{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-fundamental-theorem-calculus",
  "semantic_family_id": "sf-fundamental-theorem-calculus",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "微積分基本定理：連續前提與有向端點"
  },
  "summary": {
    "zh-Hant": "在全實數連續前提下，累積定積分的導數還原被積函數；合法原函數給有向積分端點差。"
  },
  "statement": {
    "zh-Hant": "設 f:R→R 連續，a∈R，G(x)=∫_a^x f(t)dt 使用有向定積分，則 G 在R處處可微且 G'(x)=f(x)。若 F:R→R 可微並在R滿足 F'=f，則對任意實數端點a、b，∫_a^b f(t)dt=F(b)−F(a)。這裡採全R的充分前提，避免將單側端點導數或一般可積函數混入。"
  },
  "formula": {
    "tex": "D(G,x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        },
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "c60fb86945613c31051b36b25980c5abae6a6ed2b717e26865774a205dd5ec8f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "被積函數在全R連續；題庫限定1至4個升冪安全整數係數的多項式，即至多3次。候選原函數限1至5個正分母有理係數。"
    },
    {
      "id": "a2",
      "zh-Hant": "有向積分允许反向及相同端點；G(a)=0不意味G'(a)=0。移動上下限是全R仿射式，導數為f(u(x))u'(x)−f(l(x))l'(x)。"
    },
    {
      "id": "a3",
      "zh-Hant": "候選F'=f逐係數核對，不以有限取樣宣稱相等。候選不成立不否定其他原函數存在；一般FTC未附形式Evidence。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "real_derivative_at_point",
      "role_zh": "在指定實點的導數",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "continuous_integral_accumulation",
      "role_zh": "連續函數的有向積分累積函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "real_evaluation_point",
      "role_zh": "實數求導中心",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-derivative",
      "title_zh": "實函數在一點的導數與差商",
      "reason_zh": "保留導數存在性及兩側極限。"
    },
    {
      "id": "mko-definite-integral",
      "title_zh": "有限區間上的多項式定積分",
      "reason_zh": "有向端點與連續可積切片。"
    },
    {
      "id": "mko-continuity",
      "title_zh": "實函數的點值與連續",
      "reason_zh": "全R連續是此處充分前提。"
    },
    {
      "id": "mko-antiderivative",
      "title_zh": "多項式原函數與常數選擇",
      "reason_zh": "F'=f的合法原函數前提。"
    },
    {
      "id": "mko-derivative-rules",
      "title_zh": "多項式的和、乘積與合成微分",
      "reason_zh": "移動仿射上下限的鏈式規則。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "累積函數在起點取0，導數卻取f(a)；函數值與瞬時變化不能混同。上下限在某點重合，也不保證該點導數0，因二者速度可能不同。",
      "第二部分先核對F'=f，再算端點差。加常數不改此差；端點反向改符號。題庫用精確多項式係數與有理數，未將有限代數核對升格為一般定理形式證明。"
    ],
    "common_misconception_zh": "起點積分值0就導數0、上下限重合就导數0，或忽略原函數前提。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "此為有明示充分前提的定理候選；有限多項式計算可重算，一般FTC及Riemann／Mathlib區間積分橋仍待形式證明。",
    "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": "固定Mathlib版綁定 intervalIntegral.integral_hasDerivAt_right 與 integral_eq_sub_of_hasDerivAt，補連續可積、Riemann一致性及端點方向的型別橋。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-fundamental-theorem-calculus.md 整理；2026-10-07 脈衝 R01（使用者手動接續 2026-10-09 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"
  }
}