Step-by-step derivation
S001 · implies
Before
source operation=compact-subcover-certificate;original snapshot and mathematics retained
After
explicit ordered source output fields=[selected_indices, selected_sets, covers_subset, selected_from_original, certificate]
Conditions: none
Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL parent Fin(5);parent open index=0,U=[];complete topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]]
After
ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2, 3], []], [4, [0, 1, 2, 3, 4], []], [5, [0, 1, 3], []]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S001
S003 · implies
Before
ORIGINAL parent Fin(5);parent open index=1,U=[0];complete topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2, 3], [0]], [4, [0, 1, 2, 3, 4], [0]], [5, [0, 1, 3], [0]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S002
S004 · implies
Before
ORIGINAL parent Fin(5);parent open index=2,U=[0, 1];complete topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2, 3], [0, 1]], [4, [0, 1, 2, 3, 4], [0, 1]], [5, [0, 1, 3], [0, 1]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S003
S005 · implies
Before
ORIGINAL parent Fin(5);parent open index=3,U=[0, 1, 2, 3];complete topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2, 3], []], [1, [0, 1, 2, 3], [0]], [2, [0, 1, 2, 3], [0, 1]], [3, [0, 1, 2, 3], [0, 1, 2, 3]], [4, [0, 1, 2, 3, 4], [0, 1, 2, 3]], [5, [0, 1, 2, 3], [0, 1, 3]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S004
S006 · implies
Before
ORIGINAL parent Fin(5);parent open index=4,U=[0, 1, 2, 3, 4];complete topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2, 3, 4], []], [1, [0, 1, 2, 3, 4], [0]], [2, [0, 1, 2, 3, 4], [0, 1]], [3, [0, 1, 2, 3, 4], [0, 1, 2, 3]], [4, [0, 1, 2, 3, 4], [0, 1, 2, 3, 4]], [5, [0, 1, 2, 3, 4], [0, 1, 3]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S005
S007 · implies
Before
ORIGINAL parent Fin(5);parent open index=5,U=[0, 1, 3];complete topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]]
After
ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 3], []], [1, [0, 1, 3], [0]], [2, [0, 1, 3], [0, 1]], [3, [0, 1, 2, 3], [0, 1, 3]], [4, [0, 1, 2, 3, 4], [0, 1, 3]], [5, [0, 1, 3], [0, 1, 3]]];each belongs to the SAME parent family
Conditions: none
First require the empty/full opens,then check unions and intersections for every ordered pair in the same parent family.Any union in a finite family reduces to finitely many distinct opens,so binary closure suffices;an invalid parent is rejected.
Step identifiers and dependencies
Rule: subtop_qualified_parent_all_pairs_union_intersection · Uses: G001, S006
S008 · implies
Before
ORIGINAL parent open index=0,U=[];ORIGINAL subset Y=[1, 2, 4]
After
actual U intersect Y=[];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S007
S009 · implies
Before
ORIGINAL parent open index=1,U=[0];ORIGINAL subset Y=[1, 2, 4]
After
actual U intersect Y=[];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S008
S010 · implies
Before
ORIGINAL parent open index=2,U=[0, 1];ORIGINAL subset Y=[1, 2, 4]
After
actual U intersect Y=[1];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S009
S011 · implies
Before
ORIGINAL parent open index=3,U=[0, 1, 2, 3];ORIGINAL subset Y=[1, 2, 4]
After
actual U intersect Y=[1, 2];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S010
S012 · implies
Before
ORIGINAL parent open index=4,U=[0, 1, 2, 3, 4];ORIGINAL subset Y=[1, 2, 4]
After
actual U intersect Y=[1, 2, 4];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S011
S013 · implies
Before
ORIGINAL parent open index=5,U=[0, 1, 3];ORIGINAL subset Y=[1, 2, 4]
After
actual U intersect Y=[1];retain parent index even if another open has the same intersection
Conditions: none
Intersect every parent open with Y pointwise.Deduplicate only the induced family,not its distinct ambient witnesses.U may contain points outside Y;do not require U to be a subset of Y.
Step identifiers and dependencies
Rule: subtop_every_parent_open_intersection_with_original_Y · Uses: G001, S012
S014 · calculation
Before
subtop distinct relative open count := sum(1, 1, 1, 1)
After
subtop distinct relative open count = 4
Conditions: none
Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · implies
Before
ORIGINAL Y=[1, 2, 4];ALL parent intersections=[[], [], [1], [1, 2], [1, 2, 4], [1]]
After
normalized relative topology=[[], [1], [1, 2], [1, 2, 4]];distinct count=4;empty and Y included;unions and finite intersections commute with intersect Y
Conditions: none
Intersection with Y commutes with arbitrary unions and finite intersections;parent closure gives the induced axioms.For empty Y every intersection is empty,so the topology has one open,not zero.
Step identifiers and dependencies
Rule: subtop_induced_axioms_and_empty_Y · Uses: G001, S014
S016 · implies
Before
ORIGINAL K=[1, 2, 4];COMPLETE finite relative open family=[[], [1], [1, 2], [1, 2, 4]];distinct open count=4
After
for ANY relative-open cover of K,retain one used original index per DISTINCT open;at most this many opens form a finite subcover;empty K covered by the empty subfamily
Conditions: none
This argument covers every cover,not one sample.Even infinitely repeated indices use only finitely many distinct opens;choose one original index for each used open.The source finite tag concerns distinct opens,not the number of indices in every indexed cover.
Step identifiers and dependencies
Rule: compact_finite_DISTINCT_open_family_general_cover_argument · Uses: G001, S015
S017 · implies
Before
ORIGINAL indexed cover=[[], [1], [1, 2], [1, 2, 4]];relative-open family=[[], [1], [1, 2], [1, 2, 4]];K=[1, 2, 4]
After
ALL supplied members are relative-open;full union=[1, 2, 4];covers K=Yes;outer duplicate positions retained
Conditions: none
Audit each member as relative-open and compute the full union.Retain duplicate outer positions as distinct original indices.One successful cover is not a general compactness proof;a failing list reports false.
Step identifiers and dependencies
Rule: compact_actual_indexed_cover_relative_opens_and_union · Uses: G001, S016
S018 · implies
Before
ORIGINAL cover=[[], [1], [1, 2], [1, 2, 4]];selected sorted unique ORIGINAL indices=[3]
After
selected sets=[[1, 2, 4]];their union=[1, 2, 4];covers K=Yes;selected_from_original=Yes;NO minimum/minimal claim
Conditions: none
Valid original indices guarantee provenance,not coverage.Check the selected union separately;an empty selection covers only empty K.Do not add outside opens or claim a minimum subcover.
Step identifiers and dependencies
Rule: compact_selected_ORIGINAL_indices_separate_actual_coverage · Uses: G001, S017