{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-area",
  "semantic_family_id": "sf-area",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "座標矩形與單位格的面積"
  },
  "summary": {
    "zh-Hant": "先精確處理半開矩形及有限單位格聯集，含重疊、觸邊與空矩形；不冒充一般面積理論。"
  },
  "statement": {
    "zh-Hant": "在等單位正交座標系，半開矩形 [x0,x1)×[y0,y1)（x0≤x1、y0≤y1）面積為 (x1−x0)(y1−y0)。任一邊長零時矩形為空、面積零。整數單位格 [i,i+1)×[j,j+1) 面積一，有限聯集面積為不同格子的個數；兩矩形聯集面積為各面積和減交集面積。"
  },
  "formula": {
    "tex": "A(w,h)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>A</mi><mo>(</mo><mi>w</mi><mo>,</mo><mi>h</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "w",
          "glyph": "w"
        },
        {
          "type": "symbol",
          "name": "h",
          "glyph": "h"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "fb632ab6f705e2ee13494ba63914c5262845c66ae8073bf4ca969b604b0cdbba"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫採非負整數邊長或有序整數端點；半開端點及同單位框架須明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "兩矩形交集寬高分別為 max(0,min(上界)−max(下界))；相離與僅觸邊交集面積零。"
    },
    {
      "id": "a3",
      "zh-Hant": "單位格按座標去重，重複列格不增加面積；答案單位為所用長度單位的平方。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "scoped_rectangle_area",
      "role_zh": "本輪矩形面積運算",
      "scope": "representative_formula"
    },
    {
      "token": "w",
      "role": "nonnegative_width",
      "role_zh": "非負寬",
      "scope": "representative_formula"
    },
    {
      "token": "h",
      "role": "nonnegative_height",
      "role_zh": "非負高",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-euclidean-length",
      "title_zh": "歐氏長度",
      "reason_zh": "固定長度單位及正交框架。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "精確長乘寬與加減。"
    },
    {
      "id": "mko-coordinate-plane",
      "title_zh": "座標平面",
      "reason_zh": "矩形端點和單位格位置。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "聯集、交集及表示去重。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "矩形的幾何量不是周長。兩矩形有重疊時直接相加會重算交集，必須扣回一次；相同矩形的聯集仍只有原面積。",
      "整數半開框架可把矩形分成不重複的單位格，提供與包含排除不同的有限重算方法。空矩形及空格子聯集面積都是零。",
      "本輪只建立矩形與有限格子的候選模型，不宣稱一般曲面、可測集、積分或面積存在性已驗證。共同平移或交換兩矩形只是同一計算的表示變動。"
    ],
    "common_misconception_zh": "聯集直接相加、把重複格子重算，或把零寬矩形當有面積。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限矩形／格子面積可精確重算；一般 Lebesgue 面積及積分橋未新增形式 Evidence。",
    "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": "在固定版橋接二維 box volume、空邊界與有限聯集；先維持本輪受限模型。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-area.md 整理；2026-10-05 脈衝 R06（使用者手動接續 2026-10-06 工作序列；實際日期仍為 2026-10-05），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-05"
  }
}