{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-mean-value-theorem",
  "semantic_family_id": "sf-mean-value-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "微分中值定理"
  },
  "summary": {
    "zh-Hant": "若a<b，f在[a,b]連續且在(a,b)可微，則某個內點c的導數等於端點割線斜率。"
  },
  "statement": {
    "zh-Hant": "題族參數A/B/C明確編碼f(t)=A t^2+B t+C。必須a<b、closed interval continuity與open interval differentiability；witness c嚴格位於(a,b)，且f'(c)=(f(b)-f(a))/(b-a)。c不保證唯一，affine或constant case可有多個witness。有限samples、a=b、jump或interior cusp不足。"
  },
  "formula": {
    "tex": "D(f,c)=\\frac{f(b)-f(a)}{b-a}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>D</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mi>f</mi><mo>(</mo><mi>b</mi><mo>)</mo></mrow><mo>−</mo><mrow><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "D",
          "glyph": "D"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          {
            "type": "symbol",
            "name": "c",
            "glyph": "c"
          }
        ]
      },
      "rhs": {
        "type": "fraction",
        "numerator": {
          "type": "subtraction",
          "left": {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "f",
              "glyph": "f"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "b",
                "glyph": "b"
              }
            ]
          },
          "right": {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "f",
              "glyph": "f"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "a",
                "glyph": "a"
              }
            ]
          }
        },
        "denominator": {
          "type": "subtraction",
          "left": {
            "type": "symbol",
            "name": "b",
            "glyph": "b"
          },
          "right": {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          }
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f8854e8debd0b839816cfd134eb5b1232c64c42fa65a0cbde3ca9c8a8ccc27b9"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "正向families使用明示f(t)=A t^2+B t+C的整係數一次或二次多項式，因此全域連續可微。"
    },
    {
      "id": "a2",
      "zh-Hant": "端點與witness以exact integer/rational arithmetic核對。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "derivative_evaluation",
      "role_zh": "導數值",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "real_function",
      "role_zh": "實函數",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "interior_witness",
      "role_zh": "內點witness",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "left_endpoint",
      "role_zh": "左端點",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "right_endpoint",
      "role_zh": "右端點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-derivative",
      "title_zh": "導數",
      "reason_zh": "open-interval derivative。"
    },
    {
      "id": "mko-continuity",
      "title_zh": "連續性",
      "reason_zh": "closed-interval continuity。"
    },
    {
      "id": "mko-rate-of-change",
      "title_zh": "變化率",
      "reason_zh": "secant slope。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "a<b與interior。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "A/B/C依序是quadratic、linear與constant coefficient。",
      "hypotheses作用在不同區間部分。",
      "secant denominator由a<b保證非零，且witness存在不等於唯一。"
    ],
    "common_misconception_zh": "由有限點看似平滑便套MVT，或允許c落在端點。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "exact題族限明示多項式及明示反例；不由sampling證一般可微。",
    "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與remainder。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-mean-value-theorem.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-10 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"
  }
}