{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-proof",
  "semantic_family_id": "sf-proof",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "數學證明",
    "en": "Mathematical proof"
  },
  "summary": {
    "zh-Hant": "數學證明是在明示的前提與推理規則下，建立指定結論的可檢查論證。"
  },
  "statement": {
    "zh-Hant": "給定待證命題 P，證明必須在指定語境中由已接受的前提、定義、公理或先前結果，經合法推理得到 P。"
  },
  "formula": {
    "tex": "P",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math>",
    "semantic_ast": {
      "type": "symbol",
      "name": "P",
      "glyph": "P"
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5c62e091b8c0565f1bafad0dad5934276143ae2ccef7a5381e8ada5b1a8d26d2"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "已明示待證命題、可使用的前提與推理語境。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "proof_goal",
      "role_zh": "待證命題",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-logical-connectives",
      "title_zh": "邏輯連接詞",
      "reason_zh": "證明必須能辨識待證命題及其邏輯結構。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "證明的任務不是只累積支持例子，而是說明結論為何由明示條件必然成立。單一有效反例可以否定全稱命題；有限正例若沒有證明論域有限、枚舉完整且每個案例成立，便不能單獨建立一般命題。",
      "直接證明從前提向結論推進；反證法則加入結論的否定並導出矛盾。採用哪種方法仍須尊重所處邏輯系統可用的規則。",
      "形式證明把陳述與每個推理步驟交給形式系統檢查；非形式證明也必須讓讀者能追蹤前提、推理與結論。"
    ],
    "common_misconception_zh": "常見誤解是把數值抽樣、圖形外觀、權威引用或未交代規則的直覺當成一般證明。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限測試或例子若缺少有限論域與完整枚舉的證明，只能支援指定實例；即使完成窮舉，也必須檢查完備性論證與定理聲明。自動搜尋若產生一般證明，則必須另行檢查定理聲明、前提、推理規則與證明物件。",
    "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": "選定具體定理陳述，綁定其前提、結論、Lean proof term 與可重現檢查 receipt。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-proof.md 整理；待人工審查。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-09-21"
  }
}