{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-lean-proof-term",
  "semantic_family_id": "sf-lean-proof-term",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "證明項"
  },
  "summary": {
    "zh-Hant": "證明項是在固定形式環境與 context 中具有命題 P 型別，並可由 kernel 檢查的項 t。"
  },
  "statement": {
    "zh-Hant": "在固定 Lean 版本、imports、環境 E 與 local context Γ 中，若核心項 t 具有命題 P 的型別，且 kernel 接受 Γ⊢t:P，則 t 是 P 的證明項。tactic log 只是建構過程，不等於已保存且可重播的 proof-term receipt。"
  },
  "formula": {
    "tex": "T(t)=P",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>T</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>P</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "T",
          "glyph": "T"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "t",
            "glyph": "t"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7a8a4a0f18a7f970288b35e1cdf5ca0d1af7ae16dfb328681dc88a32dcff1096"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "P 已在同一環境中 elaboration 為 Prop；term、context、imports 與 toolchain 均固定。"
    },
    {
      "id": "a2",
      "zh-Hant": "本輪不執行 Lean build；題庫使用合成小型命題型別系統及 receipt 狀態，不產生 Lean formal Evidence。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "type_assignment",
      "role_zh": "型別指派",
      "scope": "representative_formula"
    },
    {
      "token": "t",
      "role": "proof_term_candidate",
      "role_zh": "候選證明項",
      "scope": "representative_formula"
    },
    {
      "token": "P",
      "role": "proposition_type",
      "role_zh": "命題型別",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-lean-formal-statement",
      "title_zh": "證明助理中的形式陳述",
      "reason_zh": "先固定 P 與其環境身分。"
    },
    {
      "id": "mko-proof",
      "title_zh": "證明",
      "reason_zh": "證明項是形式系統中的證明實例。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "propositions-as-types 下，P 的證明是型別 P 的項。假設、函數應用、pair constructor 與 projection 都受 context 型別約束。",
      "tactic 成功且無 goals 通常產生候選 proof term；只有保存該 term 並經指定 kernel／環境檢查，才形成 receipt。",
      "kernel 接受只證形式 judgment；statement 是否忠實表達原始中文、imports 是否含不當公理仍是另外的語義與信任義務。"
    ],
    "common_misconception_zh": "把 tactic transcript、零 goals、成功編譯或模型自報等同 kernel 已接受的 proof term。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪未執行 Lean；合成型別判定不是 Lean kernel receipt，也不升格為 formal 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，為指定 theorem 固定 term、type、environment、imports、axiom inventory 與 kernel receipt。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-proof-term.md 整理；2026-10-13 脈衝 R07（使用者手動接續 2026-10-19 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"
  }
}