{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-convex-function",
  "semantic_family_id": "sf-convex-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "凸函數"
  },
  "summary": {
    "zh-Hant": "凸域上的函數為凸，若每個兩點凸組合處的函數值不超過相同權重的端點函數值。"
  },
  "statement": {
    "zh-Hant": "在凸定義域C上，f為凸若對所有x,y∈C及lambda∈[0,1]，f(lambda x+(1-lambda)y)≤lambda f(x)+(1-lambda)f(y)。本輪用區間與一元仿射／二次函數的完整代數證書。"
  },
  "formula": {
    "tex": "J(f)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>J</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "J",
        "glyph": "J"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6676521941602bad615e61f71dd8cc179165af5554f7c8edc229554a36bc50b9"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "定義域為明示非空實區間；二次函數係數與Jensen實例均精確表示。"
    },
    {
      "id": "a2",
      "zh-Hant": "單一Jensen違反足以否證；單一通過實例不能證凸，正證書需覆蓋全部x、y、lambda。"
    }
  ],
  "symbols": [
    {
      "token": "J",
      "role": "jensen_record",
      "role_zh": "Jensen凸性紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "real_function",
      "role_zh": "實值函數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-convex-set",
      "title_zh": "凸集合",
      "reason_zh": "函數凸性先要求定義域凸。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "函數在定義域上須為總映射。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "Jensen條件使用不等式比較。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "仿射函數對所有Jensen組合取等號。",
      "二次函數的Jensen gap可化為a lambda(1-lambda)(x-y)²。",
      "有限實例只負責給反例或例示，不能替代全稱證書。"
    ],
    "common_misconception_zh": "定義域非凸仍談一般凸函數，或由幾個Jensen樣本通過宣稱全域凸。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核一元區間上的仿射／二次代數證書。",
    "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": "加入嚴格凸、次梯度與高維凸函數。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-convex-function.md 整理；2026-10-13 脈衝 R02（使用者手動接續 2026-10-25 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}