{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-fraction",
  "semantic_family_id": "sf-fraction",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "分數"
  },
  "summary": {
    "zh-Hant": "分數 p/q 是有理數的一種表示；分子與分母為整數，分母必須非零。"
  },
  "statement": {
    "zh-Hant": "對整數 p、q 且 q≠0，p/q 表示有理數。兩個合法表示 p/q 與 r/s 等值，當且僅當 ps=rq；約分與正分母正規化改變表示而不改變數值。"
  },
  "formula": {
    "tex": "\\frac{p}{q}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mi>p</mi><mi>q</mi></mfrac></math>",
    "semantic_ast": {
      "type": "fraction",
      "numerator": {
        "type": "symbol",
        "name": "p",
        "glyph": "p"
      },
      "denominator": {
        "type": "symbol",
        "name": "q",
        "glyph": "q"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8c0cd9aee7ba0b87686d7f95c8f5629ea9e5c24579191af1cf0001aa1fd992c1"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "p、q、r、s 是整數，所有作為分母的數都非零。"
    }
  ],
  "symbols": [
    {
      "token": "p",
      "role": "numerator",
      "role_zh": "分子",
      "scope": "representative_formula"
    },
    {
      "token": "q",
      "role": "nonzero_denominator",
      "role_zh": "非零分母",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-rational-number",
      "title_zh": "有理數",
      "reason_zh": "分數表示有理數，表示與數值須分開。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "等值、約分與加法使用整數乘法、加法與非零除法。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "q>0 且 0≤p≤q 時，p/q 可把一個單位等分成 q 份後取 p 份。p>q 的表示延伸到多個單位；p<0 則表示 |p|/q 的相反數。",
      "零分子且分母非零時數值為零。把分子與分母同乘一個非零整數可取得等值表示；常用最簡正分母形式，零固定寫成 0/1。",
      "對非零 q、s，p/q+r/s=(ps+rq)/(qs)。這是指定有理數語境中的精確運算；文字上把分子與分母分別相加不是分數加法。"
    ],
    "common_misconception_zh": "分數的字面表示不唯一；不同分母不表示數值不同，零分母也不表示零。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "整數比值與有限加法可精確重算；這些案例不取代有理數商集建構與運算良定性的普遍證明。",
    "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 Rat 的商表示、非零分母、gcd 正規化、等值與加法的精確聲明。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-fraction.md 整理；2026-09-30 脈衝 R01，英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-09-30"
  }
}