{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-norm-induced-metric",
  "semantic_family_id": "sf-norm-induced-metric",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "範數誘導度量"
  },
  "summary": {
    "zh-Hant": "同一實賦範向量空間上的範數以向量差的範數定義度量，度量公理由範數公理全稱推出。"
  },
  "statement": {
    "zh-Hant": "若(V,norm)為已認證的實賦範向量空間，則d(x,y)=norm(x-y)滿足非負、分離、對稱與三角不等式。本輪只在R^n的命名l1與linfinity範數作精確座標切片。"
  },
  "formula": {
    "tex": "D(x,y)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        {
          "type": "symbol",
          "name": "y",
          "glyph": "y"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6b06b260d2cf1dc8367840f7cdc172142d827904c34ba335ab760eafb8fa860c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "左右點位於同一有限維實座標空間，使用同一已命名範數。"
    },
    {
      "id": "a2",
      "zh-Hant": "誘導性來自全稱公式與範數公理，不由有限點距離樣本建立。"
    }
  ],
  "symbols": [
    {
      "token": "d",
      "role": "induced_metric",
      "role_zh": "範數誘導度量",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "real_normed_vector_space",
      "role_zh": "實賦範向量空間",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "vector",
      "role_zh": "第一向量",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "vector",
      "role_zh": "第二向量",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-normed-vector-space",
      "title_zh": "實賦範向量空間",
      "reason_zh": "誘導公式依已認證範數。"
    },
    {
      "id": "mko-metric-space",
      "title_zh": "度量空間",
      "reason_zh": "結論是度量公理。"
    },
    {
      "id": "mko-vector",
      "title_zh": "向量",
      "reason_zh": "誘導公式使用同一空間中的向量差。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "映射",
      "reason_zh": "d須為V乘V到非負實數的總函數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "分離性由norm(x-y)=0當且僅當x-y=0。",
      "對稱性由norm(-(y-x))=norm(y-x)。",
      "三角不等式由x-z=(x-y)+(y-z)及範數三角不等式。"
    ],
    "common_misconception_zh": "檢查有限幾對距離後就宣稱任意metric由某個norm誘導。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理同一認證範數的誘導公式；不宣稱每個度量皆範數誘導，也不推出完備、內積、緊緻或有限維。",
    "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": "與完備度量空間共同支撐實Banach空間。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-norm-induced-metric.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-10-27 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}