{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-ratio-proportion",
  "semantic_family_id": "sf-ratio-proportion",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "比與比例"
  },
  "summary": {
    "zh-Hant": "本條目把數值比 a:b 解讀為 a/b，把比例定義為兩個合法比值相等。"
  },
  "statement": {
    "zh-Hant": "對有理數 a、b、c、d 且 b≠0、d≠0，數值比 a:b 的值為 a/b；比例 a:b=c:d 表示 a/b=c/d，等價於 ad=bc。"
  },
  "formula": {
    "tex": "\\frac{a}{b}=\\frac{c}{d}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>=</mo><mfrac><mi>c</mi><mi>d</mi></mfrac></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "fraction",
        "numerator": {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        },
        "denominator": {
          "type": "symbol",
          "name": "b",
          "glyph": "b"
        }
      },
      "rhs": {
        "type": "fraction",
        "numerator": {
          "type": "symbol",
          "name": "c",
          "glyph": "c"
        },
        "denominator": {
          "type": "symbol",
          "name": "d",
          "glyph": "d"
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "87975f5f64a27c09261a0b8a64eb02e80f931e6f6b857895688d6e04db3a6e00"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "本條目採用有理數的數值比語境，b 與 d 非零。"
    },
    {
      "id": "a2",
      "zh-Hant": "應用到量時先指定單位與可比較性；本轮不由數值比例推出幾何相似或因果關係。"
    }
  ],
  "symbols": [
    {
      "token": "a",
      "role": "first_ratio_numerator",
      "role_zh": "第一比的前項",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "first_nonzero_denominator",
      "role_zh": "第一比的非零後項",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "second_ratio_numerator",
      "role_zh": "第二比的前項",
      "scope": "representative_formula"
    },
    {
      "token": "d",
      "role": "second_nonzero_denominator",
      "role_zh": "第二比的非零後項",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-fraction",
      "title_zh": "分數",
      "reason_zh": "數值比與比例使用合法分數表示及交叉乘積。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "比的順序有意義；a:b 與 b:a 通常不同，反向比还须檢查 a≠0。",
      "兩個比形成比例前，先確認兩個後項均非零，再比較交叉乘積 ad 與 bc。",
      "比例 a/b=x/d 在 b、d 非零時有唯一有理解 x=ad/b。正比例 y=kx 的有限資料需逐點檢查同一個 k，不從單一點推論所有輸入。"
    ],
    "common_misconception_zh": "交叉乘積相等不能讓零分母的表示變得合法；相同比例也不自動建立幾何相似。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "精確比值、比例與有限點檢查僅涵蓋明示條件；一般應用模型與幾何相似另有證明義務。",
    "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": "綁定有理數非零除法、交叉乘積等價與比例未知項求解。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-ratio-proportion.md 整理；2026-09-30 脈衝 R02，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"
  }
}