{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-function-graph",
  "semantic_family_id": "sf-function-graph",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實值函數圖形"
  },
  "summary": {
    "zh-Hant": "固定直角座標系後，實值函數的圖形是全部合法輸入與其唯一輸出組成的有序對集合。"
  },
  "statement": {
    "zh-Hant": "對 D⊆R 及總函數 f:D→R，在固定平面直角座標系中定義 G_f={(x,f(x)):x∈D}。有序對 (u,v) 屬於 G_f，當且僅當 u∈D 且 v=f(u)。"
  },
  "formula": {
    "tex": "P(x,f(x))",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>x</mi><mo>,</mo><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        {
          "type": "function_call",
          "callee": {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          "arguments": [
            {
              "type": "symbol",
              "name": "x",
              "glyph": "x"
            }
          ]
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6eee8cda3526ebf943cd042cba45b70dcec12d6b0fd2507361eef6e5ca54bb5d"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "座標系固定、兩座標為有序實數對；先宣告定義域 D。"
    },
    {
      "id": "a2",
      "zh-Hant": "f 在每個 D 中輸入恰有一個實數輸出；有限表格若宣告陪域 B，所有輸出還须落在 B。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "graph_point",
      "role_zh": "圖形上的點記號",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "domain_input",
      "role_zh": "定義域輸入",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "total_real_valued_function",
      "role_zh": "實值總函數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "每個合法輸入須有唯一輸出。"
    },
    {
      "id": "mko-coordinate-plane",
      "title_zh": "座標平面",
      "reason_zh": "函數圖形的兩座標依固定座標系解讀。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "成員判定同時檢查 u∈D 和等式 v=f(u)。即使某個公式在域外也可求值，該點仍不在宣告的函數圖形內。",
      "有限圖形可依第一座標排序，重複有序對只列一次。給定 D 和 B 時，要逐輸入檢查恰有一個 B 中的輸出，也不能多出 D 外的圖形點；這是指定總函數的表格驗證。",
      "垂直線測試檢查每個輸入至多一個輸出，單独不足以檢查對宣告 D 的總定義性。示意圖不提供精確座標；圖形的單點或有限資料也不自動建立反函數、滿射或全域公式。"
    ],
    "common_misconception_zh": "只檢查 y=f(x) 而漏掉定義域，或把垂直線至多一點誤當宣告域上每點恰有輸出。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪有限明示圖形和整數仿射輸入可精確驗證；一般圖形集合與幾何表示橋仍待形式化。",
    "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 Set.graph、實數乘積座標表示、域限制及有限總函數表格語義。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-function-graph.md 整理；2026-10-03 脈衝 R06，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"
  }
}