{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-antiderivative",
  "semantic_family_id": "sf-antiderivative",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "多項式原函數與常數選擇"
  },
  "summary": {
    "zh-Hant": "原函數須在宣告區間導回f；本輪全R多項式以常數項固定0的有理係數原函數作可重播代表。"
  },
  "statement": {
    "zh-Hant": "F是f在區間上的原函數，指F在該區間可微且F'=f。本輪全R多項式f(x)=Σa_k*x^k有代表F0(x)=Σa_k*x^(k+1)/(k+1)，常數項固定0；F0+C仍為原函數。本輪多項式原函數的導數係數相同時，非零次係數相同，所以只差常數項。"
  },
  "formula": {
    "tex": "A(f)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>A</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5d640c915773a09986db716ed5b55a8d3cfa43eab75cacd73f496eaa671af19b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "多項式族f採1至4個升冪整數係數；候選F採1至5個升冪有理數對[n,d]，d>0，不以有限代入取樣判F'=f。"
    },
    {
      "id": "a2",
      "zh-Hant": "單項式族c*x^n另限n=0至5，其非零primitive可長度7；多項式x^0恆為1，包括x=0。c=0的零原函數正規表示為[0/1]，不保留無意義高次零項。"
    },
    {
      "id": "a3",
      "zh-Hant": "代表答案以升冪最簡正分母有理係數列出，常數為0；求F0(x)+C的值時，C及x是真正資料，不因求導會消掉C而刪掉它。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "zero_constant_polynomial_primitive",
      "role_zh": "本輪常數項0的多項式原函數代表",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "real_polynomial_function",
      "role_zh": "指定全R多項式",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-derivative",
      "title_zh": "實函數在一點的導數與差商",
      "reason_zh": "F'=f的原函數判準。"
    },
    {
      "id": "mko-derivative-rules",
      "title_zh": "多項式的和、乘積與合成微分",
      "reason_zh": "有限多項式求導與係數恆等。"
    },
    {
      "id": "mko-rational-number",
      "title_zh": "有理數",
      "reason_zh": "除以k+1的精確係數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "給出一個原函數和給出全部原函數不同；本輪以F0選定可重播代表，但實際常數族F0+C不被忽略。",
      "候選驗收逐係數核對導數多項式是否等於f；候選的常數項不影響此判定，卻會影響函數求值。零函數的原函數是常數，不任意給零函數一個虛假高次次數。",
      "本輪只對多項式係數核對差常數；一般可微函數的區間版本另有平均值定理義務，不能由這些有限資料升格，也不搬到不連通論域。這不是函數的反函數，也不是導數數值取倒數。"
    ],
    "common_misconception_zh": "漏掉常數族、以幾個相等測點替代多項式恆等，或把原函數當反函數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限多項式係數及條件求值可精確重算；一般原函數存在性／區間唯一差常數的形式 Evidence未新增。",
    "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": "橋接有理係數poly derivative identity及選定常數，保留全R區間與一般存在性界線。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-antiderivative.md 整理；2026-10-07 脈衝 R06（使用者手動接續 2026-10-08 工作序列；實際日期仍為 2026-10-07），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-07"
  }
}