{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-higher-order-derivative",
  "semantic_family_id": "sf-higher-order-derivative",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "高階導數"
  },
  "summary": {
    "zh-Hant": "以f^(0)=f及f^(n+1)=(f^(n))'遞迴定義每個存在的有限階導數。"
  },
  "statement": {
    "zh-Hant": "n∈N_0且每一中間階都須在宣告domain可再微分。存在第n階不推出第n+1階。多項式exact family用falling-factorial coefficients；order超過degree後為zero。高階導數存在不推出analytic或Taylor series convergence。"
  },
  "formula": {
    "tex": "D_n(f)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>D</mi><mi>n</mi></msub><mo>(</mo><mi>f</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "subscript",
        "base": {
          "type": "symbol",
          "name": "D",
          "glyph": "D"
        },
        "subscript": {
          "type": "symbol",
          "name": "n",
          "glyph": "n"
        }
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "c0631c1ef79969d9bf08152df7df7ea45b1630660675f7eee910b8fb3c2231ee"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "正向families使用low-to-high整係數多項式coefficient array。"
    },
    {
      "id": "a2",
      "zh-Hant": "order與evaluation point為bounded safe integers。"
    }
  ],
  "symbols": [
    {
      "token": "D_n",
      "role": "nth_derivative_operator",
      "role_zh": "第n階導數算子",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "function_with_recursive_derivatives",
      "role_zh": "逐階可微函數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-derivative",
      "title_zh": "導數",
      "reason_zh": "first derivative。"
    },
    {
      "id": "mko-derivative-rules",
      "title_zh": "導數運算法則",
      "reason_zh": "polynomial differentiation。"
    },
    {
      "id": "mko-mathematical-induction",
      "title_zh": "數學歸納法",
      "reason_zh": "recursive order。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "order zero回到原函數。",
      "每次derivative都以前一階存在為前提。",
      "polynomial beyond-degree derivatives為zero。"
    ],
    "common_misconception_zh": "從f^(n)存在推f^(n+1)存在，或從所有有限階存在直接推analytic。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪polynomial certificates不代表一般smooth/analytic theorem。",
    "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": "加入Taylor polynomial。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-higher-order-derivative.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-11 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"
  }
}