{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-function-mapping",
  "semantic_family_id": "sf-function-mapping",
  "type": "definition",
  "version": "0.8.0",
  "language": "zh-Hant",
  "titles": { "zh-Hant": "函數映射", "en": "Function mapping" },
  "summary": { "zh-Hant": "符號 f:X→Y 表示函數 f 從定義域 X 映射到陪域 Y。" },
  "statement": { "zh-Hant": "函數 f:X→Y 對 X 中每一個輸入指定 Y 中唯一一個輸出；陪域 Y 不必等於實際值域。" },
  "formula": {
    "tex": "f:X\\to Y",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></math>",
    "semantic_ast": {
      "type": "mapping",
      "function": { "type": "symbol", "name": "f", "glyph": "f" },
      "domain": { "type": "symbol", "name": "X", "glyph": "X" },
      "codomain": { "type": "symbol", "name": "Y", "glyph": "Y" }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "28c28a16d0b842cb3d3c986cead38d14c1fbddc06efeb3c29e6e3479896c8a21"
    }
  },
  "assumptions": [
    { "id": "a1", "zh-Hant": "X 與 Y 是已指定的集合或相容類型。" },
    { "id": "a2", "zh-Hant": "X 中每個輸入在 f 下都有且只有一個輸出。" }
  ],
  "symbols": [
    { "token": "f", "role": "function", "role_zh": "函數", "scope": "definition" },
    { "token": "X", "role": "domain", "role_zh": "定義域", "scope": "definition" },
    { "token": "Y", "role": "codomain", "role_zh": "陪域", "scope": "definition" },
    { "token": "→", "role": "mapping_arrow", "role_zh": "映射箭頭", "scope": "definition" }
  ],
  "dependencies": [
    { "id": "mko-set-membership", "title_zh": "集合隸屬", "reason_zh": "函數輸入與輸出分別屬於定義域與陪域。" }
  ],
  "explanation": {
    "paragraphs_zh": [
      "函數的核心不是一條畫出來的曲線，而是每個允許輸入都對應唯一輸出的規則。",
      "f:X→Y 宣告了定義域與陪域；實際輸出形成的值域可能只是 Y 的一部分。",
      "Lean 定理 function_total_unique 已對任意 f:X→Y 與 x:X 證明存在唯一 y:Y 使 f x=y。"
    ],
    "common_misconception_zh": "把陪域、值域與函數圖形視為同一個數學對象，或把具有副作用與失敗狀態的程式函數直接等同於純數學總函數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "formal_total_function_core_available",
    "warning_zh": "Lean 已形式化總函數對每個輸入存在唯一輸出的核心語義；部分函數、副作用、隨機性與執行失敗不在此聲明內。",
    "evidence_refs": [
      {
        "id": "evidence-sha256-d999133c1ae711b73752af8ee4b1b3ce74ca135ff7c06aabbe9fc8a3186f434b",
        "role": "formal_proof",
        "producer_id": "lean-mathlib"
      }
    ],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "active"
      }
    ]
  },
  "formalization": {
    "status": "formalized_total_function_core",
    "target_systems": ["Lean 4", "Mathlib"],
    "next_obligation_zh": "新增值域、單射、滿射、雙射與部分函數物件，明確區分型別陪域與實際輸出集合。"
  },
  "provenance": {
    "origin": "evemisslab_original_mvp",
    "note_zh": "OCME 基礎數學知識物件；敘述為原創整理。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-07-31"
  }
}