{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-open-cover-and-subcover",
  "semantic_family_id": "sf-open-cover-and-subcover",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "開覆蓋與子覆蓋"
  },
  "summary": {
    "zh-Hant": "開覆蓋由指定拓撲的開集組成且聯集包含目標集合；子覆蓋須取自原覆蓋並仍覆蓋同一目標。"
  },
  "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。"
  },
  "formula": {
    "tex": "O(A)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>O</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "O",
        "glyph": "O"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e63af2cc99446777d17b1dabfd7a8779adcb47c138eb746f6be14ec607b5b44c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限拓撲、目標子集與有序cover family。"
    },
    {
      "id": "a2",
      "zh-Hant": "subcover以原family索引選取，重複索引正規化。"
    }
  ],
  "symbols": [
    {
      "token": "O",
      "role": "open_cover_record",
      "role_zh": "開覆蓋紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "target_subset",
      "role_zh": "被覆蓋集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-topological-space",
      "title_zh": "拓撲空間",
      "reason_zh": "提供開集族。"
    },
    {
      "id": "mko-open-closed-set",
      "title_zh": "開集與閉集",
      "reason_zh": "每個cover成員須為開。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "目標及成員包含責任。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "以聯集核覆蓋。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先核cover family全部成員開。",
      "再核聯集包含目標。",
      "子覆蓋必須來自原cover；單一成功不證緊緻。"
    ],
    "common_misconception_zh": "使用不在原覆蓋中的集合，或由一個有限子覆蓋直接宣稱compact。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只驗指定有限cover與subcover，不作全稱compactness結論。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入緊緻空間與有限交性質。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-open-cover-and-subcover.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-11-01 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}