{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-convex-set",
  "semantic_family_id": "sf-convex-set",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "凸集合"
  },
  "summary": {
    "zh-Hant": "集合凸表示任兩點及每個lambda∈[0,1]形成的整條線段仍在集合內。"
  },
  "statement": {
    "zh-Hant": "實向量空間子集C為凸，若對所有x,y∈C及所有lambda∈[0,1]，lambda x+(1-lambda)y仍屬C。本輪正例只用區間、盒與半空間的完整代數證書；單一精確反例可否證凸性。"
  },
  "formula": {
    "tex": "K(C)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>K</mi><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "K",
        "glyph": "K"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "C",
          "glyph": "C"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8149eb05aecb13bd62f6414e5f27207cc7c581fe8f1a6e5b98942937efca1ae3"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "只處理非空閉區間、非空軸對齊盒、閉半空間與明示有限點集反例。"
    },
    {
      "id": "a2",
      "zh-Hant": "有限lambda抽樣不能證凸；有限點集反例失敗也不能反向證明該集合凸。"
    }
  ],
  "symbols": [
    {
      "token": "K",
      "role": "convexity_record",
      "role_zh": "凸性證書或反例",
      "scope": "representative_formula"
    },
    {
      "token": "C",
      "role": "subset_of_real_vector_space",
      "role_zh": "實向量空間子集",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-convex-combination",
      "title_zh": "有限凸組合",
      "reason_zh": "兩點凸組合描述線段。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "凸集合是實向量空間的子集。"
    },
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "線段運算依賴向量加法與純量乘法。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "區間與盒由逐座標上下界證書處理所有lambda。",
      "半空間由線性不等式對凸組合封閉。",
      "若集合含兩端點卻不含其中點，一個midpoint witness即足以否證。"
    ],
    "common_misconception_zh": "檢查幾個lambda就宣稱凸，或找不到一個反例便宣稱凸。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "正例需全稱代數證書；反例失敗不構成正證明。",
    "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": "加入凸函數、凸包與凸最佳化。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-convex-set.md 整理；2026-10-13 脈衝 R07（使用者手動接續 2026-10-24 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"
  }
}