{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-definite-integral",
  "semantic_family_id": "sf-definite-integral",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限區間上的多項式定積分"
  },
  "summary": {
    "zh-Hant": "連續多項式的有向定積分與精確矩公式；有限左端黎曼和只是有限和，不冒充一般積分或微積分基本定理完成。"
  },
  "statement": {
    "zh-Hant": "對有限實區間a≤b上的連續多項式f，Riemann定積分是所有合法標記分割和在網格趨0時的共同極限；反向端點定義I(f,a,b)=−I(f,b,a)，a=b時為0。本輪明列受限矩公式 I(x^n,a,b)=(b^(n+1)−a^(n+1))/(n+1)，對f=Σc_k*x^k直接用I(f,a,b)=Σc_k(b^(k+1)−a^(k+1))/(k+1)。"
  },
  "formula": {
    "tex": "I(f,a,b)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>I</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "I",
        "glyph": "I"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        },
        {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        },
        {
          "type": "symbol",
          "name": "b",
          "glyph": "b"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "edb5eef2042af4cbbd281f82684b1066c7ad8f21fb5a61911e37199c9560d088"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫端點及係數為安全整數；一般多項式至多3次，單項式c*x^n另限n=0至5，所有這些函數在全R連續並在有限區間可積。"
    },
    {
      "id": "a2",
      "zh-Hant": "定積分是有向、有號量，不必非負；反向區間要換號，不能一律取絕對值當幾何面積。"
    },
    {
      "id": "a3",
      "zh-Hant": "有限左端和只限a≤b、N=1至50，Δx=(b−a)/N，取i=0至N−1的左端點。零長度按0約定；有限和一般不等於積分。"
    }
  ],
  "symbols": [
    {
      "token": "I",
      "role": "signed_polynomial_interval_integral",
      "role_zh": "本輪多項式有向定積分",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "continuous_real_polynomial",
      "role_zh": "指定連續多項式",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "oriented_lower_endpoint",
      "role_zh": "有向起點",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "oriented_upper_endpoint",
      "role_zh": "有向終點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-limit",
      "title_zh": "去心實數論域的函數極限",
      "reason_zh": "極限與有限和不可混同。"
    },
    {
      "id": "mko-polynomial",
      "title_zh": "多項式",
      "reason_zh": "受限多項式與逐項矩計算。"
    },
    {
      "id": "mko-antiderivative",
      "title_zh": "多項式原函數與常數選擇",
      "reason_zh": "僅對本輪多項式矩核對primitive端點表示。"
    },
    {
      "id": "mko-area",
      "title_zh": "座標矩形與單位格的面積",
      "reason_zh": "保留有號積分與非負幾何面積之區別，不繼承一般面積proof。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "不是只讓某一種等分左端和收斂就能替任意函數定義Riemann積分；定義要求合法標記分割的共同極限。本輪只對已指定連續多項式使用明列矩公式。",
      "常數c積分為c(b−a)，單項式與有限多項式使用本物件的精確矩公式。出題端可用同一受限primitive係數端點差作另一算法，主端直接累加矩；不把尚未物化的一般微積分基本定理當作已完成前置或新形式 Evidence。",
      "仿射均勻左端和能直接精確算：Σ(mx_i+c)Δx，其中x_i=a+iΔx。產生端可用等差數列和，主端逐項有理累加。有限和即使在常數特例等於積分，也不升格成所有函數的積分證明。"
    ],
    "common_misconception_zh": "有限左端和當成積分、反向端點不換號、積分一律當非負面積，或把多項式矩驗收稱為一般FTC完成。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本物件的受限多項式矩與有限和可精確重算；Riemann／intervalIntegral橋及一般微積分基本定理仍未附形式 Evidence。",
    "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": "固定版intervalIntegral.integral_pow與有向端點橋接；Riemann／Lebesgue一致性及一般FTC仍為獨立未封閉義務。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-definite-integral.md 整理；2026-10-07 脈衝 R07（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"
  }
}