32,a,36,a)].
% 0.72/1.01 37 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].
% 0.72/1.01 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(37,a,34,a)].
% 0.72/1.01 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(37,a,35,a)].
% 0.72/1.01 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(37,a,36,a)].
% 0.72/1.02 38 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].
% 0.72/1.02 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(38,a,34,a)].
% 0.72/1.02 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(38,a,35,a)].
% 0.72/1.02 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(38,a,36,a)].
% 0.72/1.02 39 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].
% 0.72/1.02 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(39,a,34,a)].
% 0.72/1.02 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(39,a,35,a)].
% 0.72/1.02 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(39,a,36,a)].
% 0.72/1.02 40 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].
% 0.72/1.02 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(40,a,34,a)].
% 0.72/1.02 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(40,a,35,a)].
% 0.72/1.02 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(40,a,36,a)].
% 0.72/1.02 41 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].
% 0.72/1.02 42 -limit_point(A,B,C,D) | topological_space(C,D) # label(limit_point_63) # label(axiom). [assumption].
% 0.72/1.02 43 -limit_point(A,B,C,D) | subset_sets(B,C) # label(limit_point_64) # label(axiom). [assumption].
% 0.72/1.02 44 -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].
% 0.72/1.02 45 -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].
% 0.72/1.02 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(41,a,44,a)].
% 0.72/1.02 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(41,a,45,a)].
% 0.72/1.02 46 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].
% 0.72/1.02 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(46,a,44,a)].
% 0.72/1.02 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(46,a,45,a)].
% 0.72/1.02 47 hausdorff(A,B) | -topological_space(A,B) | element_of_set(f19(A,B),A) # label(hausdorff_77) # label(axiom). [assumption].
% 0.72/1.02 48 -hausdorff(A,B) | topological_space(A,B) # label(hausdorff_73) # label(axiom). [assumption].
% 0.72/1.02 49 -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].
% 0.72/1.02 50 -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].
% 0.72/1.02 51 -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].
% 0.72/1.02 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(47,a,49,a)].
% 0.72/1.02 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(47,a,50,a)].
% 0.72/1.02 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(47,a,51,a)].
% 0.72/1.02 52 hausdorff(A,B) | -topological_space(A,B) | element_of_set(f20(A,B),A) # label(hausdorff_78) # label(axiom). [assumption].
% 0.72/1.02 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(52,a,49,a)].
% 0.72/1.02 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(52,a,50,a)].
% 0.72/1.02 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(52,a,51,a)].
% 0.72/1.02 53 hausdorff(A,B) | -topological_space(A,B) | -eq_p(f19(A,B),f20(A,B)) # label(hausdorff_79) # label(axiom). [assumption].
% 0.72/1.02 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(53,a,49,a)].
% 0.72/1.02 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(53,a,50,a)].
% 0.72/1.02 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(53,a,51,a)].
% 0.72/1.02 54 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].
% 0.72/1.02 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(54,a,49,a)].
% 0.72/1.02 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(54,a,50,a)].
% 0.72/1.03 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(54,a,51,a)].
% 0.72/1.03 55 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].
% 0.72/1.03 56 -separation(A,B,C,D) | topological_space(C,D) # label(separation_81) # label(axiom). [assumption].
% 0.72/1.03 57 -separation(A,B,C,D) | -equal_sets(A,empty_set) # label(separation_82) # label(axiom). [assumption].
% 0.72/1.03 58 -separation(A,B,C,D) | -equal_sets(B,empty_set) # label(separation_83) # label(axiom). [assumption].
% 0.72/1.03 59 -separation(A,B,C,D) | element_of_collection(A,D) # label(separation_84) # label(axiom). [assumption].
% 0.72/1.03 60 -separation(A,B,C,D) | element_of_collection(B,D) # label(separation_85) # label(axiom). [assumption].
% 0.72/1.03 61 -separation(A,B,C,D) | equal_sets(union_of_sets(A,B),C) # label(separation_86) # label(axiom). [assumption].
% 0.72/1.03 62 -separation(A,B,C,D) | disjoint_s(A,B) # label(separation_87) # label(axiom). [assumption].
% 0.72/1.03 63 -connected_space(A,B) | -separation(C,D,A,B) # label(connected_space_90) # label(axiom). [assumption].
% 0.72/1.03 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(63,b,55,a)].
% 0.72/1.03 64 connected_space(A,B) | -topological_space(A,B) | separation(f21(A,B),f22(A,B),A,B) # label(connected_space_91) # label(axiom). [assumption].
% 0.72/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | -equal_sets(f21(A,B),empty_set). [resolve(64,c,57,a)].
% 0.72/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | -equal_sets(f22(A,B),empty_set). [resolve(64,c,58,a)].
% 0.72/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | element_of_collection(f21(A,B),B). [resolve(64,c,59,a)].
% 0.72/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | element_of_collection(f22(A,B),B). [resolve(64,c,60,a)].
% 0.72/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | equal_sets(union_of_sets(f21(A,B),f22(A,B)),A). [resolve(64,c,61,a)].
% 0.72/1.03 Derived: connected_space(A,B) | -topological_space(A,B) | disjoint_s(f21(A,B),f22(A,B)). [resolve(64,c,62,a)].
% 0.72/1.03 65 -connected_set(A,B,C) | connected_space(A,subspace_topology(B,C,A)) # label(connected_set_94) # label(axiom). [assumption].
% 0.72/1.03 66 -connected_space(A,B) | topological_space(A,B) # label(connected_space_89) # label(axiom). [assumption].
% 0.72/1.03 Derived: -connected_set(A,B,C) | topological_space(A,subspace_topology(B,C,A)). [resolve(65,b,66,a)].
% 0.72/1.03 67 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].
% 0.72/1.03 68 -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(63,b,55,a)].
% 0.72/1.03 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(68,a,65,b)].
% 0.72/1.03 69 connected_space(A,B) | -topological_space(A,B) | -equal_sets(f21(A,B),empty_set). [resolve(64,c,57,a)].
% 0.72/1.03 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(69,a,67,d)].
% 0.72/1.03 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(69,a,68,a)].
% 0.72/1.04 70 connected_space(A,B) | -topological_space(A,B) | -equal_sets(f22(A,B),empty_set). [resolve(64,c,58,a)].
% 0.72/1.04 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(70,a,67,d)].
% 0.72/1.04 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(70,a,68,a)].
% 0.72/1.04 71 connected_space(A,B) | -topological_space(A,B) | element_of_collection(f21(A,B),B). [resolve(64,c,59,a)].
% 0.72/1.04 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(71,a,67,d)].
% 0.72/1.04 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(71,a,68,a)].
% 0.72/1.04 72 connected_space(A,B) | -topological_space(A,B) | element_of_collection(f22(A,B),B). [resolve(64,c,60,a)].
% 0.72/1.04 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(72,a,67,d)].
% 0.72/1.04 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(72,a,68,a)].
% 0.72/1.04 73 connected_space(A,B) | -topological_space(A,B) | equal_sets(union_of_sets(f21(A,B),f22(A,B)),A). [resolve(64,c,61,a)].
% 0.72/1.04 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(73,a,67,d)].
% 0.72/1.04 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(73,a,68,a)].
% 0.72/1.04 74 connected_space(A,B) | -topological_space(A,B) | disjoint_s(f21(A,B),f22(A,B)). [resolve(64,c,62,a)].
% 0.72/1.04 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(74,a,67,d)].
% 0.72/1.04 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(74,a,68,a)].
% 0.72/1.04 75 -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(69,a,67,d)].
% 0.72/1.04 76 -connected_set(A,B,C) | topological_space(B,C) # label(connected_set_92) # label(axiom). [assumption].
% 0.72/1.04 77 -connected_set(A,B,C) | subset_sets(A,B) # label(connected_set_93) # label(axiom). [assumption].
% 0.72/1.04 78 -connected_set(A,B,C) | topological_space(A,subspace_topology(B,C,A)). [resolve(65,b,66,a)].
% 0.72/1.04 79 -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(68,a,65,b)].
% 0.72/1.04 80 -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(70,a,67,d)].
% 0.72/1.07 81 -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(71,a,67,d)].
% 0.72/1.07 82 -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(72,a,67,d)].
% 0.72/1.07 83 -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(73,a,67,d)].
% 0.72/1.07 84 -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(74,a,67,d)].
% 0.72/1.07 85 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)].
% 0.72/1.07 86 -compact_space(A,B) | -open_covering(C,A,B) | finite(f23(A,B,C)) # label(compact_space_101) # label(axiom). [assumption].
% 0.72/1.07 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(f4(C,f24(A,B)),f24(A,B)). [resolve(10,d,3,f)].
% 0.72/1.07 88 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)].
% 0.72/1.07 89 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)].
% 0.72/1.07 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(89,c,86,c)].
% 0.72/1.07 90 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)].
% 0.72/1.07 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(90,c,86,c)].
% 0.72/1.07 91 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)].
% 0.72/1.07 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(91,c,86,c)].
% 0.72/1.07 92 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].
% 0.72/1.07 93 -compact_set(A,B,C) | topological_space(B,C) # label(compact_set_106) # label(axiom). [assumption].
% 0.72/1.07 94 -compact_set(A,B,C) | subset_sets(A,B) # label(compact_set_107) # label(axiom). [assumption].
% 0.72/1.07 95 -compact_set(A,B,C) | compact_space(A,subspace_topology(B,C,A)) # label(compact_set_108) # label(axiom). [assumption].
% 0.72/1.07
% 0.72/1.07 ============================== end predicate elimination =============
% 0.72/1.07
% 0.72/1.07 Auto_denials: (non-Horn, no changes).
% 0.72/1.07
% 0.72/1.07 Term ordering decisions:
% 0.72/1.07 Function symbol KB weights: empty_set=1. cy=1. ct=1. cx=1. a=1. intersection_of_sets=1. f5=1. relative_complement_sets=1. f19=1. f20=1. f7=1. fCputime limit exceeded (core dumped)
%------------------------------------------------------------------------------