第 20549 題：上確界與下確界

原題與已知條件

非空且相應有界的實值集合或明示有界開區間。操作=supinf-lower-bound-greatestness；資料={"values":[[24,25],[-25,26],[49,27]],"candidate":[-25,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 非空且相應有界的實值集合或明示有界開區間。操作=supinf-lower-bound-greatestness；資料={"values":[[24,25],[-25,26],[49,27]],"candidate":[-25,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-supremum-and-infimum

逐步推導

source operation=supinf-lower-bound-greatestness;original snapshot and mathematics retained

explicit ordered source output fields=[candidate, is_lower_bound, infimum, equals_infimum, boundary]

ORIGINAL complete ordered nonempty finite real-value list=[24/25, -25/26, 49/27]

initialize upper/lower extremum candidates at ORIGINAL first value;duplicates retained and do not change set bounds

supinf original value1 minus previous maximum := sub(-25/26, 24/25)

supinf original value1 minus previous maximum = -1249/650

supinf original value1 minus previous minimum := sub(-25/26, 24/25)

supinf original value1 minus previous minimum = -1249/650

ORIGINAL next index=1;value=-25/26;previous extremum values=[24/25, 24/25];differences=[-1249/650, -1249/650]

prefix maximum=24/25 attained at original index0;prefix minimum=-25/26 attained at original index1

supinf original value2 minus previous maximum := sub(49/27, 24/25)

supinf original value2 minus previous maximum = 577/675

supinf original value2 minus previous minimum := sub(49/27, -25/26)

supinf original value2 minus previous minimum = 1949/702

ORIGINAL next index=2;value=49/27;previous extremum values=[24/25, -25/26];differences=[577/675, 1949/702]

prefix maximum=49/27 attained at original index2;prefix minimum=-25/26 attained at original index1

ALL original values processed;actual maximum witness=[2, 49/27];actual minimum witness=[1, -25/26]

EVERY listed value between-25/26 and49/27;ANY upper bound must>=actual maximum in the set;ANY lower bound must<=actual minimum;sup=49/27,inf=-25/26,BOTH attained

supinf candidate bound test original value0 := sub(24/25, -25/26)

supinf candidate bound test original value0 = 1249/650

supinf candidate bound test original value1 := sub(-25/26, -25/26)

supinf candidate bound test original value1 = 0

supinf candidate bound test original value2 := sub(49/27, -25/26)

supinf candidate bound test original value2 = 1949/702

ORIGINAL candidate=-25/26;lower-bound flags for EVERY original value=[是, 是, 是];separate actual extremum=-25/26

candidate is bound=是;candidate equals INF=是;a loose bound and a failed candidate are not the extremum

逐步反向核對

• S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · premise_and_result_audit — 以精確有理數重算 supinf candidate bound test original value2，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 supinf candidate bound test original value1，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 supinf candidate bound test original value0，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 supinf original value2 minus previous minimum，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 supinf original value2 minus previous maximum，不以小數近似替代。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 以精確有理數重算 supinf original value1 minus previous minimum，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 supinf original value1 minus previous maximum，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(-25/26, 是, -25/26, 是, lower_bound_and_greatestness_are_separate)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。