# Question 16127: Inclusion-Minimal Does Not Mean Minimum Cardinality [Return to catalogue page 81](https://ocme.evemisslab.com/en/questions/page/81/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations/#q-20261014-advance-20261101-r03-open-cover-and-subcover-0052) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0031-16127 · q-20261014-advance-20261101-r03-open-cover-and-subcover-0052 ## Original question and givens Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order. - G001 — Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-open-cover-and-subcover](https://ocme.evemisslab.com/en/mko/mko-open-cover-and-subcover/) ## Step-by-step derivation S001 · impliesBefore same finite carrier Fin(5);complete original ordered topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] After unique empty and whole opens=Yes;all ordered open pairs=36;first union closure failure=null;first intersection closure failure=null Conditions: none Check canonical subsets,empty and whole sets,and union/intersection of every ordered pair.Since the topology family is finite,an arbitrary union involves only finitely many distinct opens and follows by repeated binary unions.Finite intersections follow similarly with the whole set.No metric is supplied or inferred. Step identifiers and dependenciesRule: cover_qualified_complete_topology_no_silent_metric · Uses: G001 S002 · impliesBefore original cover family index=0;member=[0, 1];complete same topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] After original member open=Yes Conditions: none Check openness at every original index.Equal sets at different indices remain distinct indexed members.Do not deduplicate the input or ignore a nonopen member merely because it is unselected. Step identifiers and dependenciesRule: cover_every_original_index_open_including_unused_members · Uses: G001, S001 S003 · impliesBefore original cover family index=1;member=[0, 1, 2, 4];complete same topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] After original member open=Yes Conditions: none Check openness at every original index.Equal sets at different indices remain distinct indexed members.Do not deduplicate the input or ignore a nonopen member merely because it is unselected. Step identifiers and dependenciesRule: cover_every_original_index_open_including_unused_members · Uses: G001, S002 S004 · impliesBefore original cover family index=2;member=[0, 1, 2, 3, 4];complete same topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] After original member open=Yes Conditions: none Check openness at every original index.Equal sets at different indices remain distinct indexed members.Do not deduplicate the input or ignore a nonopen member merely because it is unselected. Step identifiers and dependenciesRule: cover_every_original_index_open_including_unused_members · Uses: G001, S003 S005 · impliesBefore same target=[0, 1, 2, 4];union of ALL original members=[0, 1, 2, 3, 4];nonopen original indices=[] After covers target=Yes;whole original open cover valid=Yes Conditions: none Openness and coverage are separate obligations.The union may exceed the target;containment,not equality,is required.Certify the original cover only when both hold. Step identifiers and dependenciesRule: cover_openness_and_target_coverage_separate_conjunction · Uses: G001, S004 S006 · impliesBefore raw selected ORIGINAL indices=[1];normalized duplicate-free indices=[1];selected original sets=[[0, 1, 2, 4]] After selected union=[0, 1, 2, 4];selected all open=Yes;selected covers target=Yes;original family qualified=Yes;valid selected subcover=Yes Conditions: none Select only original indexed members,sorting indices but rejecting duplicates.Empty selection can qualify for an empty target.This protocol additionally requires the whole original family to qualify;an otherwise good selected subfamily cannot erase an unselected bad original member. Step identifiers and dependenciesRule: cover_selected_original_indices_whole_family_premise_retained · Uses: G001, S005 S007 · impliesBefore remove original selected index=1;remaining selected indices=[];remaining union=[];same target=[0, 1, 2, 4] After removal coverage row=[1, [], No] Conditions: none Remove this one selected member and recompute coverage of the same target.Keep original indices and removal order.This is inclusion minimality,not a global minimum-cardinality search. Step identifiers and dependenciesRule: cover_every_single_member_removal_target_test · Uses: G001, S006 S008 · impliesBefore valid selected subcover=Yes;all single-removal rows=[[1, [], No]] After inclusion-minimal=Yes;inclusion_minimal_not_minimum_cardinality Conditions: none For a valid subcover,every single deletion must fail.If a proper selected subfamily covered,removing an element outside it would still cover,so these deletion tests suffice for inclusion minimality.An incomparable choice from the original family may use fewer sets;this is not generally minimum cardinality. Step identifiers and dependenciesRule: cover_inclusion_minimal_not_global_smallest_size · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: fafd702ec3c3a20eeb4e11ffcab6931bf410ac89a8fabb66f5af9195a018c89e - candidate record SHA-256: 5134b8493b4579c685bc22d3a877303b10f99bbff9e0266803f6ebd961a3d627 - semantic review receipt SHA-256: 55bb21783e2125ecf1e67e42f83fd25bec0ccd493f911f4bb937a5e9d6f9369a - question-first witness SHA-256: 754ddf2a6039c5ef4aae78a538c6e9a1e9a58d4825aee194e869a27b8a4c95bb - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261101-r03-open-cover-and-subcover-0052.json)