{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-supremum-and-infimum-01",
  "mko_id": "mko-supremum-and-infimum",
  "title": {
    "zh-Hant": "有限最大值與未取到的區間上確界",
    "en": "Finite maxima and unattained interval suprema"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-supremum-and-infimum-01",
    "source": {
      "mko_id": "mko-supremum-and-infimum",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "bb254a6cba1cbd62c80ad2e7af827c64072b05dda798122c533bdf47277f2434",
      "english_sha256": "bf93ac3e45d1cda3acf7f3835fda337bbe68e94e511b2b2c87995468091928e4"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "sets",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "求 A={-2,1/3,5/2} 與 B=(0,1) 的 sup/inf，逐一核界與最小性。",
      "en": "Find sup/inf of A={-2,1/3,5/2} and B=(0,1), checking bounds and extremality."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "A={-2,1/3,5/2} 完整列出；B={x∈R:0<x<1}。",
          "en": "A={-2,1/3,5/2} is complete; B={x in R:0<x<1}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "-2<1/3<5/2",
        "after_tex": "\\forall a\\in A:\\ -2\\le a\\le5/2",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整三元素排序給上下界。",
          "en": "Ordering all three elements supplies both bounds."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "沒有未列出的 A 成員。",
            "en": "There are no unlisted members of A."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "u\\text{ upper bound of }A,\\quad5/2\\in A",
        "after_tex": "u\\ge5/2",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個上界必至少為已取到的最大元素。",
          "en": "Every upper bound must be at least the attained largest element."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "上界與最小性各自核，不能只證一個。",
            "en": "Being an upper bound and being least are separate checks."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "\\ell\\text{ lower bound of }A,\\quad-2\\in A",
        "after_tex": "\\ell\\le-2",
        "relation": "implies",
        "rule": "sets",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個下界不大於已取到的最小元素。",
          "en": "Every lower bound is no greater than the attained smallest element."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "方向是下界≤最小值，不是相反。",
            "en": "The direction is lower bound≤minimum, not the reverse."
          }
        },
        "leaf_basis": "sets"
      },
      {
        "id": "S04",
        "uses": [
          "S01",
          "S02",
          "S03"
        ],
        "before_tex": "5/2\\text{ least upper bound},\\quad-2\\text{ greatest lower bound}",
        "after_tex": "\\sup A=\\max A=5/2,\\quad\\inf A=\\min A=-2",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "此有限集合的確界都在集合中，所以同時是最大最小。",
          "en": "These extrema belong to the finite set and are therefore maximum and minimum as well."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "成員資格已另核，不普遍假定 sup 是成員。",
            "en": "Membership was separately checked, not assumed for arbitrary suprema."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S05",
        "uses": [
          "G1"
        ],
        "before_tex": "x\\in B\\iff0<x<1",
        "after_tex": "0\\text{ is a lower bound and }1\\text{ an upper bound}",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "區間端點提供界，但依定義兩個端點都不在 B。",
          "en": "The endpoints provide bounds but neither belongs to the open interval."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不把 ≤ 的界條件與 < 的成員條件混同。",
            "en": "Bound inequalities and strict membership inequalities are kept distinct."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "u<1",
        "after_tex": "x=(1+\\max\\{u,0\\})/2\\in B,\\quad x>u",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [
          "u<1"
        ],
        "explanation": {
          "zh-Hant": "max 讓任意 u<1（含負 u）都有合法 B 成員超過它，所以 u 不可能是上界。",
          "en": "The maximum handles every u<1, including negative u, producing a member of B above u; thus u is not an upper bound."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "構造同時核 0<x<1 與 x>u，不只靠圖形。",
            "en": "The construction checks 0<x<1 and x>u, rather than a drawing."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S05"
        ],
        "before_tex": "\\ell>0",
        "after_tex": "x=\\min\\{\\ell,1\\}/2\\in B,\\quad x<\\ell",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [
          "\\ell>0"
        ],
        "explanation": {
          "zh-Hant": "任意正候選下界都被一個更小 B 成員否定。",
          "en": "Every positive candidate lower bound is refuted by a smaller member of B."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "ell≥1 與 0<ell<1 兩種情況均被 min 公式涵蓋。",
            "en": "Both ell≥1 and 0<ell<1 are covered by the minimum formula."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S08",
        "uses": [
          "S05",
          "S06",
          "S07"
        ],
        "before_tex": "1\\text{ least upper bound},\\quad0\\text{ greatest lower bound},\\quad0,1\\notin B",
        "after_tex": "\\sup B=1,\\quad\\inf B=0;\\quad B\\text{ has no maximum or minimum}",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "確界存在但未取到；若有最大最小，它們必等於已證唯一確界，與不在集合矛盾。",
          "en": "The bounds are not attained; any maximum/minimum would equal the proved unique supremum/infimum, contradicting nonmembership."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05",
            "S06",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "未將未取到端點誤判為無確界。",
            "en": "The fact that the endpoints are not attained is not mistaken for absence of a supremum or infimum."
          }
        },
        "leaf_basis": "definition_here"
      }
    ],
    "result": {
      "zh-Hant": "A 的 sup=5/2、inf=-2 且都取到；B 的 sup=1、inf=0 但都不取到。",
      "en": "A has attained sup=5/2 and inf=-2; B has unattained sup=1 and inf=0."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "B 的最大元素是 1。",
          "en": "The maximum of B is 1."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "1 不在 B。",
          "en": "1 is not in B."
        }
      },
      {
        "claim": {
          "zh-Hant": "2 是 B 的 supremum，因為它是上界。",
          "en": "2 is sup B because it is an upper bound."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "不是最小上界，1 更小且仍是上界。",
          "en": "It is not least: 1 is a smaller upper bound."
        }
      },
      {
        "claim": {
          "zh-Hant": "空集也依同一普通實數規則有最大元素。",
          "en": "The empty set has a largest element under the same real convention."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "沒有成員可取；本稿明確排除空集合版本。",
          "en": "There is no member to attain, and the empty-set version is excluded."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "s=sup A requires s to be an upper bound and no larger than any other upper bound; inf A is defined dually. Complete finite rational sets give an exact slice, and bounded open intervals demonstrate nonattained endpoints. Empty sets, unbounded sets and extended-real infinity are outside this slice."
        },
        "source": {
          "mko_id": "mko-supremum-and-infimum",
          "sha256": "bb254a6cba1cbd62c80ad2e7af827c64072b05dda798122c533bdf47277f2434",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "sets",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合的聯、交、補集、包含與函數逆像以逐元素定義計算。",
          "en": "Compute unions, intersections, complements, inclusion and preimages element by element."
        }
      },
      {
        "id": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "sup 是最小上界，inf 是最大下界；只有確界是集合成員時才同時為 maximum/minimum。",
          "en": "A supremum is a least upper bound and an infimum a greatest lower bound; it is also a maximum/minimum only when it belongs to the set."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "A-order",
        "lhs": "1/3",
        "rhs": "5/2",
        "step": "S01",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "普通實值、非空有界集合，排除空集、無界與 extended reals。",
      "en": "Ordinary real values for nonempty bounded sets, excluding empty/unbounded/extended-real cases."
    },
    "md_symbols": [
      [
        "\\sup,\\inf",
        "最小上界與最大下界",
        "least upper and greatest lower bound"
      ],
      [
        "\\max,\\min",
        "取到的極值",
        "attained extrema"
      ]
    ],
    "md_error_controls": [
      {
        "title": "兩個責任 / Two obligations",
        "zh": "每個確界都需界條件與極小／極大性。",
        "en": "For each supremum or infimum, verify both that the proposed value is a bound and that it is least or greatest among such bounds."
      },
      {
        "title": "取到另核 / Check attainment separately",
        "zh": "只有確界是成員才稱最大或最小。",
        "en": "Call it a maximum/minimum only when membership is established."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "original_source_sha256": "5943c7e129fd8edd4ae73e3d131e0463a489f64dc39ea5e81897df1c99ccccf8",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite.",
    "english_binding_revision": {
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      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c"
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    "source_quotation_revision": {
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}