# 第 26118 題：多對一歸約

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本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。

R0051-26118 · q-20261015-advance-20261115-r07-many-one-reduction-0068



## 原題與已知條件

明示decision predicates與complete total map的雙向membership equivalence。操作=mreduction-composition-direction；資料={"source_carrier":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],"middle_carrier":[0,1],"target_carrier":[10,11],"source_yes":[1,3,5,7,9,11,13,15,17,19],"middle_yes":[1],"target_yes":[11],"mapping_ab":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"mapping_bc":[10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。


- G001 — 明示decision predicates與complete total map的雙向membership equivalence。操作=mreduction-composition-direction；資料={"source_carrier":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],"middle_carrier":[0,1],"target_carrier":[10,11],"source_yes":[1,3,5,7,9,11,13,15,17,19],"middle_yes":[1],"target_yes":[11],"mapping_ab":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],"mapping_bc":[10,11]}。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-many-one-reduction](https://ocme.evemisslab.com/zh-Hant/mko/mko-many-one-reduction/)





## 逐步推導

S001 · implies操作前

source roles = [mreduction-composition-direction]

操作後

derived = [[first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate]]

條件：無

本頁直接列完整來源欄序，不需JavaScript；contract的6/4/6/6與IVT的8/7/8由原mode選，不補欄也不由答案形狀猜。

步驟識別與依賴規則：source_ordered_fields_without_JS · 依賴：G001





S002 · implies操作前

source roles = [Reduction ORIGINAL A-to-B, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]]

操作後

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [[0, 0, 是], [1, 1, 是], [2, 0, 是], [3, 1, 是], [4, 0, 是], [5, 1, 是], [6, 0, 是], [7, 1, 是], [8, 0, 是], [9, 1, 是], [10, 0, 是], [11, 1, 是], [12, 0, 是], [13, 1, 是], [14, 0, 是], [15, 1, 是], [16, 0, 是], [17, 1, 是], [18, 0, 是], [19, 1, 是]], 是, null]

條件：無

source數值排序要連image一起移動；target也數值排序。null／outside／Boolean不是合法integer target值，False不能當0；qualification保實際rawimage與首invalid row，不先捏造total map。

步驟識別與依賴規則：Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · 依賴：G001, S001





S003 · implies操作前

source roles = [Reduction ORIGINAL A-to-B, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]]

操作後

derived = [[[0, 否, 0, 否, 是], [1, 是, 1, 是, 是], [2, 否, 0, 否, 是], [3, 是, 1, 是, 是], [4, 否, 0, 否, 是], [5, 是, 1, 是, 是], [6, 否, 0, 否, 是], [7, 是, 1, 是, 是], [8, 否, 0, 否, 是], [9, 是, 1, 是, 是], [10, 否, 0, 否, 是], [11, 是, 1, 是, 是], [12, 否, 0, 否, 是], [13, 是, 1, 是, 是], [14, 否, 0, 否, 是], [15, 是, 1, 是, 是], [16, 否, 0, 否, 是], [17, 是, 1, 是, 是], [18, 否, 0, 否, 是], [19, 是, 1, 是, 是]], 是, 是]

條件：無

每個x都驗x∈A iff f(x)∈B，包含yes與no兩方向；invalid image的target predicate是null而非False。totality與membership責任合取才是此有限reduction。

步驟識別與依賴規則：Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · 依賴：G001, S002





S004 · implies操作前

source roles = [Reduction ORIGINAL B-to-C, [0, 1], [10, 11], [10, 11]]

操作後

derived = [[0, 1], [10, 11], [10, 11], [[0, 10, 是], [1, 11, 是]], 是, null]

條件：無

source數值排序要連image一起移動；target也數值排序。null／outside／Boolean不是合法integer target值，False不能當0；qualification保實際rawimage與首invalid row，不先捏造total map。

步驟識別與依賴規則：Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · 依賴：G001, S003





S005 · implies操作前

source roles = [Reduction ORIGINAL B-to-C, [1], [11], [0, 1], [10, 11], [10, 11]]

操作後

derived = [[[0, 否, 10, 否, 是], [1, 是, 11, 是, 是]], 是, 是]

條件：無

每個x都驗x∈A iff f(x)∈B，包含yes與no兩方向；invalid image的target predicate是null而非False。totality與membership責任合取才是此有限reduction。

步驟識別與依賴規則：Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · 依賴：G001, S004





S006 · implies操作前

source roles = [Reduction ACTUAL B-after-A composition, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 11], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11]]

操作後

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 11], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, 10, 是], [1, 11, 是], [2, 10, 是], [3, 11, 是], [4, 10, 是], [5, 11, 是], [6, 10, 是], [7, 11, 是], [8, 10, 是], [9, 11, 是], [10, 10, 是], [11, 11, 是], [12, 10, 是], [13, 11, 是], [14, 10, 是], [15, 11, 是], [16, 10, 是], [17, 11, 是], [18, 10, 是], [19, 11, 是]], 是, null]

條件：無

source數值排序要連image一起移動；target也數值排序。null／outside／Boolean不是合法integer target值，False不能當0；qualification保實際rawimage與首invalid row，不先捏造total map。

步驟識別與依賴規則：Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · 依賴：G001, S005





S007 · implies操作前

source roles = [Reduction ACTUAL B-after-A composition, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 11], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11]]

操作後

derived = [[[0, 否, 10, 否, 是], [1, 是, 11, 是, 是], [2, 否, 10, 否, 是], [3, 是, 11, 是, 是], [4, 否, 10, 否, 是], [5, 是, 11, 是, 是], [6, 否, 10, 否, 是], [7, 是, 11, 是, 是], [8, 否, 10, 否, 是], [9, 是, 11, 是, 是], [10, 否, 10, 否, 是], [11, 是, 11, 是, 是], [12, 否, 10, 否, 是], [13, 是, 11, 是, 是], [14, 否, 10, 否, 是], [15, 是, 11, 是, 是], [16, 否, 10, 否, 是], [17, 是, 11, 是, 是], [18, 否, 10, 否, 是], [19, 是, 11, 是, 是]], 是, 是]

條件：無

每個x都驗x∈A iff f(x)∈B，包含yes與no兩方向；invalid image的target predicate是null而非False。totality與membership責任合取才是此有限reduction。

步驟識別與依賴規則：Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · 依賴：G001, S006





S008 · implies操作前

source roles = [[[0, 否, 0, 否, 是], [1, 是, 1, 是, 是], [2, 否, 0, 否, 是], [3, 是, 1, 是, 是], [4, 否, 0, 否, 是], [5, 是, 1, 是, 是], [6, 否, 0, 否, 是], [7, 是, 1, 是, 是], [8, 否, 0, 否, 是], [9, 是, 1, 是, 是], [10, 否, 0, 否, 是], [11, 是, 1, 是, 是], [12, 否, 0, 否, 是], [13, 是, 1, 是, 是], [14, 否, 0, 否, 是], [15, 是, 1, 是, 是], [16, 否, 0, 否, 是], [17, 是, 1, 是, 是], [18, 否, 0, 否, 是], [19, 是, 1, 是, 是]], [[0, 否, 10, 否, 是], [1, 是, 11, 是, 是]]]

操作後

derived = [[10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, 否, 10, 否, 是], [1, 是, 11, 是, 是], [2, 否, 10, 否, 是], [3, 是, 11, 是, 是], [4, 否, 10, 否, 是], [5, 是, 11, 是, 是], [6, 否, 10, 否, 是], [7, 是, 11, 是, 是], [8, 否, 10, 否, 是], [9, 是, 11, 是, 是], [10, 否, 10, 否, 是], [11, 是, 11, 是, 是], [12, 否, 10, 否, 是], [13, 是, 11, 是, 是], [14, 否, 10, 否, 是], [15, 是, 11, 是, 是], [16, 否, 10, 否, 是], [17, 是, 11, 是, 是], [18, 否, 10, 否, 是], [19, 是, 11, 是, 是]], 是]

條件：無

任意x先f_AB再f_BC：x∈A iff f_AB(x)∈B iff f_BC(f_AB(x))∈C；total保每次可用。方向是A≤mB≤mC到A≤mC，不導反向或Turing reduction／無限可計算性。

步驟識別與依賴規則：Reduction_GENERAL_composition_direction_second_after_first · 依賴：G001, S007









## 逐步反向核對


- 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S006 · 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。
- 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
- S003 · 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。
- 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

([[0, 否, 0, 否, 是], [1, 是, 1, 是, 是], [2, 否, 0, 否, 是], [3, 是, 1, 是, 是], [4, 否, 0, 否, 是], [5, 是, 1, 是, 是], [6, 否, 0, 否, 是], [7, 是, 1, 是, 是], [8, 否, 0, 否, 是], [9, 是, 1, 是, 是], [10, 否, 0, 否, 是], [11, 是, 1, 是, 是], [12, 否, 0, 否, 是], [13, 是, 1, 是, 是], [14, 否, 0, 否, 是], [15, 是, 1, 是, 是], [16, 否, 0, 否, 是], [17, 是, 1, 是, 是], [18, 否, 0, 否, 是], [19, 是, 1, 是, 是]], [[0, 否, 10, 否, 是], [1, 是, 11, 是, 是]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, 否, 10, 否, 是], [1, 是, 11, 是, 是], [2, 否, 10, 否, 是], [3, 是, 11, 是, 是], [4, 否, 10, 否, 是], [5, 是, 11, 是, 是], [6, 否, 10, 否, 是], [7, 是, 11, 是, 是], [8, 否, 10, 否, 是], [9, 是, 11, 是, 是], [10, 否, 10, 否, 是], [11, 是, 11, 是, 是], [12, 否, 10, 否, 是], [13, 是, 11, 是, 是], [14, 否, 10, 否, 是], [15, 是, 11, 是, 是], [16, 否, 10, 否, 是], [17, 是, 11, 是, 是], [18, 否, 10, 否, 是], [19, 是, 11, 是, 是]], 是, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

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。

來源雜湊與覆核憑證
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: d7f82e3d40b6d531a2e66cb0c54090c1178aff766c90cd6deb06858c9166b123
- candidate record SHA-256: 3be105bb059f3412d5e13f1bdf201d48bf7d703e3ba57e0501acea70bcad8b62
- semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
- question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r07-many-one-reduction-0068.json)