# 第 25365 題：有限商群 [返回所在分頁 127](https://ocme.evemisslab.com/zh-Hant/questions/page/127/) [← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261015-advance-20261114-r02-finite-quotient-group-foundations/#q-20261015-advance-20261114-r02-finite-quotient-group-0015) 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。 R0049-25365 · q-20261015-advance-20261114-r02-finite-quotient-group-0015 ## 原題與已知條件 完整有限群、subgroup與全部coset partition及代表元責任。操作=quotient-normal-coset-partition；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],"identity":0,"subset":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。 - G001 — 完整有限群、subgroup與全部coset partition及代表元責任。操作=quotient-normal-coset-partition；資料={"group_table":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]],"identity":0,"subset":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30]}。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-finite-quotient-group](https://ocme.evemisslab.com/zh-Hant/mko/mko-finite-quotient-group/) ## 逐步推導 S001 · implies操作前 source roles = [quotient-normal-coset-partition] 操作後 derived = [[subgroup, normal, cosets, partition_certificate, coset_count, certificate]] 條件：無 本頁列完整來源欄序，不需JavaScript；LP三種mode雖同為6欄，角色仍由原mode選，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 = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]], 0] 操作後 derived = [32, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19], [20, 20, 20, 20], [21, 21, 21, 21], [22, 22, 22, 22], [23, 23, 23, 23], [24, 24, 24, 24], [25, 25, 25, 25], [26, 26, 26, 26], [27, 27, 27, 27], [28, 28, 28, 28], [29, 29, 29, 29], [30, 30, 30, 30], [31, 31, 31, 31]], 是] 條件：無 先核整個closed Cayley table與實際identity標籤，不假設0或套舊上限32。 步驟識別與依賴規則：Quotient_complete_group_table_actual_identity · 依賴：G001, S001 S003 · implies操作前 source roles = [32, 0] 操作後 derived = [[[是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 32768, [0, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]] 條件：無 每個(x,y)矩陣格核全部z的(xy)z=x(yz)，共n³責任；每個inverse同時左右乘回同一identity。完整查核不靠抽樣。 步驟識別與依賴規則：Quotient_ALL_group_associativity_triples_and_two_sided_inverses · 依賴：G001, S002 S004 · implies操作前 source roles = [0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]] 操作後 derived = [[[0, 0, 0, 是], [0, 2, 2, 是], [0, 4, 4, 是], [0, 6, 6, 是], [0, 8, 8, 是], [0, 10, 10, 是], [0, 12, 12, 是], [0, 14, 14, 是], [0, 16, 16, 是], [0, 18, 18, 是], [0, 20, 20, 是], [0, 22, 22, 是], [0, 24, 24, 是], [0, 26, 26, 是], [0, 28, 28, 是], [0, 30, 30, 是], [2, 0, 2, 是], [2, 2, 4, 是], [2, 4, 6, 是], [2, 6, 8, 是], [2, 8, 10, 是], [2, 10, 12, 是], [2, 12, 14, 是], [2, 14, 16, 是], [2, 16, 18, 是], [2, 18, 20, 是], [2, 20, 22, 是], [2, 22, 24, 是], [2, 24, 26, 是], [2, 26, 28, 是], [2, 28, 30, 是], [2, 30, 0, 是], [4, 0, 4, 是], [4, 2, 6, 是], [4, 4, 8, 是], [4, 6, 10, 是], [4, 8, 12, 是], [4, 10, 14, 是], [4, 12, 16, 是], [4, 14, 18, 是], [4, 16, 20, 是], [4, 18, 22, 是], [4, 20, 24, 是], [4, 22, 26, 是], [4, 24, 28, 是], [4, 26, 30, 是], [4, 28, 0, 是], [4, 30, 2, 是], [6, 0, 6, 是], [6, 2, 8, 是], [6, 4, 10, 是], [6, 6, 12, 是], [6, 8, 14, 是], [6, 10, 16, 是], [6, 12, 18, 是], [6, 14, 20, 是], [6, 16, 22, 是], [6, 18, 24, 是], [6, 20, 26, 是], [6, 22, 28, 是], [6, 24, 30, 是], [6, 26, 0, 是], [6, 28, 2, 是], [6, 30, 4, 是], [8, 0, 8, 是], [8, 2, 10, 是], [8, 4, 12, 是], [8, 6, 14, 是], [8, 8, 16, 是], [8, 10, 18, 是], [8, 12, 20, 是], [8, 14, 22, 是], [8, 16, 24, 是], [8, 18, 26, 是], [8, 20, 28, 是], [8, 22, 30, 是], [8, 24, 0, 是], [8, 26, 2, 是], [8, 28, 4, 是], [8, 30, 6, 是], [10, 0, 10, 是], [10, 2, 12, 是], [10, 4, 14, 是], [10, 6, 16, 是], [10, 8, 18, 是], [10, 10, 20, 是], [10, 12, 22, 是], [10, 14, 24, 是], [10, 16, 26, 是], [10, 18, 28, 是], [10, 20, 30, 是], [10, 22, 0, 是], [10, 24, 2, 是], [10, 26, 4, 是], [10, 28, 6, 是], [10, 30, 8, 是], [12, 0, 12, 是], [12, 2, 14, 是], [12, 4, 16, 是], [12, 6, 18, 是], [12, 8, 20, 是], [12, 10, 22, 是], [12, 12, 24, 是], [12, 14, 26, 是], [12, 16, 28, 是], [12, 18, 30, 是], [12, 20, 0, 是], [12, 22, 2, 是], [12, 24, 4, 是], [12, 26, 6, 是], [12, 28, 8, 是], [12, 30, 10, 是], [14, 0, 14, 是], [14, 2, 16, 是], [14, 4, 18, 是], [14, 6, 20, 是], [14, 8, 22, 是], [14, 10, 24, 是], [14, 12, 26, 是], [14, 14, 28, 是], [14, 16, 30, 是], [14, 18, 0, 是], [14, 20, 2, 是], [14, 22, 4, 是], [14, 24, 6, 是], [14, 26, 8, 是], [14, 28, 10, 是], [14, 30, 12, 是], [16, 0, 16, 是], [16, 2, 18, 是], [16, 4, 20, 是], [16, 6, 22, 是], [16, 8, 24, 是], [16, 10, 26, 是], [16, 12, 28, 是], [16, 14, 30, 是], [16, 16, 0, 是], [16, 18, 2, 是], [16, 20, 4, 是], [16, 22, 6, 是], [16, 24, 8, 是], [16, 26, 10, 是], [16, 28, 12, 是], [16, 30, 14, 是], [18, 0, 18, 是], [18, 2, 20, 是], [18, 4, 22, 是], [18, 6, 24, 是], [18, 8, 26, 是], [18, 10, 28, 是], [18, 12, 30, 是], [18, 14, 0, 是], [18, 16, 2, 是], [18, 18, 4, 是], [18, 20, 6, 是], [18, 22, 8, 是], [18, 24, 10, 是], [18, 26, 12, 是], [18, 28, 14, 是], [18, 30, 16, 是], [20, 0, 20, 是], [20, 2, 22, 是], [20, 4, 24, 是], [20, 6, 26, 是], [20, 8, 28, 是], [20, 10, 30, 是], [20, 12, 0, 是], [20, 14, 2, 是], [20, 16, 4, 是], [20, 18, 6, 是], [20, 20, 8, 是], [20, 22, 10, 是], [20, 24, 12, 是], [20, 26, 14, 是], [20, 28, 16, 是], [20, 30, 18, 是], [22, 0, 22, 是], [22, 2, 24, 是], [22, 4, 26, 是], [22, 6, 28, 是], [22, 8, 30, 是], [22, 10, 0, 是], [22, 12, 2, 是], [22, 14, 4, 是], [22, 16, 6, 是], [22, 18, 8, 是], [22, 20, 10, 是], [22, 22, 12, 是], [22, 24, 14, 是], [22, 26, 16, 是], [22, 28, 18, 是], [22, 30, 20, 是], [24, 0, 24, 是], [24, 2, 26, 是], [24, 4, 28, 是], [24, 6, 30, 是], [24, 8, 0, 是], [24, 10, 2, 是], [24, 12, 4, 是], [24, 14, 6, 是], [24, 16, 8, 是], [24, 18, 10, 是], [24, 20, 12, 是], [24, 22, 14, 是], [24, 24, 16, 是], [24, 26, 18, 是], [24, 28, 20, 是], [24, 30, 22, 是], [26, 0, 26, 是], [26, 2, 28, 是], [26, 4, 30, 是], [26, 6, 0, 是], [26, 8, 2, 是], [26, 10, 4, 是], [26, 12, 6, 是], [26, 14, 8, 是], [26, 16, 10, 是], [26, 18, 12, 是], [26, 20, 14, 是], [26, 22, 16, 是], [26, 24, 18, 是], [26, 26, 20, 是], [26, 28, 22, 是], [26, 30, 24, 是], [28, 0, 28, 是], [28, 2, 30, 是], [28, 4, 0, 是], [28, 6, 2, 是], [28, 8, 4, 是], [28, 10, 6, 是], [28, 12, 8, 是], [28, 14, 10, 是], [28, 16, 12, 是], [28, 18, 14, 是], [28, 20, 16, 是], [28, 22, 18, 是], [28, 24, 20, 是], [28, 26, 22, 是], [28, 28, 24, 是], [28, 30, 26, 是], [30, 0, 30, 是], [30, 2, 0, 是], [30, 4, 2, 是], [30, 6, 4, 是], [30, 8, 6, 是], [30, 10, 8, 是], [30, 12, 10, 是], [30, 14, 12, 是], [30, 16, 14, 是], [30, 18, 16, 是], [30, 20, 18, 是], [30, 22, 20, 是], [30, 24, 22, 是], [30, 26, 24, 是], [30, 28, 26, 是], [30, 30, 28, 是]], [[0, 0, 是], [2, 30, 是], [4, 28, 是], [6, 26, 是], [8, 24, 是], [10, 22, 是], [12, 20, 是], [14, 18, 是], [16, 16, 是], [18, 14, 是], [20, 12, 是], [22, 10, 是], [24, 8, 是], [26, 6, 是], [28, 4, 是], [30, 2, 是]], 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]]], 是] 條件：無 先含identity、全product與inverse closure成真subgroup；再對所有g核共軛H。positive branches須normal，boundary須真正nonnormal，不混同未成subgroup。 步驟識別與依賴規則：Quotient_genuine_subgroup_before_actual_normality · 依賴：G001, S003 S005 · implies操作前 source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]] 操作後 derived = [[[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31]], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], 是] 條件：無 每個元素g在gH中；若aH交bH，ah1=bh2給b^-1a∈H，從而兩完整coset相同。列全g後去重且依來源排序，元素是coset SETS，不是最小代表元本身。 步驟識別與依賴規則：Quotient_ALL_LEFT_coset_SETS_complete_disjoint_partition · 依賴：G001, S004 S006 · implies操作前 source roles = [0, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]] 操作後 derived = [[[0, 0, 0, 0], [0, 2, 2, 0], [0, 4, 4, 0], [0, 6, 6, 0], [0, 8, 8, 0], [0, 10, 10, 0], [0, 12, 12, 0], [0, 14, 14, 0], [0, 16, 16, 0], [0, 18, 18, 0], [0, 20, 20, 0], [0, 22, 22, 0], [0, 24, 24, 0], [0, 26, 26, 0], [0, 28, 28, 0], [0, 30, 30, 0], [2, 0, 2, 0], [2, 2, 4, 0], [2, 4, 6, 0], [2, 6, 8, 0], [2, 8, 10, 0], [2, 10, 12, 0], [2, 12, 14, 0], [2, 14, 16, 0], [2, 16, 18, 0], [2, 18, 20, 0], [2, 20, 22, 0], [2, 22, 24, 0], [2, 24, 26, 0], [2, 26, 28, 0], [2, 28, 30, 0], [2, 30, 0, 0], [4, 0, 4, 0], [4, 2, 6, 0], [4, 4, 8, 0], [4, 6, 10, 0], [4, 8, 12, 0], [4, 10, 14, 0], [4, 12, 16, 0], [4, 14, 18, 0], [4, 16, 20, 0], [4, 18, 22, 0], [4, 20, 24, 0], [4, 22, 26, 0], [4, 24, 28, 0], [4, 26, 30, 0], [4, 28, 0, 0], [4, 30, 2, 0], [6, 0, 6, 0], [6, 2, 8, 0], [6, 4, 10, 0], [6, 6, 12, 0], [6, 8, 14, 0], [6, 10, 16, 0], [6, 12, 18, 0], [6, 14, 20, 0], [6, 16, 22, 0], [6, 18, 24, 0], [6, 20, 26, 0], [6, 22, 28, 0], [6, 24, 30, 0], [6, 26, 0, 0], [6, 28, 2, 0], [6, 30, 4, 0], [8, 0, 8, 0], [8, 2, 10, 0], [8, 4, 12, 0], [8, 6, 14, 0], [8, 8, 16, 0], [8, 10, 18, 0], [8, 12, 20, 0], [8, 14, 22, 0], [8, 16, 24, 0], [8, 18, 26, 0], [8, 20, 28, 0], [8, 22, 30, 0], [8, 24, 0, 0], [8, 26, 2, 0], [8, 28, 4, 0], [8, 30, 6, 0], [10, 0, 10, 0], [10, 2, 12, 0], [10, 4, 14, 0], [10, 6, 16, 0], [10, 8, 18, 0], [10, 10, 20, 0], [10, 12, 22, 0], [10, 14, 24, 0], [10, 16, 26, 0], [10, 18, 28, 0], [10, 20, 30, 0], [10, 22, 0, 0], [10, 24, 2, 0], [10, 26, 4, 0], [10, 28, 6, 0], [10, 30, 8, 0], [12, 0, 12, 0], [12, 2, 14, 0], [12, 4, 16, 0], [12, 6, 18, 0], [12, 8, 20, 0], [12, 10, 22, 0], [12, 12, 24, 0], [12, 14, 26, 0], [12, 16, 28, 0], [12, 18, 30, 0], [12, 20, 0, 0], [12, 22, 2, 0], [12, 24, 4, 0], [12, 26, 6, 0], [12, 28, 8, 0], [12, 30, 10, 0], [14, 0, 14, 0], [14, 2, 16, 0], [14, 4, 18, 0], [14, 6, 20, 0], [14, 8, 22, 0], [14, 10, 24, 0], [14, 12, 26, 0], [14, 14, 28, 0], [14, 16, 30, 0], [14, 18, 0, 0], [14, 20, 2, 0], [14, 22, 4, 0], [14, 24, 6, 0], [14, 26, 8, 0], [14, 28, 10, 0], [14, 30, 12, 0], [16, 0, 16, 0], [16, 2, 18, 0], [16, 4, 20, 0], [16, 6, 22, 0], [16, 8, 24, 0], [16, 10, 26, 0], [16, 12, 28, 0], [16, 14, 30, 0], [16, 16, 0, 0], [16, 18, 2, 0], [16, 20, 4, 0], [16, 22, 6, 0], [16, 24, 8, 0], [16, 26, 10, 0], [16, 28, 12, 0], [16, 30, 14, 0], [18, 0, 18, 0], [18, 2, 20, 0], [18, 4, 22, 0], [18, 6, 24, 0], [18, 8, 26, 0], [18, 10, 28, 0], [18, 12, 30, 0], [18, 14, 0, 0], [18, 16, 2, 0], [18, 18, 4, 0], [18, 20, 6, 0], [18, 22, 8, 0], [18, 24, 10, 0], [18, 26, 12, 0], [18, 28, 14, 0], [18, 30, 16, 0], [20, 0, 20, 0], [20, 2, 22, 0], [20, 4, 24, 0], [20, 6, 26, 0], [20, 8, 28, 0], [20, 10, 30, 0], [20, 12, 0, 0], [20, 14, 2, 0], [20, 16, 4, 0], [20, 18, 6, 0], [20, 20, 8, 0], [20, 22, 10, 0], [20, 24, 12, 0], [20, 26, 14, 0], [20, 28, 16, 0], [20, 30, 18, 0], [22, 0, 22, 0], [22, 2, 24, 0], [22, 4, 26, 0], [22, 6, 28, 0], [22, 8, 30, 0], [22, 10, 0, 0], [22, 12, 2, 0], [22, 14, 4, 0], [22, 16, 6, 0], [22, 18, 8, 0], [22, 20, 10, 0], [22, 22, 12, 0], [22, 24, 14, 0], [22, 26, 16, 0], [22, 28, 18, 0], [22, 30, 20, 0], [24, 0, 24, 0], [24, 2, 26, 0], [24, 4, 28, 0], [24, 6, 30, 0], [24, 8, 0, 0], [24, 10, 2, 0], [24, 12, 4, 0], [24, 14, 6, 0], [24, 16, 8, 0], [24, 18, 10, 0], [24, 20, 12, 0], [24, 22, 14, 0], [24, 24, 16, 0], [24, 26, 18, 0], [24, 28, 20, 0], [24, 30, 22, 0], [26, 0, 26, 0], [26, 2, 28, 0], [26, 4, 30, 0], [26, 6, 0, 0], [26, 8, 2, 0], [26, 10, 4, 0], [26, 12, 6, 0], [26, 14, 8, 0], [26, 16, 10, 0], [26, 18, 12, 0], [26, 20, 14, 0], [26, 22, 16, 0], [26, 24, 18, 0], [26, 26, 20, 0], [26, 28, 22, 0], [26, 30, 24, 0], [28, 0, 28, 0], [28, 2, 30, 0], [28, 4, 0, 0], [28, 6, 2, 0], [28, 8, 4, 0], [28, 10, 6, 0], [28, 12, 8, 0], [28, 14, 10, 0], [28, 16, 12, 0], [28, 18, 14, 0], [28, 20, 16, 0], [28, 22, 18, 0], [28, 24, 20, 0], [28, 26, 22, 0], [28, 28, 24, 0], [28, 30, 26, 0], [30, 0, 30, 0], [30, 2, 0, 0], [30, 4, 2, 0], [30, 6, 4, 0], [30, 8, 6, 0], [30, 10, 8, 0], [30, 12, 10, 0], [30, 14, 12, 0], [30, 16, 14, 0], [30, 18, 16, 0], [30, 20, 18, 0], [30, 22, 20, 0], [30, 24, 22, 0], [30, 26, 24, 0], [30, 28, 26, 0], [30, 30, 28, 0]], [0], 是] 條件：無 每組coset的每對代表元都要落在同一product coset，僅用首代表元造表還不足以證良定；保完整可能index集合。 步驟識別與依賴規則：Quotient_EVERY_coset_pair_ALL_representative_products · 依賴：G001, S005 S007 · implies操作前 source roles = [0, 1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31]] 操作後 derived = [[[0, 1, 1, 1], [0, 3, 3, 1], [0, 5, 5, 1], [0, 7, 7, 1], [0, 9, 9, 1], [0, 11, 11, 1], [0, 13, 13, 1], [0, 15, 15, 1], [0, 17, 17, 1], [0, 19, 19, 1], [0, 21, 21, 1], [0, 23, 23, 1], [0, 25, 25, 1], [0, 27, 27, 1], [0, 29, 29, 1], [0, 31, 31, 1], [2, 1, 3, 1], [2, 3, 5, 1], [2, 5, 7, 1], [2, 7, 9, 1], [2, 9, 11, 1], [2, 11, 13, 1], [2, 13, 15, 1], [2, 15, 17, 1], [2, 17, 19, 1], [2, 19, 21, 1], [2, 21, 23, 1], [2, 23, 25, 1], [2, 25, 27, 1], [2, 27, 29, 1], [2, 29, 31, 1], [2, 31, 1, 1], [4, 1, 5, 1], [4, 3, 7, 1], [4, 5, 9, 1], [4, 7, 11, 1], [4, 9, 13, 1], [4, 11, 15, 1], [4, 13, 17, 1], [4, 15, 19, 1], [4, 17, 21, 1], [4, 19, 23, 1], [4, 21, 25, 1], [4, 23, 27, 1], [4, 25, 29, 1], [4, 27, 31, 1], [4, 29, 1, 1], [4, 31, 3, 1], [6, 1, 7, 1], [6, 3, 9, 1], [6, 5, 11, 1], [6, 7, 13, 1], [6, 9, 15, 1], [6, 11, 17, 1], [6, 13, 19, 1], [6, 15, 21, 1], [6, 17, 23, 1], [6, 19, 25, 1], [6, 21, 27, 1], [6, 23, 29, 1], [6, 25, 31, 1], [6, 27, 1, 1], [6, 29, 3, 1], [6, 31, 5, 1], [8, 1, 9, 1], [8, 3, 11, 1], [8, 5, 13, 1], [8, 7, 15, 1], [8, 9, 17, 1], [8, 11, 19, 1], [8, 13, 21, 1], [8, 15, 23, 1], [8, 17, 25, 1], [8, 19, 27, 1], [8, 21, 29, 1], [8, 23, 31, 1], [8, 25, 1, 1], [8, 27, 3, 1], [8, 29, 5, 1], [8, 31, 7, 1], [10, 1, 11, 1], [10, 3, 13, 1], [10, 5, 15, 1], [10, 7, 17, 1], [10, 9, 19, 1], [10, 11, 21, 1], [10, 13, 23, 1], [10, 15, 25, 1], [10, 17, 27, 1], [10, 19, 29, 1], [10, 21, 31, 1], [10, 23, 1, 1], [10, 25, 3, 1], [10, 27, 5, 1], [10, 29, 7, 1], [10, 31, 9, 1], [12, 1, 13, 1], [12, 3, 15, 1], [12, 5, 17, 1], [12, 7, 19, 1], [12, 9, 21, 1], [12, 11, 23, 1], [12, 13, 25, 1], [12, 15, 27, 1], [12, 17, 29, 1], [12, 19, 31, 1], [12, 21, 1, 1], [12, 23, 3, 1], [12, 25, 5, 1], [12, 27, 7, 1], [12, 29, 9, 1], [12, 31, 11, 1], [14, 1, 15, 1], [14, 3, 17, 1], [14, 5, 19, 1], [14, 7, 21, 1], [14, 9, 23, 1], [14, 11, 25, 1], [14, 13, 27, 1], [14, 15, 29, 1], [14, 17, 31, 1], [14, 19, 1, 1], [14, 21, 3, 1], [14, 23, 5, 1], [14, 25, 7, 1], [14, 27, 9, 1], [14, 29, 11, 1], [14, 31, 13, 1], [16, 1, 17, 1], [16, 3, 19, 1], [16, 5, 21, 1], [16, 7, 23, 1], [16, 9, 25, 1], [16, 11, 27, 1], [16, 13, 29, 1], [16, 15, 31, 1], [16, 17, 1, 1], [16, 19, 3, 1], [16, 21, 5, 1], [16, 23, 7, 1], [16, 25, 9, 1], [16, 27, 11, 1], [16, 29, 13, 1], [16, 31, 15, 1], [18, 1, 19, 1], [18, 3, 21, 1], [18, 5, 23, 1], [18, 7, 25, 1], [18, 9, 27, 1], [18, 11, 29, 1], [18, 13, 31, 1], [18, 15, 1, 1], [18, 17, 3, 1], [18, 19, 5, 1], [18, 21, 7, 1], [18, 23, 9, 1], [18, 25, 11, 1], [18, 27, 13, 1], [18, 29, 15, 1], [18, 31, 17, 1], [20, 1, 21, 1], [20, 3, 23, 1], [20, 5, 25, 1], [20, 7, 27, 1], [20, 9, 29, 1], [20, 11, 31, 1], [20, 13, 1, 1], [20, 15, 3, 1], [20, 17, 5, 1], [20, 19, 7, 1], [20, 21, 9, 1], [20, 23, 11, 1], [20, 25, 13, 1], [20, 27, 15, 1], [20, 29, 17, 1], [20, 31, 19, 1], [22, 1, 23, 1], [22, 3, 25, 1], [22, 5, 27, 1], [22, 7, 29, 1], [22, 9, 31, 1], [22, 11, 1, 1], [22, 13, 3, 1], [22, 15, 5, 1], [22, 17, 7, 1], [22, 19, 9, 1], [22, 21, 11, 1], [22, 23, 13, 1], [22, 25, 15, 1], [22, 27, 17, 1], [22, 29, 19, 1], [22, 31, 21, 1], [24, 1, 25, 1], [24, 3, 27, 1], [24, 5, 29, 1], [24, 7, 31, 1], [24, 9, 1, 1], [24, 11, 3, 1], [24, 13, 5, 1], [24, 15, 7, 1], [24, 17, 9, 1], [24, 19, 11, 1], [24, 21, 13, 1], [24, 23, 15, 1], [24, 25, 17, 1], [24, 27, 19, 1], [24, 29, 21, 1], [24, 31, 23, 1], [26, 1, 27, 1], [26, 3, 29, 1], [26, 5, 31, 1], [26, 7, 1, 1], [26, 9, 3, 1], [26, 11, 5, 1], [26, 13, 7, 1], [26, 15, 9, 1], [26, 17, 11, 1], [26, 19, 13, 1], [26, 21, 15, 1], [26, 23, 17, 1], [26, 25, 19, 1], [26, 27, 21, 1], [26, 29, 23, 1], [26, 31, 25, 1], [28, 1, 29, 1], [28, 3, 31, 1], [28, 5, 1, 1], [28, 7, 3, 1], [28, 9, 5, 1], [28, 11, 7, 1], [28, 13, 9, 1], [28, 15, 11, 1], [28, 17, 13, 1], [28, 19, 15, 1], [28, 21, 17, 1], [28, 23, 19, 1], [28, 25, 21, 1], [28, 27, 23, 1], [28, 29, 25, 1], [28, 31, 27, 1], [30, 1, 31, 1], [30, 3, 1, 1], [30, 5, 3, 1], [30, 7, 5, 1], [30, 9, 7, 1], [30, 11, 9, 1], [30, 13, 11, 1], [30, 15, 13, 1], [30, 17, 15, 1], [30, 19, 17, 1], [30, 21, 19, 1], [30, 23, 21, 1], [30, 25, 23, 1], [30, 27, 25, 1], [30, 29, 27, 1], [30, 31, 29, 1]], [1], 是] 條件：無 每組coset的每對代表元都要落在同一product coset，僅用首代表元造表還不足以證良定；保完整可能index集合。 步驟識別與依賴規則：Quotient_EVERY_coset_pair_ALL_representative_products · 依賴：G001, S006 S008 · implies操作前 source roles = [1, 0, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30]] 操作後 derived = [[[1, 0, 1, 1], [1, 2, 3, 1], [1, 4, 5, 1], [1, 6, 7, 1], [1, 8, 9, 1], [1, 10, 11, 1], [1, 12, 13, 1], [1, 14, 15, 1], [1, 16, 17, 1], [1, 18, 19, 1], [1, 20, 21, 1], [1, 22, 23, 1], [1, 24, 25, 1], [1, 26, 27, 1], [1, 28, 29, 1], [1, 30, 31, 1], [3, 0, 3, 1], [3, 2, 5, 1], [3, 4, 7, 1], [3, 6, 9, 1], [3, 8, 11, 1], [3, 10, 13, 1], [3, 12, 15, 1], [3, 14, 17, 1], [3, 16, 19, 1], [3, 18, 21, 1], [3, 20, 23, 1], [3, 22, 25, 1], [3, 24, 27, 1], [3, 26, 29, 1], [3, 28, 31, 1], [3, 30, 1, 1], [5, 0, 5, 1], [5, 2, 7, 1], [5, 4, 9, 1], [5, 6, 11, 1], [5, 8, 13, 1], [5, 10, 15, 1], [5, 12, 17, 1], [5, 14, 19, 1], [5, 16, 21, 1], [5, 18, 23, 1], [5, 20, 25, 1], [5, 22, 27, 1], [5, 24, 29, 1], [5, 26, 31, 1], [5, 28, 1, 1], [5, 30, 3, 1], [7, 0, 7, 1], [7, 2, 9, 1], [7, 4, 11, 1], [7, 6, 13, 1], [7, 8, 15, 1], [7, 10, 17, 1], [7, 12, 19, 1], [7, 14, 21, 1], [7, 16, 23, 1], [7, 18, 25, 1], [7, 20, 27, 1], [7, 22, 29, 1], [7, 24, 31, 1], [7, 26, 1, 1], [7, 28, 3, 1], [7, 30, 5, 1], [9, 0, 9, 1], [9, 2, 11, 1], [9, 4, 13, 1], [9, 6, 15, 1], [9, 8, 17, 1], [9, 10, 19, 1], [9, 12, 21, 1], [9, 14, 23, 1], [9, 16, 25, 1], [9, 18, 27, 1], [9, 20, 29, 1], [9, 22, 31, 1], [9, 24, 1, 1], [9, 26, 3, 1], [9, 28, 5, 1], [9, 30, 7, 1], [11, 0, 11, 1], [11, 2, 13, 1], [11, 4, 15, 1], [11, 6, 17, 1], [11, 8, 19, 1], [11, 10, 21, 1], [11, 12, 23, 1], [11, 14, 25, 1], [11, 16, 27, 1], [11, 18, 29, 1], [11, 20, 31, 1], [11, 22, 1, 1], [11, 24, 3, 1], [11, 26, 5, 1], [11, 28, 7, 1], [11, 30, 9, 1], [13, 0, 13, 1], [13, 2, 15, 1], [13, 4, 17, 1], [13, 6, 19, 1], [13, 8, 21, 1], [13, 10, 23, 1], [13, 12, 25, 1], [13, 14, 27, 1], [13, 16, 29, 1], [13, 18, 31, 1], [13, 20, 1, 1], [13, 22, 3, 1], [13, 24, 5, 1], [13, 26, 7, 1], [13, 28, 9, 1], [13, 30, 11, 1], [15, 0, 15, 1], [15, 2, 17, 1], [15, 4, 19, 1], [15, 6, 21, 1], [15, 8, 23, 1], [15, 10, 25, 1], [15, 12, 27, 1], [15, 14, 29, 1], [15, 16, 31, 1], [15, 18, 1, 1], [15, 20, 3, 1], [15, 22, 5, 1], [15, 24, 7, 1], [15, 26, 9, 1], [15, 28, 11, 1], [15, 30, 13, 1], [17, 0, 17, 1], [17, 2, 19, 1], [17, 4, 21, 1], [17, 6, 23, 1], [17, 8, 25, 1], [17, 10, 27, 1], [17, 12, 29, 1], [17, 14, 31, 1], [17, 16, 1, 1], [17, 18, 3, 1], [17, 20, 5, 1], [17, 22, 7, 1], [17, 24, 9, 1], [17, 26, 11, 1], [17, 28, 13, 1], [17, 30, 15, 1], [19, 0, 19, 1], [19, 2, 21, 1], [19, 4, 23, 1], [19, 6, 25, 1], [19, 8, 27, 1], [19, 10, 29, 1], [19, 12, 31, 1], [19, 14, 1, 1], [19, 16, 3, 1], [19, 18, 5, 1], [19, 20, 7, 1], [19, 22, 9, 1], [19, 24, 11, 1], [19, 26, 13, 1], [19, 28, 15, 1], [19, 30, 17, 1], [21, 0, 21, 1], [21, 2, 23, 1], [21, 4, 25, 1], [21, 6, 27, 1], [21, 8, 29, 1], [21, 10, 31, 1], [21, 12, 1, 1], [21, 14, 3, 1], [21, 16, 5, 1], [21, 18, 7, 1], [21, 20, 9, 1], [21, 22, 11, 1], [21, 24, 13, 1], [21, 26, 15, 1], [21, 28, 17, 1], [21, 30, 19, 1], [23, 0, 23, 1], [23, 2, 25, 1], [23, 4, 27, 1], [23, 6, 29, 1], [23, 8, 31, 1], [23, 10, 1, 1], [23, 12, 3, 1], [23, 14, 5, 1], [23, 16, 7, 1], [23, 18, 9, 1], [23, 20, 11, 1], [23, 22, 13, 1], [23, 24, 15, 1], [23, 26, 17, 1], [23, 28, 19, 1], [23, 30, 21, 1], [25, 0, 25, 1], [25, 2, 27, 1], [25, 4, 29, 1], [25, 6, 31, 1], [25, 8, 1, 1], [25, 10, 3, 1], [25, 12, 5, 1], [25, 14, 7, 1], [25, 16, 9, 1], [25, 18, 11, 1], [25, 20, 13, 1], [25, 22, 15, 1], [25, 24, 17, 1], [25, 26, 19, 1], [25, 28, 21, 1], [25, 30, 23, 1], [27, 0, 27, 1], [27, 2, 29, 1], [27, 4, 31, 1], [27, 6, 1, 1], [27, 8, 3, 1], [27, 10, 5, 1], [27, 12, 7, 1], [27, 14, 9, 1], [27, 16, 11, 1], [27, 18, 13, 1], [27, 20, 15, 1], [27, 22, 17, 1], [27, 24, 19, 1], [27, 26, 21, 1], [27, 28, 23, 1], [27, 30, 25, 1], [29, 0, 29, 1], [29, 2, 31, 1], [29, 4, 1, 1], [29, 6, 3, 1], [29, 8, 5, 1], [29, 10, 7, 1], [29, 12, 9, 1], [29, 14, 11, 1], [29, 16, 13, 1], [29, 18, 15, 1], [29, 20, 17, 1], [29, 22, 19, 1], [29, 24, 21, 1], [29, 26, 23, 1], [29, 28, 25, 1], [29, 30, 27, 1], [31, 0, 31, 1], [31, 2, 1, 1], [31, 4, 3, 1], [31, 6, 5, 1], [31, 8, 7, 1], [31, 10, 9, 1], [31, 12, 11, 1], [31, 14, 13, 1], [31, 16, 15, 1], [31, 18, 17, 1], [31, 20, 19, 1], [31, 22, 21, 1], [31, 24, 23, 1], [31, 26, 25, 1], [31, 28, 27, 1], [31, 30, 29, 1]], [1], 是] 條件：無 每組coset的每對代表元都要落在同一product coset，僅用首代表元造表還不足以證良定；保完整可能index集合。 步驟識別與依賴規則：Quotient_EVERY_coset_pair_ALL_representative_products · 依賴：G001, S007 S009 · implies操作前 source roles = [1, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31]] 操作後 derived = [[[1, 1, 2, 0], [1, 3, 4, 0], [1, 5, 6, 0], [1, 7, 8, 0], [1, 9, 10, 0], [1, 11, 12, 0], [1, 13, 14, 0], [1, 15, 16, 0], [1, 17, 18, 0], [1, 19, 20, 0], [1, 21, 22, 0], [1, 23, 24, 0], [1, 25, 26, 0], [1, 27, 28, 0], [1, 29, 30, 0], [1, 31, 0, 0], [3, 1, 4, 0], [3, 3, 6, 0], [3, 5, 8, 0], [3, 7, 10, 0], [3, 9, 12, 0], [3, 11, 14, 0], [3, 13, 16, 0], [3, 15, 18, 0], [3, 17, 20, 0], [3, 19, 22, 0], [3, 21, 24, 0], [3, 23, 26, 0], [3, 25, 28, 0], [3, 27, 30, 0], [3, 29, 0, 0], [3, 31, 2, 0], [5, 1, 6, 0], [5, 3, 8, 0], [5, 5, 10, 0], [5, 7, 12, 0], [5, 9, 14, 0], [5, 11, 16, 0], [5, 13, 18, 0], [5, 15, 20, 0], [5, 17, 22, 0], [5, 19, 24, 0], [5, 21, 26, 0], [5, 23, 28, 0], [5, 25, 30, 0], [5, 27, 0, 0], [5, 29, 2, 0], [5, 31, 4, 0], [7, 1, 8, 0], [7, 3, 10, 0], [7, 5, 12, 0], [7, 7, 14, 0], [7, 9, 16, 0], [7, 11, 18, 0], [7, 13, 20, 0], [7, 15, 22, 0], [7, 17, 24, 0], [7, 19, 26, 0], [7, 21, 28, 0], [7, 23, 30, 0], [7, 25, 0, 0], [7, 27, 2, 0], [7, 29, 4, 0], [7, 31, 6, 0], [9, 1, 10, 0], [9, 3, 12, 0], [9, 5, 14, 0], [9, 7, 16, 0], [9, 9, 18, 0], [9, 11, 20, 0], [9, 13, 22, 0], [9, 15, 24, 0], [9, 17, 26, 0], [9, 19, 28, 0], [9, 21, 30, 0], [9, 23, 0, 0], [9, 25, 2, 0], [9, 27, 4, 0], [9, 29, 6, 0], [9, 31, 8, 0], [11, 1, 12, 0], [11, 3, 14, 0], [11, 5, 16, 0], [11, 7, 18, 0], [11, 9, 20, 0], [11, 11, 22, 0], [11, 13, 24, 0], [11, 15, 26, 0], [11, 17, 28, 0], [11, 19, 30, 0], [11, 21, 0, 0], [11, 23, 2, 0], [11, 25, 4, 0], [11, 27, 6, 0], [11, 29, 8, 0], [11, 31, 10, 0], [13, 1, 14, 0], [13, 3, 16, 0], [13, 5, 18, 0], [13, 7, 20, 0], [13, 9, 22, 0], [13, 11, 24, 0], [13, 13, 26, 0], [13, 15, 28, 0], [13, 17, 30, 0], [13, 19, 0, 0], [13, 21, 2, 0], [13, 23, 4, 0], [13, 25, 6, 0], [13, 27, 8, 0], [13, 29, 10, 0], [13, 31, 12, 0], [15, 1, 16, 0], [15, 3, 18, 0], [15, 5, 20, 0], [15, 7, 22, 0], [15, 9, 24, 0], [15, 11, 26, 0], [15, 13, 28, 0], [15, 15, 30, 0], [15, 17, 0, 0], [15, 19, 2, 0], [15, 21, 4, 0], [15, 23, 6, 0], [15, 25, 8, 0], [15, 27, 10, 0], [15, 29, 12, 0], [15, 31, 14, 0], [17, 1, 18, 0], [17, 3, 20, 0], [17, 5, 22, 0], [17, 7, 24, 0], [17, 9, 26, 0], [17, 11, 28, 0], [17, 13, 30, 0], [17, 15, 0, 0], [17, 17, 2, 0], [17, 19, 4, 0], [17, 21, 6, 0], [17, 23, 8, 0], [17, 25, 10, 0], [17, 27, 12, 0], [17, 29, 14, 0], [17, 31, 16, 0], [19, 1, 20, 0], [19, 3, 22, 0], [19, 5, 24, 0], [19, 7, 26, 0], [19, 9, 28, 0], [19, 11, 30, 0], [19, 13, 0, 0], [19, 15, 2, 0], [19, 17, 4, 0], [19, 19, 6, 0], [19, 21, 8, 0], [19, 23, 10, 0], [19, 25, 12, 0], [19, 27, 14, 0], [19, 29, 16, 0], [19, 31, 18, 0], [21, 1, 22, 0], [21, 3, 24, 0], [21, 5, 26, 0], [21, 7, 28, 0], [21, 9, 30, 0], [21, 11, 0, 0], [21, 13, 2, 0], [21, 15, 4, 0], [21, 17, 6, 0], [21, 19, 8, 0], [21, 21, 10, 0], [21, 23, 12, 0], [21, 25, 14, 0], [21, 27, 16, 0], [21, 29, 18, 0], [21, 31, 20, 0], [23, 1, 24, 0], [23, 3, 26, 0], [23, 5, 28, 0], [23, 7, 30, 0], [23, 9, 0, 0], [23, 11, 2, 0], [23, 13, 4, 0], [23, 15, 6, 0], [23, 17, 8, 0], [23, 19, 10, 0], [23, 21, 12, 0], [23, 23, 14, 0], [23, 25, 16, 0], [23, 27, 18, 0], [23, 29, 20, 0], [23, 31, 22, 0], [25, 1, 26, 0], [25, 3, 28, 0], [25, 5, 30, 0], [25, 7, 0, 0], [25, 9, 2, 0], [25, 11, 4, 0], [25, 13, 6, 0], [25, 15, 8, 0], [25, 17, 10, 0], [25, 19, 12, 0], [25, 21, 14, 0], [25, 23, 16, 0], [25, 25, 18, 0], [25, 27, 20, 0], [25, 29, 22, 0], [25, 31, 24, 0], [27, 1, 28, 0], [27, 3, 30, 0], [27, 5, 0, 0], [27, 7, 2, 0], [27, 9, 4, 0], [27, 11, 6, 0], [27, 13, 8, 0], [27, 15, 10, 0], [27, 17, 12, 0], [27, 19, 14, 0], [27, 21, 16, 0], [27, 23, 18, 0], [27, 25, 20, 0], [27, 27, 22, 0], [27, 29, 24, 0], [27, 31, 26, 0], [29, 1, 30, 0], [29, 3, 0, 0], [29, 5, 2, 0], [29, 7, 4, 0], [29, 9, 6, 0], [29, 11, 8, 0], [29, 13, 10, 0], [29, 15, 12, 0], [29, 17, 14, 0], [29, 19, 16, 0], [29, 21, 18, 0], [29, 23, 20, 0], [29, 25, 22, 0], [29, 27, 24, 0], [29, 29, 26, 0], [29, 31, 28, 0], [31, 1, 0, 0], [31, 3, 2, 0], [31, 5, 4, 0], [31, 7, 6, 0], [31, 9, 8, 0], [31, 11, 10, 0], [31, 13, 12, 0], [31, 15, 14, 0], [31, 17, 16, 0], [31, 19, 18, 0], [31, 21, 20, 0], [31, 23, 22, 0], [31, 25, 24, 0], [31, 27, 26, 0], [31, 29, 28, 0], [31, 31, 30, 0]], [0], 是] 條件：無 每組coset的每對代表元都要落在同一product coset，僅用首代表元造表還不足以證良定；保完整可能index集合。 步驟識別與依賴規則：Quotient_EVERY_coset_pair_ALL_representative_products · 依賴：G001, S008 S010 · implies操作前 source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是, [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31]]] 操作後 derived = [[[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], 是, 否] 條件：無 若a換ah、b換bk，(ah)(bk)=ab(b^-1hb)k∈abH，normality才保代表元替換不變。associativity由G傳下，eH為identity、a^-1H為inverse。固定No只不自動提供commutativity證書，不說具體cyclic quotient非交換，也不引第一同構定理。 步驟識別與依賴規則：Quotient_GENERAL_normality_representative_replacement_and_group_laws · 依賴：G001, S009 ## 逐步反向核對 - S010 · 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。 - 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 — 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, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], 是, [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31]], 是, 2, quotient_elements_are_disjoint_complete_coset_sets_not_representatives) 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: fb498e883419c1160d72936168a51296db671bfea1c345d3820a40ce01afedab - candidate record SHA-256: 212f289fff9d2bd3a8dea8456540cb48668d53d1f0c10935854e90c3a9c2cab3 - semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3 - question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3 - Lean: NOT_RUN · human review: NOT_PERFORMED [同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r02-finite-quotient-group-0015.json)