{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-complete-metric-space",
  "semantic_family_id": "sf-complete-metric-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "完備度量空間"
  },
  "summary": {
    "zh-Hant": "每個Cauchy序列都收斂到空間內某點的度量空間稱為完備。"
  },
  "statement": {
    "zh-Hant": "度量空間(X,d)完備，若X內每個Cauchy序列都收斂到某個仍屬X的點。有限度量空間可由最小正距離證明：Cauchy尾段終將只能取同一點；抽樣若干序列不足以證一般完備性。"
  },
  "formula": {
    "tex": "C(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e1849a5c630feddeb7575d1f7a48d4caeb284f626f4bf8d15bc6472592554125"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "有限證書使用完整有限度量表且至少一個點。"
    },
    {
      "id": "a2",
      "zh-Hant": "序列題只接受最終常值證書或明示非Cauchy反例；有限前綴不冒充全稱完備證明。"
    }
  ],
  "symbols": [
    {
      "token": "X",
      "role": "metric_space",
      "role_zh": "度量空間",
      "scope": "representative_formula"
    },
    {
      "token": "d",
      "role": "metric",
      "role_zh": "度量",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "sequence",
      "role_zh": "空間內序列",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-cauchy-sequence",
      "title_zh": "Cauchy序列",
      "reason_zh": "完備性量化所有Cauchy序列。"
    },
    {
      "id": "mko-metric-convergence",
      "title_zh": "度量收斂",
      "reason_zh": "結論要求在同一空間內收斂。"
    },
    {
      "id": "mko-metric-space",
      "title_zh": "度量空間",
      "reason_zh": "完整度量提供距離責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完備性是空間性質，不是某一條序列的性質。",
      "有限metric的最小正分離距離給全稱證書。",
      "Cauchy但極限離開子空間可否證完備；本輪不以有限模擬取代該責任。"
    ],
    "common_misconception_zh": "因測過幾條序列都收斂，或序列在ambient空間有極限，就宣稱子空間完備。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "完整有限metric證書只證該有限空間；樣本序列不證一般空間完備。",
    "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": "與範數誘導度量共同加入實Banach空間。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-complete-metric-space.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-10-27 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"
  }
}