{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-linear-function",
  "semantic_family_id": "sf-linear-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "仿射函數與一次函數"
  },
  "summary": {
    "zh-Hant": "實函數 f(x)=mx+b 稱仿射函數；m≠0 時為一次函數，m=0 時為常數函數。"
  },
  "statement": {
    "zh-Hant": "對 m、b∈R，f:R→R 由 f(x)=mx+b 定義為仿射函數。當 m≠0 時多項式次數為一；m=0 時為常數函數。線性代數中的 R 線性映射另要求 b=0，不能與初等教材的 linear function 稱呼混同。"
  },
  "formula": {
    "tex": "f:R\\to R",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow></math>",
    "semantic_ast": {
      "type": "mapping",
      "function": {
        "type": "symbol",
        "name": "f",
        "glyph": "f"
      },
      "domain": {
        "type": "symbol",
        "name": "R",
        "glyph": "R"
      },
      "codomain": {
        "type": "symbol",
        "name": "R",
        "glyph": "R"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "62d5897688306dde8a9f1f279eec7b651e2aa333f341d44c77dd972ae1271beb"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "函數的定義域與陪域固定為 R，m、b 是實常數。"
    },
    {
      "id": "a2",
      "zh-Hant": "兩點斜率只在 x_1≠x_2 時定義；單點比較只主張該點輸出相同。"
    }
  ],
  "symbols": [
    {
      "token": "f",
      "role": "affine_real_function",
      "role_zh": "實仿射函數",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "real_domain_and_codomain",
      "role_zh": "實數定義域與陪域",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "輸入、輸出及全域函數身份需明示。"
    },
    {
      "id": "mko-linear-equation",
      "title_zh": "一元一次方程式",
      "reason_zh": "m≠0 時零點由一次方程式求得。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "m 是每一單位輸入變化對應的輸出變化，b=f(0)。兩個不同橫座標的圖形點給出 m=(y_2−y_1)/(x_2−x_1)，不能對垂直線除以零。",
      "m≠0 時零點唯一為 −b/m；m=0、b=0 時所有實數皆為零點，m=0、b≠0 時無零點。常數函數仍屬仿射族，但不稱次數一的函數。",
      "若比較 f(x_0) 與 g(x_0)，輸出相等僅是單點事實。全實數域上兩個仿射函數相同當且僅當其斜率與截距都相同；本輪題族的單點問題不可擴張成全域相等。"
    ],
    "common_misconception_zh": "把所有 mx+b 都叫一次函數，或把帶非零截距的仿射函數當成線性代數的線性映射。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "整數係數與兩點有理斜率可精確重算；全域仿射結構與線性映射橋仍待形式化。",
    "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 R 上仿射函數、m≠0 一次性、常數退化零點集與線性代數同質性邊界。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-linear-function.md 整理；2026-10-03 脈衝 R05，英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}