{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-absolute-value",
  "semantic_family_id": "sf-absolute-value",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "絕對值",
    "en": "Absolute value"
  },
  "summary": {
    "zh-Hant": "實數 x 的絕對值是 x 到 0 的非負距離，並依 x 的正負號分段取 x 或 -x。"
  },
  "statement": {
    "zh-Hant": "函數 A:R→R 定義為：若 x≥0，則 A(x)=x；若 x<0，則 A(x)=-x。慣用記號為 |x|，且 |x|≥0。"
  },
  "formula": {
    "tex": "A(x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "c4e08f36576ff9cb77a540736ecf21a595f9a662768a1137ccb5966edda4f67a"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "x 是帶標準線性序與加法逆元的實數。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "absolute_value_function",
      "role_zh": "絕對值函數的代表記號",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "real_input",
      "role_zh": "實數輸入",
      "scope": "definition"
    },
    {
      "token": "|x|",
      "role": "absolute_value_notation",
      "role_zh": "x 的絕對值慣用記號",
      "scope": "explanation"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "本條目在實數線與標準線性序上定義絕對值。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "負分支使用加法逆元 -x，距離例子使用差。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "若 x≥0，絕對值保留 x；若 x<0，絕對值取其相反數 -x。因此 |x| 永遠非負，且只有 x=0 時 |x|=0。",
      "在數線上，|x| 是 x 到 0 的距離；兩個實數 a 與 b 之間的距離可表示為 |a-b|。距離不因交換端點而改變。",
      "三角不等式 |x+y|≤|x|+|y|、一般不等式解法、複數模與範數都是另需聲明與證明的後續內容，不是這個分段定義本身。"
    ],
    "common_misconception_zh": "常見誤解是把 |x| 一律寫成 x、把絕對值當成刪除負號的文字規則，或認為 |x| 可以是負數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "整數案例可由分段規則精確重算，但有限案例不構成全部實數性質、三角不等式或抽象絕對值公理的證明。",
    "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 Mathlib 的 abs、abs_cases、abs_nonneg、abs_eq_zero 與實數線距離 |a-b| 的精確聲明及版本證據。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-absolute-value.md 整理；待人工與形式化審查。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-09-28"
  }
}