{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-lean-formal-statement",
  "semantic_family_id": "sf-lean-formal-statement",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "證明助理中的形式陳述"
  },
  "summary": {
    "zh-Hant": "形式陳述是在固定版本、imports、環境與 local context 中成功 elaboration，且型別為 Prop 的核心項。"
  },
  "statement": {
    "zh-Hant": "在固定 Lean 版本、imports、名稱環境與 local context Γ 中，source syntax s 若成功 elaboration 為核心項 S，且 Γ⊢S:Prop，則 s 在該環境中表達一個形式陳述。陳述成立與存在已檢查證明項是分離狀態。"
  },
  "formula": {
    "tex": "S\\in P",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>S</mi><mo>∈</mo><mi>P</mi></mrow></math>",
    "semantic_ast": {
      "type": "membership",
      "element": {
        "type": "symbol",
        "name": "S",
        "glyph": "S"
      },
      "set": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "66d341dc2798f54961fb26faab4dba215ef84e3e740e6e7dec2fec3fd0ce3a27"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "Lean 版本、imports、已宣告名稱與 local context 都是陳述身分的一部分。"
    },
    {
      "id": "a2",
      "zh-Hant": "本輪只記錄明示 pipeline receipt；使用者要求暫停 Lean build，因此不把合成狀態當實際 elaboration 或 kernel check。"
    }
  ],
  "symbols": [
    {
      "token": "S",
      "role": "elaborated_statement",
      "role_zh": "elaboration 後的核心陳述項",
      "scope": "representative_formula"
    },
    {
      "token": "P",
      "role": "proposition_sort",
      "role_zh": "命題型別 Prop 的代表",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-proposition",
      "title_zh": "命題",
      "reason_zh": "形式陳述的核心項必須具有命題型別。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "解析、elaboration、型別檢查、證明項建立與 kernel checking 是不同階段。字串看起來像 Lean，不代表它在指定環境可 elaboration。",
      "同一段 syntax 在 imports、namespace、notation 或 local variables 不同時可能解析或 elaboration 成不同項，也可能失敗；receipt 必須保存環境身分。",
      "一個已 elaboration 的 Prop 只是可被證明的陳述；只有另有對應 proof term 並經 kernel check，才能宣稱該形式陳述已有形式證明。"
    ],
    "common_misconception_zh": "把成功解析、成功 elaboration、型別為 Prop 與已通過 kernel 的證明視為同一件事。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪未執行 Lean build；題目只核對明示的合成 pipeline 記錄與狀態邏輯，不產生或聲稱形式 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": "在使用者恢復 Lean 工作後，固定 toolchain/import closure，為每個 statement 建立 theorem identity、elaboration 與 kernel receipts。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-formal-statement.md 整理；2026-10-13 脈衝 R07（使用者手動接續 2026-10-18 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"
  }
}