{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-open-cover-and-subcover-01",
  "mko_id": "mko-open-cover-and-subcover",
  "title": {
    "zh-Hant": "從原覆蓋索引選取子覆蓋",
    "en": "Selecting a subcover by original indices"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-open-cover-and-subcover-01",
    "source": {
      "mko_id": "mko-open-cover-and-subcover",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "bd5a852d24868cb59747e9030f8da58e604c78a87a2442558dc2643a01d8ecb1",
      "english_sha256": "a58afb586a37df65445b878eb9611ccd4f1ad7f03e2d7e5ef5d651f785f43d00"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "sets",
      "topology",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "核對 K={a,b} 的指定開覆蓋與子覆蓋索引。",
      "en": "Audit a supplied open cover of K={a,b} and its subcover indices."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X={a,b,c}，tau=全部 X 的子集；K={a,b}。",
          "en": "X={a,b,c}, tau is the entire power set of X; K={a,b}."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "有序族 U_1={a}、U_2={b}、U_3={a,b}、U_4={c}。原始選取索引 (3,3) 正規化為 {3}。",
          "en": "The ordered family is U_1={a}, U_2={b}, U_3={a,b}, U_4={c}. Normalize the raw selection (3,3) to {3}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "\\tau=\\mathcal P(X)",
        "after_tex": "\\tau=\\{\\varnothing,\\{a\\},\\{b\\},\\{c\\},\\{a,b\\},\\{a,c\\},\\{b,c\\},X\\}",
        "relation": "implies",
        "rule": "topology",
        "conditions": [],
        "explanation": {
          "zh-Hant": "列出全部八個子集。任意聯與有限交仍為 X 子集，因此確為拓撲。",
          "en": "List all eight subsets. Every union and finite intersection remains a subset of X, so this is a topology."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不是把供給的四個覆蓋成員當完整拓撲。",
            "en": "The four cover members are not being substituted for the complete topology."
          }
        },
        "leaf_basis": "topology"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "G2",
          "S01"
        ],
        "before_tex": "i\\in\\{1,2,3,4\\}",
        "after_tex": "U_i\\in\\tau",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐個成員都在八項開集清單，含不與 K 相交的 U_4 也合法。",
          "en": "Each member occurs in the eight-open list; U_4 may be disjoint from K and still be legal."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "開覆蓋不要求每個成員包含於 K，也不要求每個都必要。",
            "en": "An open cover need not have every member inside K or make every member necessary."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S03",
        "uses": [
          "G2"
        ],
        "before_tex": "\\bigcup_{i=1}^4 U_i",
        "after_tex": "\\{a\\}\\cup\\{b\\}\\cup\\{a,b\\}\\cup\\{c\\}=X",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "按原有序族取聯，覆蓋了 X 因而也覆蓋 K。",
          "en": "The union of the supplied ordered family is X and therefore covers K."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G2"
          ],
          "explanation": {
            "zh-Hant": "覆蓋目標只需 K 包含於聯集，不需恰等。",
            "en": "Coverage requires inclusion of K, not equality with K."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S04",
        "uses": [
          "G2"
        ],
        "before_tex": "(3,3)",
        "after_tex": "I=\\{3\\}\\subseteq\\{1,2,3,4\\}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "去除重複索引不會改變選取集合的聯。",
          "en": "Removing a duplicate index does not change the selected union."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G2"
          ],
          "explanation": {
            "zh-Hant": "索引 3 存在於原族；沒有加入新集合。",
            "en": "Index 3 belongs to the original family; no new set has been introduced."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "G2",
          "S04"
        ],
        "before_tex": "\\bigcup_{i\\in I}U_i",
        "after_tex": "U_3=\\{a,b\\}=K",
        "relation": "equivalent",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "原族中的第三個集合單獨已覆蓋目標。",
          "en": "The third original set alone covers the target."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "G1",
            "G2",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "將正規化後索引回讀原族而不是直接憑目標造一個集合。",
            "en": "Read the normalized index back from the original family instead of inventing a set from the target."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S06",
        "uses": [
          "S02",
          "S04",
          "S05"
        ],
        "before_tex": "I=\\{3\\},\\quad K\\subseteq\\bigcup_{i\\in I}U_i",
        "after_tex": "(U_i)_{i\\in I}\\text{ is a finite open subcover}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "開性由原成員繼承，且一個索引是有限集合。",
          "en": "Openness is inherited from the original member, and a singleton index set is finite."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "需要開性、來源索引與覆蓋三件事，先前步驟各自提供。",
            "en": "Openness, original-index membership and coverage are separate obligations supplied earlier."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "G2"
        ],
        "before_tex": "J=\\{1\\}",
        "after_tex": "\\bigcup_{i\\in J}U_i=\\{a\\}\\not\\supseteq K",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以漏掉 b 的選取作負向核對；原覆蓋合法不代表任意子族仍覆蓋。",
          "en": "A selection missing b is a negative check: a valid cover does not make every subfamily a cover."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2"
          ],
          "explanation": {
            "zh-Hant": "失敗原因是缺 b，不是 U_1 不開。",
            "en": "The failure is missing b, not lack of openness of U_1."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S08",
        "uses": [
          "S06"
        ],
        "before_tex": "\\exists I\\text{ finite for this supplied cover}",
        "after_tex": "\\text{this supplied cover has a finite subcover}",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "結論只保留已核的覆蓋；每個開覆蓋的全稱命題是另一責任。",
          "en": "Conclude only for the audited cover; the universal statement over every open cover is a separate obligation."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S06"
          ],
          "explanation": {
            "zh-Hant": "沒有把存在一個成功選取反轉成對所有覆蓋的證明。",
            "en": "No reversal from one successful selection to all covers has occurred."
          }
        },
        "leaf_basis": "sets"
      }
    ],
    "result": {
      "zh-Hant": "{3} 是合法有限子覆蓋索引，而 {1} 不是。",
      "en": "{3} indexes a valid finite subcover; {1} does not."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "選 {a,b,c} 也算原覆蓋的子覆蓋。",
          "en": "Selecting {a,b,c} counts as a subcover of the original family."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "X 雖然開但不在原族中，子覆蓋不能新增成員。",
          "en": "X is open but is not a member of the original family; a subcover cannot add it."
        }
      },
      {
        "claim": {
          "zh-Hant": "{1} 覆蓋 K。",
          "en": "{1} covers K."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "聯集漏掉 b。",
          "en": "Its union misses b."
        }
      },
      {
        "claim": {
          "zh-Hant": "這一次成功已證所有空間緊緻。",
          "en": "This success proves all spaces compact."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "只核一個供給覆蓋，沒有全稱量詞。",
          "en": "Only one supplied cover was checked, not the universal quantifier."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "完整核每個成員為開集、目標包含與選取索引。某一供給覆蓋有有限子覆蓋只證該覆蓋，不證空間緊緻。",
          "en": "Check openness of every supplied member, target inclusion in the union and the selection indices completely. A finite subcover for one supplied cover certifies that cover only; it is not by itself the universal compactness obligation."
        },
        "source": {
          "mko_id": "mko-open-cover-and-subcover",
          "sha256": "bd5a852d24868cb59747e9030f8da58e604c78a87a2442558dc2643a01d8ecb1",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "sets",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合的聯、交、補集、包含與函數逆像以逐元素定義計算。",
          "en": "Compute unions, intersections, complements, inclusion and preimages element by element."
        }
      },
      {
        "id": "topology",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "拓撲含空集與全集，對任意聯及有限交封閉；有限集合族可完整核對。",
          "en": "A topology contains empty and whole sets and is closed under arbitrary unions and finite intersections; a finite family admits exhaustive checks."
        }
      },
      {
        "id": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "開覆蓋各成員開且聯集包含目標；子覆蓋取自原索引子集且仍覆蓋同一目標。",
          "en": "An open cover has open members whose union contains the target; a subcover uses a subset of the original indices and covers the same target."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只核這個覆蓋，不以單一成功推出一般緊緻性。",
      "en": "This audits one cover only, not universal compactness."
    },
    "md_symbols": [
      [
        "K",
        "覆蓋目標",
        "covered target"
      ],
      [
        "U_i",
        "原覆蓋成員",
        "original cover member"
      ],
      [
        "I",
        "正規化索引集",
        "normalized index set"
      ]
    ],
    "md_error_controls": [
      {
        "title": "索引溯源 / Index provenance",
        "zh": "每次選取都由原有序族讀回成員。",
        "en": "Read every selected member from the original ordered family."
      },
      {
        "title": "量詞不升格 / No quantifier promotion",
        "zh": "一次子覆蓋成功與每個開覆蓋均成功分開。",
        "en": "Separate one successful subcover from success for every open cover."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "5da7ac00e9a6cf40783fb48632b4b9388c717c3012a47aff481fcdd179f5f222",
    "original_source_sha256": "5da7ac00e9a6cf40783fb48632b4b9388c717c3012a47aff481fcdd179f5f222",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}