{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-metric-space",
  "semantic_family_id": "sf-metric-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "度量空間"
  },
  "summary": {
    "zh-Hant": "度量空間由集合與距離函數組成，距離非負、只在相同點為零、對稱且滿足三角不等式。"
  },
  "statement": {
    "zh-Hant": "集合X上的度量d:X×X→R滿足非負與分離、對稱及三角不等式。完整有限距離表可逐格及逐三元組核對指定候選。"
  },
  "formula": {
    "tex": "d(x,y)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "d",
        "glyph": "d"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        {
          "type": "symbol",
          "name": "y",
          "glyph": "y"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "c83bc97623e7296211353b466a760a4f88bc981d0f0204647fcc81883b76c4ef"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫使用0到n-1的有限載體及完整整數距離表。"
    },
    {
      "id": "a2",
      "zh-Hant": "有限表驗收只證指定模型，不把窮舉外推到任意無限度量空間。"
    }
  ],
  "symbols": [
    {
      "token": "d",
      "role": "metric_candidate",
      "role_zh": "距離函數候選",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "point",
      "role_zh": "有限載體點",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "point",
      "role_zh": "有限載體點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "度量有固定載體。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "距離對每個有序點對給唯一實值。"
    },
    {
      "id": "mko-relation",
      "title_zh": "二元關係與集合投影",
      "reason_zh": "距離表以點對索引。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "距離值為非負實數。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "非負與三角不等式使用序。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "對角線須為零，非對角線不得為零，二者共同構成分離性。",
      "對稱性要求d(i,j)=d(j,i)。",
      "三角不等式必須對每個i,j,k核對d(i,k)≤d(i,j)+d(j,k)。"
    ],
    "common_misconception_zh": "只核對非負與對稱就叫度量，或漏掉非對角零值與三角不等式。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只驗完整有限表；不外推任意函數空間或無限載體。",
    "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": "加入開球、度量收斂、Cauchy序列與完備性。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-metric-space.md 整理；2026-10-13 脈衝 R06（使用者手動接續 2026-10-22 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}