{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-intermediate-value-theorem",
  "semantic_family_id": "sf-intermediate-value-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "介值定理"
  },
  "summary": {
    "zh-Hant": "連續實函數在閉區間端點值之間取到每個中間值。"
  },
  "statement": {
    "zh-Hant": "固定a<b、完整interval domain與continuous polynomial certificate，target y須inclusive介於f(a)、f(b)。exact families提供明示c∈[a,b]且f(c)=y；endpoint root保留。IVT只保證存在，不保證唯一、rational closed form或有限bisection已收斂。明示discontinuous step可有sign change但沒有root；有限samples不證continuity。"
  },
  "formula": {
    "tex": "I(f,a,b,y)=c",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>I</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>c</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "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"
          },
          {
            "type": "symbol",
            "name": "y",
            "glyph": "y"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "c",
        "glyph": "c"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8700d6a45a86289693e54723a7bfe4ca6807874b0b6b3ab021603266cd6ffb2f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "exact rational polynomial用作可驗continuous instance。"
    },
    {
      "id": "a2",
      "zh-Hant": "root witness與endpoint/bracket均exact核對。"
    }
  ],
  "symbols": [
    {
      "token": "I",
      "role": "intermediate_value_operator",
      "role_zh": "介值定理",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "continuous_real_function",
      "role_zh": "連續實函數",
      "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"
    },
    {
      "token": "y",
      "role": "target_value",
      "role_zh": "目標值",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "intermediate_point",
      "role_zh": "介值點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-continuity",
      "title_zh": "連續性",
      "reason_zh": "核心假設。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "interval與values。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "between relation。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "domain/codomain。"
    },
    {
      "id": "mko-supremum-and-infimum",
      "title_zh": "上確界與下確界",
      "reason_zh": "一般證明背景。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "interval、continuity與bracket完整資格核對。",
      "exact root witness不等同唯一性。",
      "discontinuous反例保護continuity責任。"
    ],
    "common_misconception_zh": "只由endpoint sign change但未證continuity便宣稱有root。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "polynomial instances不取代一般IVT proof；有限bisection不等於exact convergence。",
    "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root-finding convergence boundaries。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-intermediate-value-theorem.md 整理；2026-10-15 脈衝 R02（使用者手動接續 2026-11-16 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}