{
  "$schema": "../../../schemas/mko-v0.3.schema.json",
  "schema_version": "mko-v0.3",
  "id": "mko-euclidean-length",
  "semantic_family_id": "sf-euclidean-length",
  "type": "definition",
  "version": "0.5.0",
  "language": "zh-Hant",
  "titles": { "zh-Hant": "歐幾里得長度", "en": "Euclidean length" },
  "summary": { "zh-Hant": "平面上兩點之間由座標差平方和的正平方根所定義的距離。" },
  "statement": { "zh-Hant": "在歐幾里得平面中，若 P=(x₁,y₁)、Q=(x₂,y₂)，則兩點距離定義為座標差平方和的正平方根。" },
  "formula": {
    "tex": "d(P,Q)=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>d</mi><mo>(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><msup><mrow><mo>(</mo><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": { "type": "symbol", "name": "d", "glyph": "d" },
        "arguments": [
          { "type": "symbol", "name": "P", "glyph": "P" },
          { "type": "symbol", "name": "Q", "glyph": "Q" }
        ]
      },
      "rhs": {
        "type": "square_root",
        "radicand": {
          "type": "addition",
          "operands": [
            {
              "type": "power",
              "base": {
                "type": "group",
                "body": {
                  "type": "subtraction",
                  "left": { "type": "subscript", "base": { "type": "symbol", "name": "x", "glyph": "x" }, "subscript": { "type": "number", "value": 2 } },
                  "right": { "type": "subscript", "base": { "type": "symbol", "name": "x", "glyph": "x" }, "subscript": { "type": "number", "value": 1 } }
                }
              },
              "exponent": { "type": "number", "value": 2 }
            },
            {
              "type": "power",
              "base": {
                "type": "group",
                "body": {
                  "type": "subtraction",
                  "left": { "type": "subscript", "base": { "type": "symbol", "name": "y", "glyph": "y" }, "subscript": { "type": "number", "value": 2 } },
                  "right": { "type": "subscript", "base": { "type": "symbol", "name": "y", "glyph": "y" }, "subscript": { "type": "number", "value": 1 } }
                }
              },
              "exponent": { "type": "number", "value": 2 }
            }
          ]
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.3.0",
      "source_sha256": "1e1c9f11e53bf6639ab84fb44c67d06df41db6672cd37e61aba68e70e35c42ac"
    }
  },
  "assumptions": [
    { "id": "a1", "zh-Hant": "P 與 Q 是二維歐幾里得平面中的點。" },
    { "id": "a2", "zh-Hant": "座標 x₁、y₁、x₂、y₂ 為實數。" },
    { "id": "a3", "zh-Hant": "平方根取非負主值，因此距離必為非負。" }
  ],
  "symbols": [
    { "token": "d", "role": "distance_function", "role_zh": "距離函數", "scope": "definition" },
    { "token": "P", "role": "point", "role_zh": "第一個平面點", "scope": "definition" },
    { "token": "Q", "role": "point", "role_zh": "第二個平面點", "scope": "definition" },
    { "token": "x₁,y₁", "role": "coordinates", "role_zh": "P 的座標", "scope": "definition" },
    { "token": "x₂,y₂", "role": "coordinates", "role_zh": "Q 的座標", "scope": "definition" }
  ],
  "dependencies": [],
  "explanation": {
    "paragraphs_zh": [
      "距離公式把水平位移與垂直位移視為互相垂直的兩個分量，再取平方和的正平方根。",
      "當 P 與 Q 相同時，兩個座標差皆為零，因此距離為零；交換 P 與 Q 不會改變距離。",
      "程式可用 hypot 或平方根計算具體距離，但浮點表示與數學中的實數並不完全相同。"
    ],
    "common_misconception_zh": "忽略平方根的非負主值，或將座標差直接相加而沒有先平方。"
  },
  "proofs": [],
  "computational_companions": [
    {
      "id": "cc-python-euclidean-length-001",
      "language": "python",
      "relation": "numeric_distance_evaluator",
      "status": "reference_only",
      "path": "reference/python/euclidean_length.py",
      "warning": "此程式在有限浮點精度下計算具體座標距離，不等同於實數度量空間中的完整定義。",
      "source": "from math import hypot\n\ndef euclidean_distance(p, q):\n    x1, y1 = p\n    x2, y2 = q\n    return hypot(x2 - x1, y2 - y1)",
      "non_identity": {
        "preserved": ["二維座標差", "平方和", "非負距離"],
        "approximated": ["實數運算由有限精度浮點數近似"],
        "omitted": ["度量公理的普遍證明", "任意維度的一般化"]
      }
    }
  ],
  "verification": {
    "computational_status": "finite_cases_passed",
    "warning_zh": "具體座標測試只驗證參考實作，不等於證明距離函數對所有實數點滿足全部度量公理。",
    "evidence_refs": [
      {
        "id": "evidence-sha256-de740ccfca05c5027587999af29adeead880cfe6ebaa28063d95ad584875e31e",
        "role": "finite_computational_check",
        "producer_id": "ocme-python-suite"
      }
    ],
    "producers": [
      {
        "id": "ocme-python-suite",
        "adapter": "native_python",
        "config_path": "reference/python/verify_all.py",
        "status": "active"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": ["Lean 4"],
    "next_obligation_zh": "對接 Mathlib 的 EuclideanSpace 與 dist 定義，並記錄二維座標公式與抽象距離之間的定理。"
  },
  "provenance": {
    "origin": "evemisslab_original_mvp",
    "note_zh": "此定義物件為 OCME v0.2 的原生架構示範。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-07-25"
  }
}