{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-supremum-and-infimum",
  "semantic_family_id": "sf-supremum-and-infimum",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "上確界與下確界"
  },
  "summary": {
    "zh-Hant": "非空且有界實數集合的supremum是最小上界，infimum是最大下界，兩者未必屬於集合。"
  },
  "statement": {
    "zh-Hant": "s=sup A需同時證s為upper bound且不大於任何其他upper bound；inf對偶。有限有理集合提供exact slice，開區間提供未取到端點的邊界。空集、無界集合與extended-real infinity不在本輪。"
  },
  "formula": {
    "tex": "s=sup(A)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>s</mi><mo>=</mo><mrow><mi>sup</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "symbol",
        "name": "s",
        "glyph": "s"
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "sup",
          "glyph": "sup"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "A",
            "glyph": "A"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f9d54bb320ff6720e96af6a5431e3319712c99f85552af02751d24ae0084bd67"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "普通實值版本只處理非空且相應有界集合。"
    },
    {
      "id": "a2",
      "zh-Hant": "全部有限值與區間端點以exact rationals表示。"
    }
  ],
  "symbols": [
    {
      "token": "sup",
      "role": "supremum",
      "role_zh": "上確界",
      "scope": "representative_formula"
    },
    {
      "token": "inf",
      "role": "infimum",
      "role_zh": "下確界",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "順序完備carrier。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "上下界。"
    },
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "被界定集合。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "least/greatest全稱責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "upper bound與leastness必須同時成立。",
      "sup/inf不必是member。",
      "只有實際取到時才是maximum/minimum。"
    ],
    "common_misconception_zh": "把supremum一律當集合中的最大元素。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不處理空集、無界集合或extended reals。",
    "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monotone convergence、Bolzano-Weierstrass與IVT。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-supremum-and-infimum.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-07 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}