{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-order-relation",
  "semantic_family_id": "sf-order-relation",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實數大小與順序"
  },
  "summary": {
    "zh-Hant": "在實數論域比較大小；精確有理數交叉乘積、有限極值及含端點區間是本輪切片。"
  },
  "statement": {
    "zh-Hant": "實數的通常順序 ≤ 滿足自反、反對稱、傳遞及任兩數可比較；a<b 表示 a≤b 且 a≠b。任意實數 a、b、c 都有 a<b 當且僅當 a+c<b+c。非空有限實數集合有最小與最大元素；閉區間 [l,u]（l≤u）包含兩端點。"
  },
  "formula": {
    "tex": "a\\in R",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>a</mi><mo>∈</mo><mi>R</mi></mrow></math>",
    "semantic_ast": {
      "type": "membership",
      "element": {
        "type": "symbol",
        "name": "a",
        "glyph": "a"
      },
      "set": {
        "type": "symbol",
        "name": "R",
        "glyph": "R"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "329fc5b2bedb8992e257274dd98f6d5252000f725fa281586f76de38c63cf34f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "只談實數的通常全序，不把任意關係或複數宣告為同一種順序。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫採安全整數或正分母有理數；比較 p/q 與 r/s 時 q、s>0，可比較 ps 與 rq。"
    },
    {
      "id": "a3",
      "zh-Hant": "有限極值集合非空；閉區間先驗下界不大於上界。代表公式只標示實數論域。"
    }
  ],
  "symbols": [
    {
      "token": "a",
      "role": "real_value",
      "role_zh": "待比較的實數",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "real_domain",
      "role_zh": "實數論域",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "順序論域明示為實數。"
    },
    {
      "id": "mko-rational-number",
      "title_zh": "有理數",
      "reason_zh": "精確分數表示及正分母規約。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "實數的三分比較輸出 <、= 或 >。乘正分母保留順序；如果任意乘負數，必須反向，不能把本輪的正分母前提刪掉。",
      "有限非空集合按值決定極值，不按輸入列出的先後。空集合在此不指定最小或最大元素；單元素集合的兩個極值相同。",
      "閉區間會員條件是 l≤x≤u；l=u 時只有一個值。兩數同加 c 不改變比較，因此平移參數不被拿來充作新的實質題型。"
    ],
    "common_misconception_zh": "忽略分母正負、漏掉閉區間端點，或對空集合給任意極值。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "精確有限實例可重算；通常實數全序及平移等價仍未新增形式 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": "依專案固定版 Mathlib 綁定實數線性順序、加法保序與非空有限極值。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-order-relation.md 整理；2026-10-05 脈衝 R01（使用者手動接續 2026-10-06 工作序列；實際日期仍為 2026-10-05），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-05"
  }
}