{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-natural-divisibility",
  "semantic_family_id": "sf-natural-divisibility",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "自然數整除"
  },
  "summary": {
    "zh-Hant": "自然數 a 整除 b，表示存在自然數乘數 k 使 b=a·k；零的兩側行為必須分開。"
  },
  "statement": {
    "zh-Hant": "對自然數 a、b，a∣b 當且僅當存在自然數 k 使 b=a·k。每個 a 都整除 0；0∣b 當且僅當 b=0。"
  },
  "formula": {
    "tex": "b=m(a,k)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>b</mi><mo>=</mo><mrow><mi>m</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "symbol",
        "name": "b",
        "glyph": "b"
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "m",
          "glyph": "m"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          },
          {
            "type": "symbol",
            "name": "k",
            "glyph": "k"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "d588b10f8dfc16b5d41259c20d6539b468d6ba62f17293f1efff248ce783ce7b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "自然數包含 0，乘法為自然數乘法。"
    },
    {
      "id": "a2",
      "zh-Hant": "整除的見證 k 必須是自然數；不得以非整數商作見證。"
    }
  ],
  "symbols": [
    {
      "token": "a",
      "role": "candidate_divisor",
      "role_zh": "候選除數",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "candidate_multiple",
      "role_zh": "被整除數",
      "scope": "representative_formula"
    },
    {
      "token": "m",
      "role": "natural_multiplication",
      "role_zh": "自然數乘法",
      "scope": "representative_formula"
    },
    {
      "token": "k",
      "role": "natural_witness",
      "role_zh": "自然數乘法見證",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-natural-number",
      "title_zh": "自然數",
      "reason_zh": "論域與零邊界來自自然數。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "整除見證以精確乘法等式表示。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "檢查 a∣b 時先處理 a=0：只有 b=0 時成立。a>0 時可用商餘或直接搜尋 k，但餘數為零才提供整除見證。",
      "a∣0 對所有自然數 a 成立，因為 0=a·0；0∣0 也成立，而且見證不唯一。",
      "整除具有自反性與傳遞性；有限題目會保存明示的乘數見證，不把浮點除法或近似相等當成證據。"
    ],
    "common_misconception_zh": "看到 b/a 是數值就宣稱 a∣b，或把除數為零一律判成沒有整除關係。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限整數乘法與商餘可精確重算；案例核對不構成全部整除定理的形式證明。",
    "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": "綁定 Nat.Dvd 的存在乘數定義、dvd_zero、zero_dvd 與傳遞性。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-natural-divisibility.md 整理；2026-10-13 脈衝 R02（使用者手動接續 2026-10-18 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"
  }
}