Step-by-step derivation
S001 · implies
Before
source operation=topbasis-existing-topology-audit;original snapshot and mathematics retained
After
explicit ordered source output fields=[topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology]
Conditions: none
Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
original basis candidate:ORIGINAL complete candidate family=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]];carrier Fin(5)
After
within-set numeric normalization=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]];no duplicate points or duplicate FAMILY elements;WHOLE original family already in required key order;empty family allowed as candidate
Conditions: none
A topological basis is a family of subsets,not a vector-space basis;its size may exceed n.Normalize point order within each set without changing original family indices or deleting duplicates to repair data.An empty candidate family can still be audited.
Step identifiers and dependencies
Rule: topbasis_canonical_candidate_family_not_semantic_reordering · Uses: G001, S001
S003 · implies
Before
COMPLETE original candidate basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]];ALL carrier points0..4
After
ALL [point,containing_basis_indices] rows=[[0, [0, 1, 2]], [1, [1, 3, 4]], [2, [2, 4, 5]], [3, [6, 7]], [4, [7, 8]]];covers WHOLE carrier=Yes
Conditions: none
Every carrier point needs a containing candidate element for coverage.This does not replace local intersection refinement.With n≥1 an empty family fails coverage rather than being discarded as an unauditable candidate.
Step identifiers and dependencies
Rule: topbasis_every_point_cover_separate_from_refinement · Uses: G001, S002
S004 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 0];actual basis elements=[[0], [0]];FULL intersection=[0]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S003
S005 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 1];actual basis elements=[[0], [0, 1]];FULL intersection=[0]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S004
S006 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 2];actual basis elements=[[0], [0, 2]];FULL intersection=[0]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S005
S007 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 3];actual basis elements=[[0], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S006
S008 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 4];actual basis elements=[[0], [1, 2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S007
S009 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 5];actual basis elements=[[0], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S008
S010 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 6];actual basis elements=[[0], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S009
S011 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 7];actual basis elements=[[0], [3, 4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S010
S012 · implies
Before
ORIGINAL ordered basis positions(left,right)=[0, 8];actual basis elements=[[0], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S011
S013 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 0];actual basis elements=[[0, 1], [0]];FULL intersection=[0]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S012
S014 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 1];actual basis elements=[[0, 1], [0, 1]];FULL intersection=[0, 1]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0, 1]], [1, [1, 3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S013
S015 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 2];actual basis elements=[[0, 1], [0, 2]];FULL intersection=[0]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S014
S016 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 3];actual basis elements=[[0, 1], [1]];FULL intersection=[1]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[1, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S015
S017 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 4];actual basis elements=[[0, 1], [1, 2]];FULL intersection=[1]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[1, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S016
S018 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 5];actual basis elements=[[0, 1], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S017
S019 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 6];actual basis elements=[[0, 1], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S018
S020 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 7];actual basis elements=[[0, 1], [3, 4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S019
S021 · implies
Before
ORIGINAL ordered basis positions(left,right)=[1, 8];actual basis elements=[[0, 1], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S020
S022 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 0];actual basis elements=[[0, 2], [0]];FULL intersection=[0]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S021
S023 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 1];actual basis elements=[[0, 2], [0, 1]];FULL intersection=[0]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S022
S024 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 2];actual basis elements=[[0, 2], [0, 2]];FULL intersection=[0, 2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[0, [0, 2]], [2, [2, 5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S023
S025 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 3];actual basis elements=[[0, 2], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S024
S026 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 4];actual basis elements=[[0, 2], [1, 2]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S025
S027 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 5];actual basis elements=[[0, 2], [2]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S026
S028 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 6];actual basis elements=[[0, 2], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S027
S029 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 7];actual basis elements=[[0, 2], [3, 4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S028
S030 · implies
Before
ORIGINAL ordered basis positions(left,right)=[2, 8];actual basis elements=[[0, 2], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S029
S031 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 0];actual basis elements=[[1], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S030
S032 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 1];actual basis elements=[[1], [0, 1]];FULL intersection=[1]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[1, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S031
S033 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 2];actual basis elements=[[1], [0, 2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S032
S034 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 3];actual basis elements=[[1], [1]];FULL intersection=[1]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[1, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S033
S035 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 4];actual basis elements=[[1], [1, 2]];FULL intersection=[1]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[1, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S034
S036 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 5];actual basis elements=[[1], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S035
S037 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 6];actual basis elements=[[1], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S036
S038 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 7];actual basis elements=[[1], [3, 4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S037
S039 · implies
Before
ORIGINAL ordered basis positions(left,right)=[3, 8];actual basis elements=[[1], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S038
S040 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 0];actual basis elements=[[1, 2], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S039
S041 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 1];actual basis elements=[[1, 2], [0, 1]];FULL intersection=[1]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[1, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S040
S042 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 2];actual basis elements=[[1, 2], [0, 2]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S041
S043 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 3];actual basis elements=[[1, 2], [1]];FULL intersection=[1]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[1, [3]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S042
S044 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 4];actual basis elements=[[1, 2], [1, 2]];FULL intersection=[1, 2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[1, [3, 4]], [2, [4, 5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S043
S045 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 5];actual basis elements=[[1, 2], [2]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S044
S046 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 6];actual basis elements=[[1, 2], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S045
S047 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 7];actual basis elements=[[1, 2], [3, 4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S046
S048 · implies
Before
ORIGINAL ordered basis positions(left,right)=[4, 8];actual basis elements=[[1, 2], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S047
S049 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 0];actual basis elements=[[2], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S048
S050 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 1];actual basis elements=[[2], [0, 1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S049
S051 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 2];actual basis elements=[[2], [0, 2]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S050
S052 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 3];actual basis elements=[[2], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S051
S053 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 4];actual basis elements=[[2], [1, 2]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S052
S054 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 5];actual basis elements=[[2], [2]];FULL intersection=[2]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[2, [5]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S053
S055 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 6];actual basis elements=[[2], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S054
S056 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 7];actual basis elements=[[2], [3, 4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S055
S057 · implies
Before
ORIGINAL ordered basis positions(left,right)=[5, 8];actual basis elements=[[2], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S056
S058 · implies
Before
ORIGINAL ordered basis positions(left,right)=[6, 0];actual basis elements=[[3], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S057
S059 · implies
Before
ORIGINAL ordered basis positions(left,right)=[6, 1];actual basis elements=[[3], [0, 1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S058
S060 · implies
Before
ORIGINAL ordered basis positions(left,right)=[6, 2];actual basis elements=[[3], [0, 2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S059
S061 · implies
Before
ORIGINAL ordered basis positions(left,right)=[6, 3];actual basis elements=[[3], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S060
S062 · implies
Before
ORIGINAL ordered basis positions(left,right)=[6, 4];actual basis elements=[[3], [1, 2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S061
S063 · implies
Before
ORIGINAL ordered basis positions(left,right)=[6, 5];actual basis elements=[[3], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S062
S064 · implies
Before
ORIGINAL ordered basis positions(left,right)=[6, 6];actual basis elements=[[3], [3]];FULL intersection=[3]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[3, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S063
S065 · implies
Before
ORIGINAL ordered basis positions(left,right)=[6, 7];actual basis elements=[[3], [3, 4]];FULL intersection=[3]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[3, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S064
S066 · implies
Before
ORIGINAL ordered basis positions(left,right)=[6, 8];actual basis elements=[[3], [4]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S065
S067 · implies
Before
ORIGINAL ordered basis positions(left,right)=[7, 0];actual basis elements=[[3, 4], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S066
S068 · implies
Before
ORIGINAL ordered basis positions(left,right)=[7, 1];actual basis elements=[[3, 4], [0, 1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S067
S069 · implies
Before
ORIGINAL ordered basis positions(left,right)=[7, 2];actual basis elements=[[3, 4], [0, 2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S068
S070 · implies
Before
ORIGINAL ordered basis positions(left,right)=[7, 3];actual basis elements=[[3, 4], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S069
S071 · implies
Before
ORIGINAL ordered basis positions(left,right)=[7, 4];actual basis elements=[[3, 4], [1, 2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S070
S072 · implies
Before
ORIGINAL ordered basis positions(left,right)=[7, 5];actual basis elements=[[3, 4], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S071
S073 · implies
Before
ORIGINAL ordered basis positions(left,right)=[7, 6];actual basis elements=[[3, 4], [3]];FULL intersection=[3]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[3, [6]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S072
S074 · implies
Before
ORIGINAL ordered basis positions(left,right)=[7, 7];actual basis elements=[[3, 4], [3, 4]];FULL intersection=[3, 4]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[3, [6, 7]], [4, [7, 8]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S073
S075 · implies
Before
ORIGINAL ordered basis positions(left,right)=[7, 8];actual basis elements=[[3, 4], [4]];FULL intersection=[4]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[4, [8]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S074
S076 · implies
Before
ORIGINAL ordered basis positions(left,right)=[8, 0];actual basis elements=[[4], [0]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S075
S077 · implies
Before
ORIGINAL ordered basis positions(left,right)=[8, 1];actual basis elements=[[4], [0, 1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S076
S078 · implies
Before
ORIGINAL ordered basis positions(left,right)=[8, 2];actual basis elements=[[4], [0, 2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S077
S079 · implies
Before
ORIGINAL ordered basis positions(left,right)=[8, 3];actual basis elements=[[4], [1]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S078
S080 · implies
Before
ORIGINAL ordered basis positions(left,right)=[8, 4];actual basis elements=[[4], [1, 2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S079
S081 · implies
Before
ORIGINAL ordered basis positions(left,right)=[8, 5];actual basis elements=[[4], [2]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S080
S082 · implies
Before
ORIGINAL ordered basis positions(left,right)=[8, 6];actual basis elements=[[4], [3]];FULL intersection=[]
After
ALL [intersection_point,ALL refining original basis indices] rows=[];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S081
S083 · implies
Before
ORIGINAL ordered basis positions(left,right)=[8, 7];actual basis elements=[[4], [3, 4]];FULL intersection=[4]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[4, [8]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S082
S084 · implies
Before
ORIGINAL ordered basis positions(left,right)=[8, 8];actual basis elements=[[4], [4]];FULL intersection=[4]
After
ALL [intersection_point,ALL refining original basis indices] rows=[[4, [8]]];empty intersection gives VACUOUS local obligation;first failure retained in source order
Conditions: none
For each intersection point x require some candidate basis element containing x and contained in the intersection.The entire intersection need not itself be a basis element.Check all refiners before declaring absence,and keep the first left/right/point failure in source order.
Step identifiers and dependencies
Rule: topbasis_full_local_point_refinement_not_intersection_membership · Uses: G001, S083
S085 · implies
Before
ACTUAL cover=Yes;ALL local point obligations refined=Yes;first failure[left,right,point,intersection]=null
After
abstract basis validity=Yes is cover AND LOCAL refinement;not closure of the family under intersections and not a given-topology certificate yet
Conditions: none
Their conjunction gives an abstract basis.Coverage alone is insufficient,and a valid basis need not be intersection-closed.A basis for a separately supplied topology also needs openness and generation of every given open.
Step identifiers and dependencies
Rule: topbasis_cover_and_local_refinement_conjunction · Uses: G001, S084
S086 · implies
Before
given topology candidate:ORIGINAL complete candidate family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]];carrier Fin(5)
After
within-set numeric normalization=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]];no duplicate points or duplicate FAMILY elements;WHOLE original family already in required key order;empty family allowed as candidate
Conditions: none
A topological basis is a family of subsets,not a vector-space basis;its size may exceed n.Normalize point order within each set without changing original family indices or deleting duplicates to repair data.An empty candidate family can still be audited.
Step identifiers and dependencies
Rule: topbasis_canonical_candidate_family_not_semantic_reordering · Uses: G001, S085
S087 · implies
Before
COMPLETE ORIGINAL given topology CANDIDATE=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]];carrier Fin(5)
After
empty-set listed=Yes;full-carrier listed=Yes;candidate not assumed to satisfy topology axioms
Conditions: none
This operation audits a given candidate family,retaining false flags for invalid candidates.Structural qualification alone does not establish a topology,and blanket rejection would omit the requested audit.
Step identifiers and dependencies
Rule: topbasis_given_topology_candidate_not_blanket_assumed_valid · Uses: G001, S086
S088 · implies
Before
given candidate left-open position=0;set=[];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[], 0, [], 0], [[0], 1, [], 0], [[0, 1], 2, [], 0], [[0, 1, 2], 3, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 4], 9, [], 0], [[0, 2], 10, [], 0], [[0, 2, 3], 11, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 4], 13, [], 0], [[0, 3], 14, [], 0], [[0, 3, 4], 15, [], 0], [[0, 4], 16, [], 0], [[1], 17, [], 0], [[1, 2], 18, [], 0], [[1, 2, 3], 19, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 4], 21, [], 0], [[1, 3], 22, [], 0], [[1, 3, 4], 23, [], 0], [[1, 4], 24, [], 0], [[2], 25, [], 0], [[2, 3], 26, [], 0], [[2, 3, 4], 27, [], 0], [[2, 4], 28, [], 0], [[3], 29, [], 0], [[3, 4], 30, [], 0], [[4], 31, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S087
S089 · implies
Before
given candidate left-open position=1;set=[0];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0], 1, [], 0], [[0], 1, [0], 1], [[0, 1], 2, [0], 1], [[0, 1, 2], 3, [0], 1], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 3], 7, [0], 1], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 4], 9, [0], 1], [[0, 2], 10, [0], 1], [[0, 2, 3], 11, [0], 1], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 4], 13, [0], 1], [[0, 3], 14, [0], 1], [[0, 3, 4], 15, [0], 1], [[0, 4], 16, [0], 1], [[0, 1], 2, [], 0], [[0, 1, 2], 3, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 4], 9, [], 0], [[0, 2], 10, [], 0], [[0, 2, 3], 11, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 4], 13, [], 0], [[0, 3], 14, [], 0], [[0, 3, 4], 15, [], 0], [[0, 4], 16, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S088
S090 · implies
Before
given candidate left-open position=2;set=[0, 1];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1], 2, [], 0], [[0, 1], 2, [0], 1], [[0, 1], 2, [0, 1], 2], [[0, 1, 2], 3, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 4], 6, [0, 1], 2], [[0, 1, 3], 7, [0, 1], 2], [[0, 1, 3, 4], 8, [0, 1], 2], [[0, 1, 4], 9, [0, 1], 2], [[0, 1, 2], 3, [0], 1], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 3], 7, [0], 1], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 4], 9, [0], 1], [[0, 1], 2, [1], 17], [[0, 1, 2], 3, [1], 17], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 3], 7, [1], 17], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 4], 9, [1], 17], [[0, 1, 2], 3, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 4], 9, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S089
S091 · implies
Before
given candidate left-open position=3;set=[0, 1, 2];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 2], 3, [], 0], [[0, 1, 2], 3, [0], 1], [[0, 1, 2], 3, [0, 1], 2], [[0, 1, 2], 3, [0, 1, 2], 3], [[0, 1, 2, 3], 4, [0, 1, 2], 3], [[0, 1, 2, 3, 4], 5, [0, 1, 2], 3], [[0, 1, 2, 4], 6, [0, 1, 2], 3], [[0, 1, 2, 3], 4, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 4], 6, [0, 1], 2], [[0, 1, 2], 3, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 4], 6, [0, 2], 10], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2], 3, [1], 17], [[0, 1, 2], 3, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 4], 6, [1, 2], 18], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2], 3, [2], 25], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S090
S092 · implies
Before
given candidate left-open position=4;set=[0, 1, 2, 3];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3], 4, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 1, 2], 3], [[0, 1, 2, 3], 4, [0, 1, 2, 3], 4], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 3], 4], [[0, 1, 2, 3, 4], 5, [0, 1, 2], 3], [[0, 1, 2, 3], 4, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3], 4, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3], 4, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S091
S093 · implies
Before
given candidate left-open position=5;set=[0, 1, 2, 3, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1, 2], 3], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 3], 4], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 3, 4], 5], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 4], 6], [[0, 1, 2, 3, 4], 5, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1, 3, 4], 8], [[0, 1, 2, 3, 4], 5, [0, 1, 4], 9], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2, 3, 4], 12], [[0, 1, 2, 3, 4], 5, [0, 2, 4], 13], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3, 4], 15], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2, 3, 4], 20], [[0, 1, 2, 3, 4], 5, [1, 2, 4], 21], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3, 4], 23], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3, 4], 27], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S092
S094 · implies
Before
given candidate left-open position=6;set=[0, 1, 2, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2, 4], 6, [0, 1], 2], [[0, 1, 2, 4], 6, [0, 1, 2], 3], [[0, 1, 2, 3, 4], 5, [0, 1, 2], 3], [[0, 1, 2, 3, 4], 5, [0, 1, 2, 4], 6], [[0, 1, 2, 4], 6, [0, 1, 2, 4], 6], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1, 4], 9], [[0, 1, 2, 4], 6, [0, 1, 4], 9], [[0, 1, 2, 4], 6, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2, 4], 13], [[0, 1, 2, 4], 6, [0, 2, 4], 13], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 4], 6, [0, 4], 16], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2, 4], 6, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2, 4], 21], [[0, 1, 2, 4], 6, [1, 2, 4], 21], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 4], 6, [1, 4], 24], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 4], 6, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S093
S095 · implies
Before
given candidate left-open position=7;set=[0, 1, 3];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 3], 7, [], 0], [[0, 1, 3], 7, [0], 1], [[0, 1, 3], 7, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 1], 2], [[0, 1, 2, 3], 4, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 3], 7, [0, 1, 3], 7], [[0, 1, 3, 4], 8, [0, 1, 3], 7], [[0, 1, 3, 4], 8, [0, 1], 2], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3], 4, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 3], 7, [0, 3], 14], [[0, 1, 3, 4], 8, [0, 3], 14], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 3], 7, [1], 17], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3], 4, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 3], 7, [1, 3], 22], [[0, 1, 3, 4], 8, [1, 3], 22], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 3], 7, [3], 29], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S094
S096 · implies
Before
given candidate left-open position=8;set=[0, 1, 3, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 3, 4], 8, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1, 3], 7], [[0, 1, 2, 3, 4], 5, [0, 1, 3, 4], 8], [[0, 1, 2, 3, 4], 5, [0, 1, 4], 9], [[0, 1, 3, 4], 8, [0, 1, 3], 7], [[0, 1, 3, 4], 8, [0, 1, 3, 4], 8], [[0, 1, 3, 4], 8, [0, 1, 4], 9], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3, 4], 15], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 3, 4], 8, [0, 3], 14], [[0, 1, 3, 4], 8, [0, 3, 4], 15], [[0, 1, 3, 4], 8, [0, 4], 16], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3, 4], 23], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 3, 4], 8, [1, 3], 22], [[0, 1, 3, 4], 8, [1, 3, 4], 23], [[0, 1, 3, 4], 8, [1, 4], 24], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [3, 4], 30], [[0, 1, 3, 4], 8, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S095
S097 · implies
Before
given candidate left-open position=9;set=[0, 1, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 1, 4], 9, [], 0], [[0, 1, 4], 9, [0], 1], [[0, 1, 4], 9, [0, 1], 2], [[0, 1, 2, 4], 6, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1], 2], [[0, 1, 2, 3, 4], 5, [0, 1, 4], 9], [[0, 1, 2, 4], 6, [0, 1, 4], 9], [[0, 1, 3, 4], 8, [0, 1], 2], [[0, 1, 3, 4], 8, [0, 1, 4], 9], [[0, 1, 4], 9, [0, 1, 4], 9], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 4], 6, [0, 4], 16], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 3, 4], 8, [0, 4], 16], [[0, 1, 4], 9, [0, 4], 16], [[0, 1, 4], 9, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 4], 6, [1, 4], 24], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 3, 4], 8, [1, 4], 24], [[0, 1, 4], 9, [1, 4], 24], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [4], 31], [[0, 1, 4], 9, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S096
S098 · implies
Before
given candidate left-open position=10;set=[0, 2];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 2], 10, [], 0], [[0, 2], 10, [0], 1], [[0, 1, 2], 3, [0], 1], [[0, 1, 2], 3, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 4], 6, [0, 2], 10], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 2], 10, [0, 2], 10], [[0, 2, 3], 11, [0, 2], 10], [[0, 2, 3, 4], 12, [0, 2], 10], [[0, 2, 4], 13, [0, 2], 10], [[0, 2, 3], 11, [0], 1], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 4], 13, [0], 1], [[0, 1, 2], 3, [], 0], [[0, 1, 2], 3, [2], 25], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 2], 10, [2], 25], [[0, 2, 3], 11, [2], 25], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 4], 13, [2], 25], [[0, 2, 3], 11, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 4], 13, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S097
S099 · implies
Before
given candidate left-open position=11;set=[0, 2, 3];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 2, 3], 11, [], 0], [[0, 2, 3], 11, [0], 1], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3], 4, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3], 4, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 2, 3], 11, [0, 2], 10], [[0, 2, 3], 11, [0, 2, 3], 11], [[0, 2, 3, 4], 12, [0, 2, 3], 11], [[0, 2, 3, 4], 12, [0, 2], 10], [[0, 2, 3], 11, [0, 3], 14], [[0, 2, 3, 4], 12, [0, 3], 14], [[0, 2, 3, 4], 12, [0], 1], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3], 4, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 2, 3], 11, [2], 25], [[0, 2, 3], 11, [2, 3], 26], [[0, 2, 3, 4], 12, [2, 3], 26], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3], 11, [3], 29], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S098
S100 · implies
Before
given candidate left-open position=12;set=[0, 2, 3, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2, 3], 11], [[0, 1, 2, 3, 4], 5, [0, 2, 3, 4], 12], [[0, 1, 2, 3, 4], 5, [0, 2, 4], 13], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3, 4], 15], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 2, 3, 4], 12, [0, 2], 10], [[0, 2, 3, 4], 12, [0, 2, 3], 11], [[0, 2, 3, 4], 12, [0, 2, 3, 4], 12], [[0, 2, 3, 4], 12, [0, 2, 4], 13], [[0, 2, 3, 4], 12, [0, 3], 14], [[0, 2, 3, 4], 12, [0, 3, 4], 15], [[0, 2, 3, 4], 12, [0, 4], 16], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3, 4], 27], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3, 4], 12, [2, 3], 26], [[0, 2, 3, 4], 12, [2, 3, 4], 27], [[0, 2, 3, 4], 12, [2, 4], 28], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [3, 4], 30], [[0, 2, 3, 4], 12, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S099
S101 · implies
Before
given candidate left-open position=13;set=[0, 2, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 2, 4], 13, [], 0], [[0, 2, 4], 13, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2, 4], 6, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2], 10], [[0, 1, 2, 3, 4], 5, [0, 2, 4], 13], [[0, 1, 2, 4], 6, [0, 2, 4], 13], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 4], 6, [0, 4], 16], [[0, 2, 4], 13, [0, 2], 10], [[0, 2, 3, 4], 12, [0, 2], 10], [[0, 2, 3, 4], 12, [0, 2, 4], 13], [[0, 2, 4], 13, [0, 2, 4], 13], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 3, 4], 12, [0, 4], 16], [[0, 2, 4], 13, [0, 4], 16], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 4], 6, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 2, 4], 13, [2], 25], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3, 4], 12, [2, 4], 28], [[0, 2, 4], 13, [2, 4], 28], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [4], 31], [[0, 2, 4], 13, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S100
S102 · implies
Before
given candidate left-open position=14;set=[0, 3];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 3], 14, [], 0], [[0, 3], 14, [0], 1], [[0, 1, 3], 7, [0], 1], [[0, 1, 2, 3], 4, [0], 1], [[0, 1, 2, 3], 4, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 3], 7, [0, 3], 14], [[0, 1, 3, 4], 8, [0, 3], 14], [[0, 1, 3, 4], 8, [0], 1], [[0, 2, 3], 11, [0], 1], [[0, 2, 3], 11, [0, 3], 14], [[0, 2, 3, 4], 12, [0, 3], 14], [[0, 2, 3, 4], 12, [0], 1], [[0, 3], 14, [0, 3], 14], [[0, 3, 4], 15, [0, 3], 14], [[0, 3, 4], 15, [0], 1], [[0, 1, 3], 7, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 3], 7, [3], 29], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [], 0], [[0, 2, 3], 11, [], 0], [[0, 2, 3], 11, [3], 29], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [], 0], [[0, 3], 14, [3], 29], [[0, 3, 4], 15, [3], 29], [[0, 3, 4], 15, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S101
S103 · implies
Before
given candidate left-open position=15;set=[0, 3, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 3, 4], 15, [], 0], [[0, 3, 4], 15, [0], 1], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 3], 14], [[0, 1, 2, 3, 4], 5, [0, 3, 4], 15], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 3, 4], 8, [0, 3], 14], [[0, 1, 3, 4], 8, [0, 3, 4], 15], [[0, 1, 3, 4], 8, [0, 4], 16], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 3, 4], 12, [0, 3], 14], [[0, 2, 3, 4], 12, [0, 3, 4], 15], [[0, 2, 3, 4], 12, [0, 4], 16], [[0, 3, 4], 15, [0, 3], 14], [[0, 3, 4], 15, [0, 3, 4], 15], [[0, 3, 4], 15, [0, 4], 16], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [3, 4], 30], [[0, 1, 3, 4], 8, [4], 31], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [3, 4], 30], [[0, 2, 3, 4], 12, [4], 31], [[0, 3, 4], 15, [3], 29], [[0, 3, 4], 15, [3, 4], 30], [[0, 3, 4], 15, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S102
S104 · implies
Before
given candidate left-open position=16;set=[0, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[0, 4], 16, [], 0], [[0, 4], 16, [0], 1], [[0, 1, 4], 9, [0], 1], [[0, 1, 2, 4], 6, [0], 1], [[0, 1, 2, 3, 4], 5, [0], 1], [[0, 1, 2, 3, 4], 5, [0, 4], 16], [[0, 1, 2, 4], 6, [0, 4], 16], [[0, 1, 3, 4], 8, [0], 1], [[0, 1, 3, 4], 8, [0, 4], 16], [[0, 1, 4], 9, [0, 4], 16], [[0, 2, 4], 13, [0], 1], [[0, 2, 3, 4], 12, [0], 1], [[0, 2, 3, 4], 12, [0, 4], 16], [[0, 2, 4], 13, [0, 4], 16], [[0, 3, 4], 15, [0], 1], [[0, 3, 4], 15, [0, 4], 16], [[0, 4], 16, [0, 4], 16], [[0, 1, 4], 9, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [4], 31], [[0, 1, 4], 9, [4], 31], [[0, 2, 4], 13, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [4], 31], [[0, 2, 4], 13, [4], 31], [[0, 3, 4], 15, [], 0], [[0, 3, 4], 15, [4], 31], [[0, 4], 16, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S103
S105 · implies
Before
given candidate left-open position=17;set=[1];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1], 17, [], 0], [[0, 1], 2, [], 0], [[0, 1], 2, [1], 17], [[0, 1, 2], 3, [1], 17], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 3], 7, [1], 17], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 4], 9, [1], 17], [[0, 1, 2], 3, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 4], 9, [], 0], [[1], 17, [1], 17], [[1, 2], 18, [1], 17], [[1, 2, 3], 19, [1], 17], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 4], 21, [1], 17], [[1, 3], 22, [1], 17], [[1, 3, 4], 23, [1], 17], [[1, 4], 24, [1], 17], [[1, 2], 18, [], 0], [[1, 2, 3], 19, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 4], 21, [], 0], [[1, 3], 22, [], 0], [[1, 3, 4], 23, [], 0], [[1, 4], 24, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S104
S106 · implies
Before
given candidate left-open position=18;set=[1, 2];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 2], 18, [], 0], [[0, 1, 2], 3, [], 0], [[0, 1, 2], 3, [1], 17], [[0, 1, 2], 3, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 4], 6, [1, 2], 18], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2], 3, [2], 25], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[1, 2], 18, [1], 17], [[1, 2], 18, [1, 2], 18], [[1, 2, 3], 19, [1, 2], 18], [[1, 2, 3, 4], 20, [1, 2], 18], [[1, 2, 4], 21, [1, 2], 18], [[1, 2, 3], 19, [1], 17], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 4], 21, [1], 17], [[1, 2], 18, [2], 25], [[1, 2, 3], 19, [2], 25], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 4], 21, [2], 25], [[1, 2, 3], 19, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 4], 21, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S105
S107 · implies
Before
given candidate left-open position=19;set=[1, 2, 3];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 2, 3], 19, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3], 4, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3], 4, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3], 4, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[1, 2, 3], 19, [1], 17], [[1, 2, 3], 19, [1, 2], 18], [[1, 2, 3], 19, [1, 2, 3], 19], [[1, 2, 3, 4], 20, [1, 2, 3], 19], [[1, 2, 3, 4], 20, [1, 2], 18], [[1, 2, 3], 19, [1, 3], 22], [[1, 2, 3, 4], 20, [1, 3], 22], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3], 19, [2], 25], [[1, 2, 3], 19, [2, 3], 26], [[1, 2, 3, 4], 20, [2, 3], 26], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3], 19, [3], 29], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S106
S108 · implies
Before
given candidate left-open position=20;set=[1, 2, 3, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 2, 3, 4], 20, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2, 3], 19], [[0, 1, 2, 3, 4], 5, [1, 2, 3, 4], 20], [[0, 1, 2, 3, 4], 5, [1, 2, 4], 21], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3, 4], 23], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3, 4], 27], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3, 4], 20, [1, 2], 18], [[1, 2, 3, 4], 20, [1, 2, 3], 19], [[1, 2, 3, 4], 20, [1, 2, 3, 4], 20], [[1, 2, 3, 4], 20, [1, 2, 4], 21], [[1, 2, 3, 4], 20, [1, 3], 22], [[1, 2, 3, 4], 20, [1, 3, 4], 23], [[1, 2, 3, 4], 20, [1, 4], 24], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3, 4], 20, [2, 3], 26], [[1, 2, 3, 4], 20, [2, 3, 4], 27], [[1, 2, 3, 4], 20, [2, 4], 28], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [3, 4], 30], [[1, 2, 3, 4], 20, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S107
S109 · implies
Before
given candidate left-open position=21;set=[1, 2, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 2, 4], 21, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2, 4], 6, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2], 18], [[0, 1, 2, 3, 4], 5, [1, 2, 4], 21], [[0, 1, 2, 4], 6, [1, 2, 4], 21], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 4], 6, [1, 4], 24], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 4], 6, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[1, 2, 4], 21, [1], 17], [[1, 2, 4], 21, [1, 2], 18], [[1, 2, 3, 4], 20, [1, 2], 18], [[1, 2, 3, 4], 20, [1, 2, 4], 21], [[1, 2, 4], 21, [1, 2, 4], 21], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3, 4], 20, [1, 4], 24], [[1, 2, 4], 21, [1, 4], 24], [[1, 2, 4], 21, [2], 25], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3, 4], 20, [2, 4], 28], [[1, 2, 4], 21, [2, 4], 28], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [4], 31], [[1, 2, 4], 21, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S108
S110 · implies
Before
given candidate left-open position=22;set=[1, 3];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 3], 22, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 3], 7, [1], 17], [[0, 1, 2, 3], 4, [1], 17], [[0, 1, 2, 3], 4, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 3], 7, [1, 3], 22], [[0, 1, 3, 4], 8, [1, 3], 22], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 3], 7, [3], 29], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [], 0], [[1, 3], 22, [1], 17], [[1, 2, 3], 19, [1], 17], [[1, 2, 3], 19, [1, 3], 22], [[1, 2, 3, 4], 20, [1, 3], 22], [[1, 2, 3, 4], 20, [1], 17], [[1, 3], 22, [1, 3], 22], [[1, 3, 4], 23, [1, 3], 22], [[1, 3, 4], 23, [1], 17], [[1, 2, 3], 19, [], 0], [[1, 2, 3], 19, [3], 29], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [], 0], [[1, 3], 22, [3], 29], [[1, 3, 4], 23, [3], 29], [[1, 3, 4], 23, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S109
S111 · implies
Before
given candidate left-open position=23;set=[1, 3, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 3, 4], 23, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 3], 22], [[0, 1, 2, 3, 4], 5, [1, 3, 4], 23], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 3, 4], 8, [1, 3], 22], [[0, 1, 3, 4], 8, [1, 3, 4], 23], [[0, 1, 3, 4], 8, [1, 4], 24], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [3, 4], 30], [[0, 1, 3, 4], 8, [4], 31], [[1, 3, 4], 23, [1], 17], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3, 4], 20, [1, 3], 22], [[1, 2, 3, 4], 20, [1, 3, 4], 23], [[1, 2, 3, 4], 20, [1, 4], 24], [[1, 3, 4], 23, [1, 3], 22], [[1, 3, 4], 23, [1, 3, 4], 23], [[1, 3, 4], 23, [1, 4], 24], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [3, 4], 30], [[1, 2, 3, 4], 20, [4], 31], [[1, 3, 4], 23, [3], 29], [[1, 3, 4], 23, [3, 4], 30], [[1, 3, 4], 23, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S110
S112 · implies
Before
given candidate left-open position=24;set=[1, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[1, 4], 24, [], 0], [[0, 1, 4], 9, [], 0], [[0, 1, 4], 9, [1], 17], [[0, 1, 2, 4], 6, [1], 17], [[0, 1, 2, 3, 4], 5, [1], 17], [[0, 1, 2, 3, 4], 5, [1, 4], 24], [[0, 1, 2, 4], 6, [1, 4], 24], [[0, 1, 3, 4], 8, [1], 17], [[0, 1, 3, 4], 8, [1, 4], 24], [[0, 1, 4], 9, [1, 4], 24], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [4], 31], [[0, 1, 4], 9, [4], 31], [[1, 4], 24, [1], 17], [[1, 2, 4], 21, [1], 17], [[1, 2, 3, 4], 20, [1], 17], [[1, 2, 3, 4], 20, [1, 4], 24], [[1, 2, 4], 21, [1, 4], 24], [[1, 3, 4], 23, [1], 17], [[1, 3, 4], 23, [1, 4], 24], [[1, 4], 24, [1, 4], 24], [[1, 2, 4], 21, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [4], 31], [[1, 2, 4], 21, [4], 31], [[1, 3, 4], 23, [], 0], [[1, 3, 4], 23, [4], 31], [[1, 4], 24, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S111
S113 · implies
Before
given candidate left-open position=25;set=[2];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[2], 25, [], 0], [[0, 2], 10, [], 0], [[0, 1, 2], 3, [], 0], [[0, 1, 2], 3, [2], 25], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 2], 10, [2], 25], [[0, 2, 3], 11, [2], 25], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 4], 13, [2], 25], [[0, 2, 3], 11, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 4], 13, [], 0], [[1, 2], 18, [], 0], [[1, 2], 18, [2], 25], [[1, 2, 3], 19, [2], 25], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 4], 21, [2], 25], [[1, 2, 3], 19, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 4], 21, [], 0], [[2], 25, [2], 25], [[2, 3], 26, [2], 25], [[2, 3, 4], 27, [2], 25], [[2, 4], 28, [2], 25], [[2, 3], 26, [], 0], [[2, 3, 4], 27, [], 0], [[2, 4], 28, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S112
S114 · implies
Before
given candidate left-open position=26;set=[2, 3];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[2, 3], 26, [], 0], [[0, 2, 3], 11, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [2], 25], [[0, 1, 2, 3], 4, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 2, 3], 11, [2], 25], [[0, 2, 3], 11, [2, 3], 26], [[0, 2, 3, 4], 12, [2, 3], 26], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3], 11, [3], 29], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [], 0], [[1, 2, 3], 19, [], 0], [[1, 2, 3], 19, [2], 25], [[1, 2, 3], 19, [2, 3], 26], [[1, 2, 3, 4], 20, [2, 3], 26], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3], 19, [3], 29], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [], 0], [[2, 3], 26, [2], 25], [[2, 3], 26, [2, 3], 26], [[2, 3, 4], 27, [2, 3], 26], [[2, 3, 4], 27, [2], 25], [[2, 3], 26, [3], 29], [[2, 3, 4], 27, [3], 29], [[2, 3, 4], 27, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S113
S115 · implies
Before
given candidate left-open position=27;set=[2, 3, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[2, 3, 4], 27, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 3], 26], [[0, 1, 2, 3, 4], 5, [2, 3, 4], 27], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3, 4], 12, [2, 3], 26], [[0, 2, 3, 4], 12, [2, 3, 4], 27], [[0, 2, 3, 4], 12, [2, 4], 28], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [3, 4], 30], [[0, 2, 3, 4], 12, [4], 31], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3, 4], 20, [2, 3], 26], [[1, 2, 3, 4], 20, [2, 3, 4], 27], [[1, 2, 3, 4], 20, [2, 4], 28], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [3, 4], 30], [[1, 2, 3, 4], 20, [4], 31], [[2, 3, 4], 27, [2], 25], [[2, 3, 4], 27, [2, 3], 26], [[2, 3, 4], 27, [2, 3, 4], 27], [[2, 3, 4], 27, [2, 4], 28], [[2, 3, 4], 27, [3], 29], [[2, 3, 4], 27, [3, 4], 30], [[2, 3, 4], 27, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S114
S116 · implies
Before
given candidate left-open position=28;set=[2, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[2, 4], 28, [], 0], [[0, 2, 4], 13, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 4], 6, [2], 25], [[0, 1, 2, 3, 4], 5, [2], 25], [[0, 1, 2, 3, 4], 5, [2, 4], 28], [[0, 1, 2, 4], 6, [2, 4], 28], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 2, 4], 13, [2], 25], [[0, 2, 3, 4], 12, [2], 25], [[0, 2, 3, 4], 12, [2, 4], 28], [[0, 2, 4], 13, [2, 4], 28], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [4], 31], [[0, 2, 4], 13, [4], 31], [[1, 2, 4], 21, [], 0], [[1, 2, 4], 21, [2], 25], [[1, 2, 3, 4], 20, [2], 25], [[1, 2, 3, 4], 20, [2, 4], 28], [[1, 2, 4], 21, [2, 4], 28], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [4], 31], [[1, 2, 4], 21, [4], 31], [[2, 4], 28, [2], 25], [[2, 3, 4], 27, [2], 25], [[2, 3, 4], 27, [2, 4], 28], [[2, 4], 28, [2, 4], 28], [[2, 3, 4], 27, [], 0], [[2, 3, 4], 27, [4], 31], [[2, 4], 28, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S115
S117 · implies
Before
given candidate left-open position=29;set=[3];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[3], 29, [], 0], [[0, 3], 14, [], 0], [[0, 1, 3], 7, [], 0], [[0, 1, 2, 3], 4, [], 0], [[0, 1, 2, 3], 4, [3], 29], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 3], 7, [3], 29], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [], 0], [[0, 2, 3], 11, [], 0], [[0, 2, 3], 11, [3], 29], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [], 0], [[0, 3], 14, [3], 29], [[0, 3, 4], 15, [3], 29], [[0, 3, 4], 15, [], 0], [[1, 3], 22, [], 0], [[1, 2, 3], 19, [], 0], [[1, 2, 3], 19, [3], 29], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [], 0], [[1, 3], 22, [3], 29], [[1, 3, 4], 23, [3], 29], [[1, 3, 4], 23, [], 0], [[2, 3], 26, [], 0], [[2, 3], 26, [3], 29], [[2, 3, 4], 27, [3], 29], [[2, 3, 4], 27, [], 0], [[3], 29, [3], 29], [[3, 4], 30, [3], 29], [[3, 4], 30, [], 0]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S116
S118 · implies
Before
given candidate left-open position=30;set=[3, 4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[3, 4], 30, [], 0], [[0, 3, 4], 15, [], 0], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [3], 29], [[0, 1, 2, 3, 4], 5, [3, 4], 30], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 3, 4], 8, [3], 29], [[0, 1, 3, 4], 8, [3, 4], 30], [[0, 1, 3, 4], 8, [4], 31], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [3], 29], [[0, 2, 3, 4], 12, [3, 4], 30], [[0, 2, 3, 4], 12, [4], 31], [[0, 3, 4], 15, [3], 29], [[0, 3, 4], 15, [3, 4], 30], [[0, 3, 4], 15, [4], 31], [[1, 3, 4], 23, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [3], 29], [[1, 2, 3, 4], 20, [3, 4], 30], [[1, 2, 3, 4], 20, [4], 31], [[1, 3, 4], 23, [3], 29], [[1, 3, 4], 23, [3, 4], 30], [[1, 3, 4], 23, [4], 31], [[2, 3, 4], 27, [], 0], [[2, 3, 4], 27, [3], 29], [[2, 3, 4], 27, [3, 4], 30], [[2, 3, 4], 27, [4], 31], [[3, 4], 30, [3], 29], [[3, 4], 30, [3, 4], 30], [[3, 4], 30, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S117
S119 · implies
Before
given candidate left-open position=31;set=[4];ALL right candidate opens in original family order
After
ALL [union,its_candidate_index,intersection,its_candidate_index] rows=[[[4], 31, [], 0], [[0, 4], 16, [], 0], [[0, 1, 4], 9, [], 0], [[0, 1, 2, 4], 6, [], 0], [[0, 1, 2, 3, 4], 5, [], 0], [[0, 1, 2, 3, 4], 5, [4], 31], [[0, 1, 2, 4], 6, [4], 31], [[0, 1, 3, 4], 8, [], 0], [[0, 1, 3, 4], 8, [4], 31], [[0, 1, 4], 9, [4], 31], [[0, 2, 4], 13, [], 0], [[0, 2, 3, 4], 12, [], 0], [[0, 2, 3, 4], 12, [4], 31], [[0, 2, 4], 13, [4], 31], [[0, 3, 4], 15, [], 0], [[0, 3, 4], 15, [4], 31], [[0, 4], 16, [4], 31], [[1, 4], 24, [], 0], [[1, 2, 4], 21, [], 0], [[1, 2, 3, 4], 20, [], 0], [[1, 2, 3, 4], 20, [4], 31], [[1, 2, 4], 21, [4], 31], [[1, 3, 4], 23, [], 0], [[1, 3, 4], 23, [4], 31], [[1, 4], 24, [4], 31], [[2, 4], 28, [], 0], [[2, 3, 4], 27, [], 0], [[2, 3, 4], 27, [4], 31], [[2, 4], 28, [4], 31], [[3, 4], 30, [], 0], [[3, 4], 30, [4], 31], [[4], 31, [4], 31]];null is an actual missing required set
Conditions: none
Check binary unions/intersections against the complete original table,not samples.Display missing sets explicitly rather than repairing the table by adding them.
Step identifiers and dependencies
Rule: topbasis_given_topology_all_binary_axioms_actual_missing_sets · Uses: G001, S118
S120 · implies
Before
given candidate empty/full flags=[Yes, Yes];ALL binary closure flags=[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]
After
topology_valid=Yes;IF valid,arbitrary indexed unions reduce to finitely many distinct opens,so binary union closure suffices;empty intersections use full carrier
Conditions: none
A finite family has only finitely many distinct opens,so arbitrary indexed unions reduce to finite unions.Apply that topology certificate only when empty/full sets and all closure obligations pass.For false candidates retain the actual failure without changing data.
Step identifiers and dependencies
Rule: topbasis_given_topology_full_axiom_conjunction_not_only_binary_slogan · Uses: G001, S119
S121 · implies
Before
ORIGINAL complete candidate basis in fixed family-index order=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]];carrier Fin(5)
After
prefix0 has EXACTLY the empty-subfamily union:empty set,with empty index witness;this mathematical union exists even when basis axioms fail
Conditions: none
The empty subfamily always has empty union,without requiring the empty set to be a basis element.This step constructs unions only and does not call an invalid basis's unions a topology.
Step identifiers and dependencies
Rule: topbasis_empty_union_initial_state_not_missing_empty_set · Uses: G001, S120
S122 · implies
Before
next ORIGINAL basis index=0;basis set=[0];ALL previous-prefix unions in numeric POINT-mask order=[[]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [0], [0]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S121
S123 · implies
Before
next ORIGINAL basis index=1;basis set=[0, 1];ALL previous-prefix unions in numeric POINT-mask order=[[], [0]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [0, 1], [1]], [[0], [0, 1], [0, 1]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S122
S124 · implies
Before
next ORIGINAL basis index=2;basis set=[0, 2];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [0, 1]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [0, 2], [2]], [[0], [0, 2], [0, 2]], [[0, 1], [0, 1, 2], [1, 2]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 2]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S123
S125 · implies
Before
next ORIGINAL basis index=3;basis set=[1];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [0, 1], [0, 2], [0, 1, 2]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [1], [3]], [[0], [0, 1], [0, 3]], [[0, 1], [0, 1], [1, 3]], [[0, 2], [0, 1, 2], [2, 3]], [[0, 1, 2], [0, 1, 2], [1, 2, 3]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S124
S126 · implies
Before
next ORIGINAL basis index=4;basis set=[1, 2];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [0, 2], [0, 1, 2]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [1, 2], [4]], [[0], [0, 1, 2], [0, 4]], [[1], [1, 2], [3, 4]], [[0, 1], [0, 1, 2], [1, 4]], [[0, 2], [0, 1, 2], [2, 4]], [[0, 1, 2], [0, 1, 2], [1, 2, 4]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S125
S127 · implies
Before
next ORIGINAL basis index=5;basis set=[2];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [0, 2], [1, 2], [0, 1, 2]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [2], [5]], [[0], [0, 2], [0, 5]], [[1], [1, 2], [3, 5]], [[0, 1], [0, 1, 2], [1, 5]], [[0, 2], [0, 2], [2, 5]], [[1, 2], [1, 2], [4, 5]], [[0, 1, 2], [0, 1, 2], [1, 2, 5]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S126
S128 · implies
Before
next ORIGINAL basis index=6;basis set=[3];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3], [6]], [[0], [0, 3], [0, 6]], [[1], [1, 3], [3, 6]], [[0, 1], [0, 1, 3], [1, 6]], [[2], [2, 3], [5, 6]], [[0, 2], [0, 2, 3], [2, 6]], [[1, 2], [1, 2, 3], [4, 6]], [[0, 1, 2], [0, 1, 2, 3], [1, 2, 6]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 3], [0, 2], [0, 2, 3], [0, 3], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S127
S129 · implies
Before
next ORIGINAL basis index=7;basis set=[3, 4];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [3, 4], [7]], [[0], [0, 3, 4], [0, 7]], [[1], [1, 3, 4], [3, 7]], [[0, 1], [0, 1, 3, 4], [1, 7]], [[2], [2, 3, 4], [5, 7]], [[0, 2], [0, 2, 3, 4], [2, 7]], [[1, 2], [1, 2, 3, 4], [4, 7]], [[0, 1, 2], [0, 1, 2, 3, 4], [1, 2, 7]], [[3], [3, 4], [6, 7]], [[0, 3], [0, 3, 4], [0, 6, 7]], [[1, 3], [1, 3, 4], [3, 6, 7]], [[0, 1, 3], [0, 1, 3, 4], [1, 6, 7]], [[2, 3], [2, 3, 4], [5, 6, 7]], [[0, 2, 3], [0, 2, 3, 4], [2, 6, 7]], [[1, 2, 3], [1, 2, 3, 4], [4, 6, 7]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 6, 7]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 3], [0, 3, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 3], [1, 3, 4], [2], [2, 3], [2, 3, 4], [3], [3, 4]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S128
S130 · implies
Before
next ORIGINAL basis index=8;basis set=[4];ALL previous-prefix unions in numeric POINT-mask order=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3], [3, 4], [0, 3, 4], [1, 3, 4], [0, 1, 3, 4], [2, 3, 4], [0, 2, 3, 4], [1, 2, 3, 4], [0, 1, 2, 3, 4]]
After
ALL [previous_union,union_with_new_basis,index_witness] rows=[[[], [4], [8]], [[0], [0, 4], [0, 8]], [[1], [1, 4], [3, 8]], [[0, 1], [0, 1, 4], [1, 8]], [[2], [2, 4], [5, 8]], [[0, 2], [0, 2, 4], [2, 8]], [[1, 2], [1, 2, 4], [4, 8]], [[0, 1, 2], [0, 1, 2, 4], [1, 2, 8]], [[3], [3, 4], [6, 8]], [[0, 3], [0, 3, 4], [0, 6, 8]], [[1, 3], [1, 3, 4], [3, 6, 8]], [[0, 1, 3], [0, 1, 3, 4], [1, 6, 8]], [[2, 3], [2, 3, 4], [5, 6, 8]], [[0, 2, 3], [0, 2, 3, 4], [2, 6, 8]], [[1, 2, 3], [1, 2, 3, 4], [4, 6, 8]], [[0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 6, 8]], [[3, 4], [3, 4], [7, 8]], [[0, 3, 4], [0, 3, 4], [0, 7, 8]], [[1, 3, 4], [1, 3, 4], [3, 7, 8]], [[0, 1, 3, 4], [0, 1, 3, 4], [1, 7, 8]], [[2, 3, 4], [2, 3, 4], [5, 7, 8]], [[0, 2, 3, 4], [0, 2, 3, 4], [2, 7, 8]], [[1, 2, 3, 4], [1, 2, 3, 4], [4, 7, 8]], [[0, 1, 2, 3, 4], [0, 1, 2, 3, 4], [1, 2, 7, 8]]];keep both omit/include choices,then deduplicate EQUAL UNIONS;new canonical union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]
Conditions: none
Any subfamily of the new prefix either omits the new element or includes it with a union from the previous prefix.These choices are complete and every state has an actual index witness.Deduplicate equal unions,not original basis positions.There are at most2^n distinct states,not a sample of the2^basisCount subfamilies.
Step identifiers and dependencies
Rule: topbasis_complete_prefix_union_DP_not_sampled_subfamilies · Uses: G001, S129
S131 · implies
Before
ALL original basis positions processed;distinct union states=32<=2^5;canonical ALL-union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]
After
prefix induction covers EVERY original-index subfamily and every state is its actual union;mathematical all-union family COMPLETE including empty union
Conditions: none
Omit/include induction puts every subfamily union into the states,and every state has an actual subfamily witness,proving both directions.Do not misreport the smaller distinct-state count as the number of subfamilies explicitly enumerated.
Step identifiers and dependencies
Rule: topbasis_all_subfamily_unions_complete_not_exponential_enumeration_claim · Uses: G001, S130
S132 · implies
Before
JUST-VERIFIED cover and ALL local refinement for ORIGINAL basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]];COMPLETE all-union family=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]
After
empty/full carrier present;ANY union of basis-unions flattens to a basis-union;for x in U intersection V choose basis B,C around x then its verified local refiner inside B intersection C,so U intersection V is itself a basis-union;this family IS a topology on Fin(5)
Conditions: none
Local refinement supplies a basis element around each point inside U∩V,and their union equals that intersection.This proves intersection closure without requiring the basis family itself intersection-closed.Arbitrary-union flattening,coverage of the full carrier and the empty union complete the topology axioms.
Step identifiers and dependencies
Rule: topbasis_basis_to_topology_arbitrary_union_and_local_intersection_proof · Uses: G001, S131
S133 · implies
Before
ORIGINAL basis membership rows[basis_index,given_open_index_or_null]=[[0, 1], [1, 2], [2, 10], [3, 17], [4, 18], [5, 25], [6, 29], [7, 30], [8, 31]];given candidate-open generation rows[open_index,certified_union_index_or_null]=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [6, 6], [7, 7], [8, 8], [9, 9], [10, 10], [11, 11], [12, 12], [13, 13], [14, 14], [15, 15], [16, 16], [17, 17], [18, 18], [19, 19], [20, 20], [21, 21], [22, 22], [23, 23], [24, 24], [25, 25], [26, 26], [27, 27], [28, 28], [29, 29], [30, 30], [31, 31]]
After
basis_elements_open=Yes;every_given_open_is_basis_union=Yes;source component agrees with actual mathematical union predicate=Yes
Conditions: none
Basis elements being listed as given opens and every given open being generated by the basis are opposite-direction obligations.A basis for this supplied topology needs both,not just abstract basis validity.
Step identifiers and dependencies
Rule: topbasis_given_openness_and_every_open_generation_separate · Uses: G001, S132
S134 · implies
Before
ACTUAL topology_valid=Yes,abstract_basis_valid=Yes,basis_elements_open=Yes,every_open_generated=Yes
After
basis_for_THIS_given_topology=Yes is the FOUR-WAY conjunction;any passing component alone is insufficient
Conditions: none
Only their conjunction establishes a basis for this given topology.Retain the original candidate families and flags without promoting a passing component to an arbitrary-space conclusion.
Step identifiers and dependencies
Rule: topbasis_basis_for_given_topology_all_responsibilities_conjunction · Uses: G001, S133