{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-tends-to-relation",
  "semantic_family_id": "sf-tends-to-relation",
  "type": "concept",
  "version": "0.8.0",
  "language": "zh-Hant",
  "titles": { "zh-Hant": "趨近關係", "en": "Tends-to relation" },
  "summary": { "zh-Hant": "符號 x→a 表示變數或索引所描述的狀態朝目標 a 趨近。" },
  "statement": { "zh-Hant": "x→a 是極限語言中的方向性關係；它本身尚未指定函數值是否收斂，也不等同於 x 最終必須等於 a。" },
  "formula": {
    "tex": "x\\to a",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>x</mi><mo>→</mo><mi>a</mi></mrow></math>",
    "semantic_ast": {
      "type": "tends_to",
      "expression": { "type": "symbol", "name": "x", "glyph": "x" },
      "target": { "type": "symbol", "name": "a", "glyph": "a" }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b02e34b2d9f06f10e61905032b56c36ef9eb448186055de79456ee841817c3cb"
    }
  },
  "assumptions": [
    { "id": "a1", "zh-Hant": "已指定可討論鄰近與趨近的數學結構，例如度量或拓撲。" }
  ],
  "symbols": [
    { "token": "x", "role": "varying_expression", "role_zh": "變動表達式或變數", "scope": "concept" },
    { "token": "a", "role": "target", "role_zh": "趨近目標", "scope": "concept" },
    { "token": "→", "role": "tends_to_relation", "role_zh": "趨近關係", "scope": "concept" }
  ],
  "dependencies": [
    { "id": "mko-function-mapping", "title_zh": "函數映射", "reason_zh": "函數極限會在輸入趨近時研究映射值的行為。" }
  ],
  "explanation": {
    "paragraphs_zh": [
      "趨近描述的是一個過程或關係，不是單次代入。x 可以任意接近 a，而不必在討論中的每一步都等於 a。",
      "完整函數極限還需要另外描述 f(x) 的目標值、量詞順序與鄰域條件；本物件只提供其中的趨近語義。",
      "Mathlib 使用 Filter.Tendsto f l₁ l₂ 表達函數相對於來源與目標濾子的趨近，並在定義上等價於 Filter.map f l₁≤l₂。"
    ],
    "common_misconception_zh": "把 x→a 直接解讀成 x=a，或在尚未指定來源與目標濾子時，宣稱裸箭頭已等同於完整 Filter.Tendsto 或 ε–δ 定義。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "formal_filter_tendsto_core_available",
    "warning_zh": "Lean 已形式化 Filter.Tendsto 的核心濾子語義；裸符號 x→a 仍必須補上函數與來源、目標濾子，且本證據不是特定函數極限。",
    "evidence_refs": [
      {
        "id": "evidence-sha256-38b848281997cbb1a2c1ea5559c6937dbb67ee03ac0be338756320a99e0f8b1d",
        "role": "formal_proof",
        "producer_id": "lean-mathlib"
      }
    ],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "active"
      }
    ]
  },
  "formalization": {
    "status": "formalized_filter_tendsto_core",
    "target_systems": ["Lean 4", "Mathlib"],
    "next_obligation_zh": "建立 atTop、nhds、nhdsWithin 等常用來源與目標濾子物件，再分別連接數列極限與函數點極限。"
  },
  "provenance": {
    "origin": "evemisslab_original_mvp",
    "note_zh": "OCME 極限基礎知識物件；敘述為原創整理。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-07-31"
  }
}