{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261114-r02-finite-quotient-group-0072", "pilot_number": "R0049-25397", "number": 25397, "family_id": "qf-mod45-quotient-representative-well-definedness", "title": { "zh-Hant": "第 25397 題：有限商群", "en": "Question 25397: Every Representative Pair Must Land in the Same Declared Product Coset" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261114-r02-finite-quotient-group-0072", "batch_id": "batch-20261015-advance-20261114-r02-finite-quotient-group-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-finite-quotient-group" ], "title_zh": "有限商群", "stem_zh": "完整有限群、subgroup與全部coset partition及代表元責任。操作=quotient-representative-well-definedness；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]], [[0, 0, [0], 是], [0, 1, [1], 是], [1, 0, [1], 是], [1, 1, [0], 是]], 是, every_representative_pair_lands_in_one_declared_product_coset)", "explanation_zh": "Wave6第八層核對=([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]], [[0, 0, [0], 是], [0, 1, [1], 是], [1, 0, [1], 是], [1, 1, [0], 是]], 是, every_representative_pair_lands_in_one_declared_product_coset)；依完整明示資料獨立重算；quotient元素是coset sets；乘法對所有代表元替換不變並保存identity與inverse；nonnormal代表元依賴會阻止商群；不自動commutative或宣稱第一同構定理。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "有限商群" ] }, "variant": { "family_id": "qf-mod45-quotient-representative-well-definedness", "seed": 202611142072, "parameters": { "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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45 ], [ 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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 ], [ 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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 ], [ 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32 ], [ 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33 ], [ 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34 ], [ 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35 ], [ 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36 ], [ 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37 ], [ 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38 ], [ 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39 ], [ 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40 ], [ 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41 ], [ 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42 ], [ 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43 ], [ 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44 ] ], "identity": 0, "subset": [ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44 ], "domain": "Complete_finite_quotient_group", "operation": "quotient-representative-well-definedness", "semantic_template": "qf-mod45-quotient-representative-well-definedness/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261114-r02-finite-quotient-group-0072", "batch_id": "batch-20261015-advance-20261114-r02-finite-quotient-group-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "e90cfd20c5b0e62ee87d48c51645aabec11ee5f0dc940992d7e875973f8a59a2", "family_id": "qf-mod45-quotient-representative-well-definedness", "seed": 202611142072, "semantic_template": "qf-mod45-quotient-representative-well-definedness/{\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44]}" }, "fields": { "title": "Every Representative Pair Must Land in the Same Declared Product Coset", "stem": "Use complete finite 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (cosets, representative_rows, well_defined, certificate).", "answer": { "kind": "expression", "value": "([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]], [[0, 0, [0], Yes], [0, 1, [1], Yes], [1, 0, [1], Yes], [1, 1, [0], Yes]], Yes, every_representative_pair_lands_in_one_declared_product_coset)", "explanation": "For every ordered coset pair, list [left index,right index,all possible product-coset indices,unique]. Exhaust every representative combination. A table produced from only the least representatives is not a substitute for well-definedness. The exact result, retaining the requested order, is ([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]], [[0, 0, [0], Yes], [0, 1, [1], Yes], [1, 0, [1], Yes], [1, 1, [0], Yes]], Yes, every_representative_pair_lands_in_one_declared_product_coset)." }, "skills": [ "Every Representative Pair Must Land in the Same Declared Product Coset" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "完整有限群、subgroup與全部coset partition及代表元責任。操作=quotient-representative-well-definedness；資料={\"group_table\":[[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0],[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1],[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2],[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3],[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4],[6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5],[7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6],[8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7],[9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8],[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9],[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10],[12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11],[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12],[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13],[15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],[16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],[20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],[23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],[25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],[26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],[27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27],[29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],[30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29],[31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],[32,33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],[33,34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],[34,35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33],[35,36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],[36,37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35],[37,38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],[38,39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],[39,40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],[40,41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39],[41,42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40],[42,43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],[43,44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42],[44,45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],[45,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]],\"identity\":0,\"subset\":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use complete finite 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (cosets, representative_rows, well_defined, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [quotient-representative-well-definedness]", "after": "derived = [[cosets, representative_rows, well_defined, certificate]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]], 0]", "after": "derived = [46, [[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], [32, 32, 32, 32], [33, 33, 33, 33], [34, 34, 34, 34], [35, 35, 35, 35], [36, 36, 36, 36], [37, 37, 37, 37], [38, 38, 38, 38], [39, 39, 39, 39], [40, 40, 40, 40], [41, 41, 41, 41], [42, 42, 42, 42], [43, 43, 43, 43], [44, 44, 44, 44], [45, 45, 45, 45]], 是]", "relation": "implies", "rule_id": "Quotient_complete_group_table_actual_identity", "conditions": [], "explanation": { "zh-Hant": "先核整個closed Cayley table與實際identity標籤，不假設0或套舊上限32。", "en": "Validate the complete closed Cayley table and actual identity label,without assuming0 or the old32-element cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [46, 0]", "after": "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, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 97336, [0, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 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]]", "relation": "implies", "rule_id": "Quotient_ALL_group_associativity_triples_and_two_sided_inverses", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Each pair-matrix entry checks associativity for every z,covering n^3 obligations;each inverse returns the same identity from both sides.This is exhaustive,not sampling." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]]", "after": "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], [0, 32, 32, 是], [0, 34, 34, 是], [0, 36, 36, 是], [0, 38, 38, 是], [0, 40, 40, 是], [0, 42, 42, 是], [0, 44, 44, 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, 32, 是], [2, 32, 34, 是], [2, 34, 36, 是], [2, 36, 38, 是], [2, 38, 40, 是], [2, 40, 42, 是], [2, 42, 44, 是], [2, 44, 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, 32, 是], [4, 30, 34, 是], [4, 32, 36, 是], [4, 34, 38, 是], [4, 36, 40, 是], [4, 38, 42, 是], [4, 40, 44, 是], [4, 42, 0, 是], [4, 44, 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, 32, 是], [6, 28, 34, 是], [6, 30, 36, 是], [6, 32, 38, 是], [6, 34, 40, 是], [6, 36, 42, 是], [6, 38, 44, 是], [6, 40, 0, 是], [6, 42, 2, 是], [6, 44, 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, 32, 是], [8, 26, 34, 是], [8, 28, 36, 是], [8, 30, 38, 是], [8, 32, 40, 是], [8, 34, 42, 是], [8, 36, 44, 是], [8, 38, 0, 是], [8, 40, 2, 是], [8, 42, 4, 是], [8, 44, 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, 32, 是], [10, 24, 34, 是], [10, 26, 36, 是], [10, 28, 38, 是], [10, 30, 40, 是], [10, 32, 42, 是], [10, 34, 44, 是], [10, 36, 0, 是], [10, 38, 2, 是], [10, 40, 4, 是], [10, 42, 6, 是], [10, 44, 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, 32, 是], [12, 22, 34, 是], [12, 24, 36, 是], [12, 26, 38, 是], [12, 28, 40, 是], [12, 30, 42, 是], [12, 32, 44, 是], [12, 34, 0, 是], [12, 36, 2, 是], [12, 38, 4, 是], [12, 40, 6, 是], [12, 42, 8, 是], [12, 44, 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, 32, 是], [14, 20, 34, 是], [14, 22, 36, 是], [14, 24, 38, 是], [14, 26, 40, 是], [14, 28, 42, 是], [14, 30, 44, 是], [14, 32, 0, 是], [14, 34, 2, 是], [14, 36, 4, 是], [14, 38, 6, 是], [14, 40, 8, 是], [14, 42, 10, 是], [14, 44, 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, 32, 是], [16, 18, 34, 是], [16, 20, 36, 是], [16, 22, 38, 是], [16, 24, 40, 是], [16, 26, 42, 是], [16, 28, 44, 是], [16, 30, 0, 是], [16, 32, 2, 是], [16, 34, 4, 是], [16, 36, 6, 是], [16, 38, 8, 是], [16, 40, 10, 是], [16, 42, 12, 是], [16, 44, 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, 32, 是], [18, 16, 34, 是], [18, 18, 36, 是], [18, 20, 38, 是], [18, 22, 40, 是], [18, 24, 42, 是], [18, 26, 44, 是], [18, 28, 0, 是], [18, 30, 2, 是], [18, 32, 4, 是], [18, 34, 6, 是], [18, 36, 8, 是], [18, 38, 10, 是], [18, 40, 12, 是], [18, 42, 14, 是], [18, 44, 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, 32, 是], [20, 14, 34, 是], [20, 16, 36, 是], [20, 18, 38, 是], [20, 20, 40, 是], [20, 22, 42, 是], [20, 24, 44, 是], [20, 26, 0, 是], [20, 28, 2, 是], [20, 30, 4, 是], [20, 32, 6, 是], [20, 34, 8, 是], [20, 36, 10, 是], [20, 38, 12, 是], [20, 40, 14, 是], [20, 42, 16, 是], [20, 44, 18, 是], [22, 0, 22, 是], [22, 2, 24, 是], [22, 4, 26, 是], [22, 6, 28, 是], [22, 8, 30, 是], [22, 10, 32, 是], [22, 12, 34, 是], [22, 14, 36, 是], [22, 16, 38, 是], [22, 18, 40, 是], [22, 20, 42, 是], [22, 22, 44, 是], [22, 24, 0, 是], [22, 26, 2, 是], [22, 28, 4, 是], [22, 30, 6, 是], [22, 32, 8, 是], [22, 34, 10, 是], [22, 36, 12, 是], [22, 38, 14, 是], [22, 40, 16, 是], [22, 42, 18, 是], [22, 44, 20, 是], [24, 0, 24, 是], [24, 2, 26, 是], [24, 4, 28, 是], [24, 6, 30, 是], [24, 8, 32, 是], [24, 10, 34, 是], [24, 12, 36, 是], [24, 14, 38, 是], [24, 16, 40, 是], [24, 18, 42, 是], [24, 20, 44, 是], [24, 22, 0, 是], [24, 24, 2, 是], [24, 26, 4, 是], [24, 28, 6, 是], [24, 30, 8, 是], [24, 32, 10, 是], [24, 34, 12, 是], [24, 36, 14, 是], [24, 38, 16, 是], [24, 40, 18, 是], [24, 42, 20, 是], [24, 44, 22, 是], [26, 0, 26, 是], [26, 2, 28, 是], [26, 4, 30, 是], [26, 6, 32, 是], [26, 8, 34, 是], [26, 10, 36, 是], [26, 12, 38, 是], [26, 14, 40, 是], [26, 16, 42, 是], [26, 18, 44, 是], [26, 20, 0, 是], [26, 22, 2, 是], [26, 24, 4, 是], [26, 26, 6, 是], [26, 28, 8, 是], [26, 30, 10, 是], [26, 32, 12, 是], [26, 34, 14, 是], [26, 36, 16, 是], [26, 38, 18, 是], [26, 40, 20, 是], [26, 42, 22, 是], [26, 44, 24, 是], [28, 0, 28, 是], [28, 2, 30, 是], [28, 4, 32, 是], [28, 6, 34, 是], [28, 8, 36, 是], [28, 10, 38, 是], [28, 12, 40, 是], [28, 14, 42, 是], [28, 16, 44, 是], [28, 18, 0, 是], [28, 20, 2, 是], [28, 22, 4, 是], [28, 24, 6, 是], [28, 26, 8, 是], [28, 28, 10, 是], [28, 30, 12, 是], [28, 32, 14, 是], [28, 34, 16, 是], [28, 36, 18, 是], [28, 38, 20, 是], [28, 40, 22, 是], [28, 42, 24, 是], [28, 44, 26, 是], [30, 0, 30, 是], [30, 2, 32, 是], [30, 4, 34, 是], [30, 6, 36, 是], [30, 8, 38, 是], [30, 10, 40, 是], [30, 12, 42, 是], [30, 14, 44, 是], [30, 16, 0, 是], [30, 18, 2, 是], [30, 20, 4, 是], [30, 22, 6, 是], [30, 24, 8, 是], [30, 26, 10, 是], [30, 28, 12, 是], [30, 30, 14, 是], [30, 32, 16, 是], [30, 34, 18, 是], [30, 36, 20, 是], [30, 38, 22, 是], [30, 40, 24, 是], [30, 42, 26, 是], [30, 44, 28, 是], [32, 0, 32, 是], [32, 2, 34, 是], [32, 4, 36, 是], [32, 6, 38, 是], [32, 8, 40, 是], [32, 10, 42, 是], [32, 12, 44, 是], [32, 14, 0, 是], [32, 16, 2, 是], [32, 18, 4, 是], [32, 20, 6, 是], [32, 22, 8, 是], [32, 24, 10, 是], [32, 26, 12, 是], [32, 28, 14, 是], [32, 30, 16, 是], [32, 32, 18, 是], [32, 34, 20, 是], [32, 36, 22, 是], [32, 38, 24, 是], [32, 40, 26, 是], [32, 42, 28, 是], [32, 44, 30, 是], [34, 0, 34, 是], [34, 2, 36, 是], [34, 4, 38, 是], [34, 6, 40, 是], [34, 8, 42, 是], [34, 10, 44, 是], [34, 12, 0, 是], [34, 14, 2, 是], [34, 16, 4, 是], [34, 18, 6, 是], [34, 20, 8, 是], [34, 22, 10, 是], [34, 24, 12, 是], [34, 26, 14, 是], [34, 28, 16, 是], [34, 30, 18, 是], [34, 32, 20, 是], [34, 34, 22, 是], [34, 36, 24, 是], [34, 38, 26, 是], [34, 40, 28, 是], [34, 42, 30, 是], [34, 44, 32, 是], [36, 0, 36, 是], [36, 2, 38, 是], [36, 4, 40, 是], [36, 6, 42, 是], [36, 8, 44, 是], [36, 10, 0, 是], [36, 12, 2, 是], [36, 14, 4, 是], [36, 16, 6, 是], [36, 18, 8, 是], [36, 20, 10, 是], [36, 22, 12, 是], [36, 24, 14, 是], [36, 26, 16, 是], [36, 28, 18, 是], [36, 30, 20, 是], [36, 32, 22, 是], [36, 34, 24, 是], [36, 36, 26, 是], [36, 38, 28, 是], [36, 40, 30, 是], [36, 42, 32, 是], [36, 44, 34, 是], [38, 0, 38, 是], [38, 2, 40, 是], [38, 4, 42, 是], [38, 6, 44, 是], [38, 8, 0, 是], [38, 10, 2, 是], [38, 12, 4, 是], [38, 14, 6, 是], [38, 16, 8, 是], [38, 18, 10, 是], [38, 20, 12, 是], [38, 22, 14, 是], [38, 24, 16, 是], [38, 26, 18, 是], [38, 28, 20, 是], [38, 30, 22, 是], [38, 32, 24, 是], [38, 34, 26, 是], [38, 36, 28, 是], [38, 38, 30, 是], [38, 40, 32, 是], [38, 42, 34, 是], [38, 44, 36, 是], [40, 0, 40, 是], [40, 2, 42, 是], [40, 4, 44, 是], [40, 6, 0, 是], [40, 8, 2, 是], [40, 10, 4, 是], [40, 12, 6, 是], [40, 14, 8, 是], [40, 16, 10, 是], [40, 18, 12, 是], [40, 20, 14, 是], [40, 22, 16, 是], [40, 24, 18, 是], [40, 26, 20, 是], [40, 28, 22, 是], [40, 30, 24, 是], [40, 32, 26, 是], [40, 34, 28, 是], [40, 36, 30, 是], [40, 38, 32, 是], [40, 40, 34, 是], [40, 42, 36, 是], [40, 44, 38, 是], [42, 0, 42, 是], [42, 2, 44, 是], [42, 4, 0, 是], [42, 6, 2, 是], [42, 8, 4, 是], [42, 10, 6, 是], [42, 12, 8, 是], [42, 14, 10, 是], [42, 16, 12, 是], [42, 18, 14, 是], [42, 20, 16, 是], [42, 22, 18, 是], [42, 24, 20, 是], [42, 26, 22, 是], [42, 28, 24, 是], [42, 30, 26, 是], [42, 32, 28, 是], [42, 34, 30, 是], [42, 36, 32, 是], [42, 38, 34, 是], [42, 40, 36, 是], [42, 42, 38, 是], [42, 44, 40, 是], [44, 0, 44, 是], [44, 2, 0, 是], [44, 4, 2, 是], [44, 6, 4, 是], [44, 8, 6, 是], [44, 10, 8, 是], [44, 12, 10, 是], [44, 14, 12, 是], [44, 16, 14, 是], [44, 18, 16, 是], [44, 20, 18, 是], [44, 22, 20, 是], [44, 24, 22, 是], [44, 26, 24, 是], [44, 28, 26, 是], [44, 30, 28, 是], [44, 32, 30, 是], [44, 34, 32, 是], [44, 36, 34, 是], [44, 38, 36, 是], [44, 40, 38, 是], [44, 42, 40, 是], [44, 44, 42, 是]], [[0, 0, 是], [2, 44, 是], [4, 42, 是], [6, 40, 是], [8, 38, 是], [10, 36, 是], [12, 34, 是], [14, 32, 是], [16, 30, 是], [18, 28, 是], [20, 26, 是], [22, 24, 是], [24, 22, 是], [26, 20, 是], [28, 18, 是], [30, 16, 是], [32, 14, 是], [34, 12, 是], [36, 10, 是], [38, 8, 是], [40, 6, 是], [42, 4, 是], [44, 2, 是]], 是, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]]], 是]", "relation": "implies", "rule_id": "Quotient_genuine_subgroup_before_actual_normality", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "First qualify identity,all products and inverse closure as a genuine subgroup,then test conjugation for every g.Positive branches require normality and the boundary requires actual nonnormality,not failed subgroup qualification." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [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, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]]", "after": "derived = [[[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]], [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, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], 是]", "relation": "implies", "rule_id": "Quotient_ALL_LEFT_coset_SETS_complete_disjoint_partition", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Every g belongs to gH.If aH meets bH,ah1=bh2 implies b^-1a in H and the complete cosets coincide.Enumerate every g,deduplicate and retain source order.Quotient elements are coset SETS,not least representatives." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [0, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]]", "after": "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], [0, 32, 32, 0], [0, 34, 34, 0], [0, 36, 36, 0], [0, 38, 38, 0], [0, 40, 40, 0], [0, 42, 42, 0], [0, 44, 44, 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, 32, 0], [2, 32, 34, 0], [2, 34, 36, 0], [2, 36, 38, 0], [2, 38, 40, 0], [2, 40, 42, 0], [2, 42, 44, 0], [2, 44, 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, 32, 0], [4, 30, 34, 0], [4, 32, 36, 0], [4, 34, 38, 0], [4, 36, 40, 0], [4, 38, 42, 0], [4, 40, 44, 0], [4, 42, 0, 0], [4, 44, 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, 32, 0], [6, 28, 34, 0], [6, 30, 36, 0], [6, 32, 38, 0], [6, 34, 40, 0], [6, 36, 42, 0], [6, 38, 44, 0], [6, 40, 0, 0], [6, 42, 2, 0], [6, 44, 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, 32, 0], [8, 26, 34, 0], [8, 28, 36, 0], [8, 30, 38, 0], [8, 32, 40, 0], [8, 34, 42, 0], [8, 36, 44, 0], [8, 38, 0, 0], [8, 40, 2, 0], [8, 42, 4, 0], [8, 44, 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, 32, 0], [10, 24, 34, 0], [10, 26, 36, 0], [10, 28, 38, 0], [10, 30, 40, 0], [10, 32, 42, 0], [10, 34, 44, 0], [10, 36, 0, 0], [10, 38, 2, 0], [10, 40, 4, 0], [10, 42, 6, 0], [10, 44, 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, 32, 0], [12, 22, 34, 0], [12, 24, 36, 0], [12, 26, 38, 0], [12, 28, 40, 0], [12, 30, 42, 0], [12, 32, 44, 0], [12, 34, 0, 0], [12, 36, 2, 0], [12, 38, 4, 0], [12, 40, 6, 0], [12, 42, 8, 0], [12, 44, 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, 32, 0], [14, 20, 34, 0], [14, 22, 36, 0], [14, 24, 38, 0], [14, 26, 40, 0], [14, 28, 42, 0], [14, 30, 44, 0], [14, 32, 0, 0], [14, 34, 2, 0], [14, 36, 4, 0], [14, 38, 6, 0], [14, 40, 8, 0], [14, 42, 10, 0], [14, 44, 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, 32, 0], [16, 18, 34, 0], [16, 20, 36, 0], [16, 22, 38, 0], [16, 24, 40, 0], [16, 26, 42, 0], [16, 28, 44, 0], [16, 30, 0, 0], [16, 32, 2, 0], [16, 34, 4, 0], [16, 36, 6, 0], [16, 38, 8, 0], [16, 40, 10, 0], [16, 42, 12, 0], [16, 44, 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, 32, 0], [18, 16, 34, 0], [18, 18, 36, 0], [18, 20, 38, 0], [18, 22, 40, 0], [18, 24, 42, 0], [18, 26, 44, 0], [18, 28, 0, 0], [18, 30, 2, 0], [18, 32, 4, 0], [18, 34, 6, 0], [18, 36, 8, 0], [18, 38, 10, 0], [18, 40, 12, 0], [18, 42, 14, 0], [18, 44, 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, 32, 0], [20, 14, 34, 0], [20, 16, 36, 0], [20, 18, 38, 0], [20, 20, 40, 0], [20, 22, 42, 0], [20, 24, 44, 0], [20, 26, 0, 0], [20, 28, 2, 0], [20, 30, 4, 0], [20, 32, 6, 0], [20, 34, 8, 0], [20, 36, 10, 0], [20, 38, 12, 0], [20, 40, 14, 0], [20, 42, 16, 0], [20, 44, 18, 0], [22, 0, 22, 0], [22, 2, 24, 0], [22, 4, 26, 0], [22, 6, 28, 0], [22, 8, 30, 0], [22, 10, 32, 0], [22, 12, 34, 0], [22, 14, 36, 0], [22, 16, 38, 0], [22, 18, 40, 0], [22, 20, 42, 0], [22, 22, 44, 0], [22, 24, 0, 0], [22, 26, 2, 0], [22, 28, 4, 0], [22, 30, 6, 0], [22, 32, 8, 0], [22, 34, 10, 0], [22, 36, 12, 0], [22, 38, 14, 0], [22, 40, 16, 0], [22, 42, 18, 0], [22, 44, 20, 0], [24, 0, 24, 0], [24, 2, 26, 0], [24, 4, 28, 0], [24, 6, 30, 0], [24, 8, 32, 0], [24, 10, 34, 0], [24, 12, 36, 0], [24, 14, 38, 0], [24, 16, 40, 0], [24, 18, 42, 0], [24, 20, 44, 0], [24, 22, 0, 0], [24, 24, 2, 0], [24, 26, 4, 0], [24, 28, 6, 0], [24, 30, 8, 0], [24, 32, 10, 0], [24, 34, 12, 0], [24, 36, 14, 0], [24, 38, 16, 0], [24, 40, 18, 0], [24, 42, 20, 0], [24, 44, 22, 0], [26, 0, 26, 0], [26, 2, 28, 0], [26, 4, 30, 0], [26, 6, 32, 0], [26, 8, 34, 0], [26, 10, 36, 0], [26, 12, 38, 0], [26, 14, 40, 0], [26, 16, 42, 0], [26, 18, 44, 0], [26, 20, 0, 0], [26, 22, 2, 0], [26, 24, 4, 0], [26, 26, 6, 0], [26, 28, 8, 0], [26, 30, 10, 0], [26, 32, 12, 0], [26, 34, 14, 0], [26, 36, 16, 0], [26, 38, 18, 0], [26, 40, 20, 0], [26, 42, 22, 0], [26, 44, 24, 0], [28, 0, 28, 0], [28, 2, 30, 0], [28, 4, 32, 0], [28, 6, 34, 0], [28, 8, 36, 0], [28, 10, 38, 0], [28, 12, 40, 0], [28, 14, 42, 0], [28, 16, 44, 0], [28, 18, 0, 0], [28, 20, 2, 0], [28, 22, 4, 0], [28, 24, 6, 0], [28, 26, 8, 0], [28, 28, 10, 0], [28, 30, 12, 0], [28, 32, 14, 0], [28, 34, 16, 0], [28, 36, 18, 0], [28, 38, 20, 0], [28, 40, 22, 0], [28, 42, 24, 0], [28, 44, 26, 0], [30, 0, 30, 0], [30, 2, 32, 0], [30, 4, 34, 0], [30, 6, 36, 0], [30, 8, 38, 0], [30, 10, 40, 0], [30, 12, 42, 0], [30, 14, 44, 0], [30, 16, 0, 0], [30, 18, 2, 0], [30, 20, 4, 0], [30, 22, 6, 0], [30, 24, 8, 0], [30, 26, 10, 0], [30, 28, 12, 0], [30, 30, 14, 0], [30, 32, 16, 0], [30, 34, 18, 0], [30, 36, 20, 0], [30, 38, 22, 0], [30, 40, 24, 0], [30, 42, 26, 0], [30, 44, 28, 0], [32, 0, 32, 0], [32, 2, 34, 0], [32, 4, 36, 0], [32, 6, 38, 0], [32, 8, 40, 0], [32, 10, 42, 0], [32, 12, 44, 0], [32, 14, 0, 0], [32, 16, 2, 0], [32, 18, 4, 0], [32, 20, 6, 0], [32, 22, 8, 0], [32, 24, 10, 0], [32, 26, 12, 0], [32, 28, 14, 0], [32, 30, 16, 0], [32, 32, 18, 0], [32, 34, 20, 0], [32, 36, 22, 0], [32, 38, 24, 0], [32, 40, 26, 0], [32, 42, 28, 0], [32, 44, 30, 0], [34, 0, 34, 0], [34, 2, 36, 0], [34, 4, 38, 0], [34, 6, 40, 0], [34, 8, 42, 0], [34, 10, 44, 0], [34, 12, 0, 0], [34, 14, 2, 0], [34, 16, 4, 0], [34, 18, 6, 0], [34, 20, 8, 0], [34, 22, 10, 0], [34, 24, 12, 0], [34, 26, 14, 0], [34, 28, 16, 0], [34, 30, 18, 0], [34, 32, 20, 0], [34, 34, 22, 0], [34, 36, 24, 0], [34, 38, 26, 0], [34, 40, 28, 0], [34, 42, 30, 0], [34, 44, 32, 0], [36, 0, 36, 0], [36, 2, 38, 0], [36, 4, 40, 0], [36, 6, 42, 0], [36, 8, 44, 0], [36, 10, 0, 0], [36, 12, 2, 0], [36, 14, 4, 0], [36, 16, 6, 0], [36, 18, 8, 0], [36, 20, 10, 0], [36, 22, 12, 0], [36, 24, 14, 0], [36, 26, 16, 0], [36, 28, 18, 0], [36, 30, 20, 0], [36, 32, 22, 0], [36, 34, 24, 0], [36, 36, 26, 0], [36, 38, 28, 0], [36, 40, 30, 0], [36, 42, 32, 0], [36, 44, 34, 0], [38, 0, 38, 0], [38, 2, 40, 0], [38, 4, 42, 0], [38, 6, 44, 0], [38, 8, 0, 0], [38, 10, 2, 0], [38, 12, 4, 0], [38, 14, 6, 0], [38, 16, 8, 0], [38, 18, 10, 0], [38, 20, 12, 0], [38, 22, 14, 0], [38, 24, 16, 0], [38, 26, 18, 0], [38, 28, 20, 0], [38, 30, 22, 0], [38, 32, 24, 0], [38, 34, 26, 0], [38, 36, 28, 0], [38, 38, 30, 0], [38, 40, 32, 0], [38, 42, 34, 0], [38, 44, 36, 0], [40, 0, 40, 0], [40, 2, 42, 0], [40, 4, 44, 0], [40, 6, 0, 0], [40, 8, 2, 0], [40, 10, 4, 0], [40, 12, 6, 0], [40, 14, 8, 0], [40, 16, 10, 0], [40, 18, 12, 0], [40, 20, 14, 0], [40, 22, 16, 0], [40, 24, 18, 0], [40, 26, 20, 0], [40, 28, 22, 0], [40, 30, 24, 0], [40, 32, 26, 0], [40, 34, 28, 0], [40, 36, 30, 0], [40, 38, 32, 0], [40, 40, 34, 0], [40, 42, 36, 0], [40, 44, 38, 0], [42, 0, 42, 0], [42, 2, 44, 0], [42, 4, 0, 0], [42, 6, 2, 0], [42, 8, 4, 0], [42, 10, 6, 0], [42, 12, 8, 0], [42, 14, 10, 0], [42, 16, 12, 0], [42, 18, 14, 0], [42, 20, 16, 0], [42, 22, 18, 0], [42, 24, 20, 0], [42, 26, 22, 0], [42, 28, 24, 0], [42, 30, 26, 0], [42, 32, 28, 0], [42, 34, 30, 0], [42, 36, 32, 0], [42, 38, 34, 0], [42, 40, 36, 0], [42, 42, 38, 0], [42, 44, 40, 0], [44, 0, 44, 0], [44, 2, 0, 0], [44, 4, 2, 0], [44, 6, 4, 0], [44, 8, 6, 0], [44, 10, 8, 0], [44, 12, 10, 0], [44, 14, 12, 0], [44, 16, 14, 0], [44, 18, 16, 0], [44, 20, 18, 0], [44, 22, 20, 0], [44, 24, 22, 0], [44, 26, 24, 0], [44, 28, 26, 0], [44, 30, 28, 0], [44, 32, 30, 0], [44, 34, 32, 0], [44, 36, 34, 0], [44, 38, 36, 0], [44, 40, 38, 0], [44, 42, 40, 0], [44, 44, 42, 0]], [0], 是]", "relation": "implies", "rule_id": "Quotient_EVERY_coset_pair_ALL_representative_products", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [0, 1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]]", "after": "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], [0, 33, 33, 1], [0, 35, 35, 1], [0, 37, 37, 1], [0, 39, 39, 1], [0, 41, 41, 1], [0, 43, 43, 1], [0, 45, 45, 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, 33, 1], [2, 33, 35, 1], [2, 35, 37, 1], [2, 37, 39, 1], [2, 39, 41, 1], [2, 41, 43, 1], [2, 43, 45, 1], [2, 45, 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, 33, 1], [4, 31, 35, 1], [4, 33, 37, 1], [4, 35, 39, 1], [4, 37, 41, 1], [4, 39, 43, 1], [4, 41, 45, 1], [4, 43, 1, 1], [4, 45, 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, 33, 1], [6, 29, 35, 1], [6, 31, 37, 1], [6, 33, 39, 1], [6, 35, 41, 1], [6, 37, 43, 1], [6, 39, 45, 1], [6, 41, 1, 1], [6, 43, 3, 1], [6, 45, 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, 33, 1], [8, 27, 35, 1], [8, 29, 37, 1], [8, 31, 39, 1], [8, 33, 41, 1], [8, 35, 43, 1], [8, 37, 45, 1], [8, 39, 1, 1], [8, 41, 3, 1], [8, 43, 5, 1], [8, 45, 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, 33, 1], [10, 25, 35, 1], [10, 27, 37, 1], [10, 29, 39, 1], [10, 31, 41, 1], [10, 33, 43, 1], [10, 35, 45, 1], [10, 37, 1, 1], [10, 39, 3, 1], [10, 41, 5, 1], [10, 43, 7, 1], [10, 45, 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, 33, 1], [12, 23, 35, 1], [12, 25, 37, 1], [12, 27, 39, 1], [12, 29, 41, 1], [12, 31, 43, 1], [12, 33, 45, 1], [12, 35, 1, 1], [12, 37, 3, 1], [12, 39, 5, 1], [12, 41, 7, 1], [12, 43, 9, 1], [12, 45, 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, 33, 1], [14, 21, 35, 1], [14, 23, 37, 1], [14, 25, 39, 1], [14, 27, 41, 1], [14, 29, 43, 1], [14, 31, 45, 1], [14, 33, 1, 1], [14, 35, 3, 1], [14, 37, 5, 1], [14, 39, 7, 1], [14, 41, 9, 1], [14, 43, 11, 1], [14, 45, 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, 33, 1], [16, 19, 35, 1], [16, 21, 37, 1], [16, 23, 39, 1], [16, 25, 41, 1], [16, 27, 43, 1], [16, 29, 45, 1], [16, 31, 1, 1], [16, 33, 3, 1], [16, 35, 5, 1], [16, 37, 7, 1], [16, 39, 9, 1], [16, 41, 11, 1], [16, 43, 13, 1], [16, 45, 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, 33, 1], [18, 17, 35, 1], [18, 19, 37, 1], [18, 21, 39, 1], [18, 23, 41, 1], [18, 25, 43, 1], [18, 27, 45, 1], [18, 29, 1, 1], [18, 31, 3, 1], [18, 33, 5, 1], [18, 35, 7, 1], [18, 37, 9, 1], [18, 39, 11, 1], [18, 41, 13, 1], [18, 43, 15, 1], [18, 45, 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, 33, 1], [20, 15, 35, 1], [20, 17, 37, 1], [20, 19, 39, 1], [20, 21, 41, 1], [20, 23, 43, 1], [20, 25, 45, 1], [20, 27, 1, 1], [20, 29, 3, 1], [20, 31, 5, 1], [20, 33, 7, 1], [20, 35, 9, 1], [20, 37, 11, 1], [20, 39, 13, 1], [20, 41, 15, 1], [20, 43, 17, 1], [20, 45, 19, 1], [22, 1, 23, 1], [22, 3, 25, 1], [22, 5, 27, 1], [22, 7, 29, 1], [22, 9, 31, 1], [22, 11, 33, 1], [22, 13, 35, 1], [22, 15, 37, 1], [22, 17, 39, 1], [22, 19, 41, 1], [22, 21, 43, 1], [22, 23, 45, 1], [22, 25, 1, 1], [22, 27, 3, 1], [22, 29, 5, 1], [22, 31, 7, 1], [22, 33, 9, 1], [22, 35, 11, 1], [22, 37, 13, 1], [22, 39, 15, 1], [22, 41, 17, 1], [22, 43, 19, 1], [22, 45, 21, 1], [24, 1, 25, 1], [24, 3, 27, 1], [24, 5, 29, 1], [24, 7, 31, 1], [24, 9, 33, 1], [24, 11, 35, 1], [24, 13, 37, 1], [24, 15, 39, 1], [24, 17, 41, 1], [24, 19, 43, 1], [24, 21, 45, 1], [24, 23, 1, 1], [24, 25, 3, 1], [24, 27, 5, 1], [24, 29, 7, 1], [24, 31, 9, 1], [24, 33, 11, 1], [24, 35, 13, 1], [24, 37, 15, 1], [24, 39, 17, 1], [24, 41, 19, 1], [24, 43, 21, 1], [24, 45, 23, 1], [26, 1, 27, 1], [26, 3, 29, 1], [26, 5, 31, 1], [26, 7, 33, 1], [26, 9, 35, 1], [26, 11, 37, 1], [26, 13, 39, 1], [26, 15, 41, 1], [26, 17, 43, 1], [26, 19, 45, 1], [26, 21, 1, 1], [26, 23, 3, 1], [26, 25, 5, 1], [26, 27, 7, 1], [26, 29, 9, 1], [26, 31, 11, 1], [26, 33, 13, 1], [26, 35, 15, 1], [26, 37, 17, 1], [26, 39, 19, 1], [26, 41, 21, 1], [26, 43, 23, 1], [26, 45, 25, 1], [28, 1, 29, 1], [28, 3, 31, 1], [28, 5, 33, 1], [28, 7, 35, 1], [28, 9, 37, 1], [28, 11, 39, 1], [28, 13, 41, 1], [28, 15, 43, 1], [28, 17, 45, 1], [28, 19, 1, 1], [28, 21, 3, 1], [28, 23, 5, 1], [28, 25, 7, 1], [28, 27, 9, 1], [28, 29, 11, 1], [28, 31, 13, 1], [28, 33, 15, 1], [28, 35, 17, 1], [28, 37, 19, 1], [28, 39, 21, 1], [28, 41, 23, 1], [28, 43, 25, 1], [28, 45, 27, 1], [30, 1, 31, 1], [30, 3, 33, 1], [30, 5, 35, 1], [30, 7, 37, 1], [30, 9, 39, 1], [30, 11, 41, 1], [30, 13, 43, 1], [30, 15, 45, 1], [30, 17, 1, 1], [30, 19, 3, 1], [30, 21, 5, 1], [30, 23, 7, 1], [30, 25, 9, 1], [30, 27, 11, 1], [30, 29, 13, 1], [30, 31, 15, 1], [30, 33, 17, 1], [30, 35, 19, 1], [30, 37, 21, 1], [30, 39, 23, 1], [30, 41, 25, 1], [30, 43, 27, 1], [30, 45, 29, 1], [32, 1, 33, 1], [32, 3, 35, 1], [32, 5, 37, 1], [32, 7, 39, 1], [32, 9, 41, 1], [32, 11, 43, 1], [32, 13, 45, 1], [32, 15, 1, 1], [32, 17, 3, 1], [32, 19, 5, 1], [32, 21, 7, 1], [32, 23, 9, 1], [32, 25, 11, 1], [32, 27, 13, 1], [32, 29, 15, 1], [32, 31, 17, 1], [32, 33, 19, 1], [32, 35, 21, 1], [32, 37, 23, 1], [32, 39, 25, 1], [32, 41, 27, 1], [32, 43, 29, 1], [32, 45, 31, 1], [34, 1, 35, 1], [34, 3, 37, 1], [34, 5, 39, 1], [34, 7, 41, 1], [34, 9, 43, 1], [34, 11, 45, 1], [34, 13, 1, 1], [34, 15, 3, 1], [34, 17, 5, 1], [34, 19, 7, 1], [34, 21, 9, 1], [34, 23, 11, 1], [34, 25, 13, 1], [34, 27, 15, 1], [34, 29, 17, 1], [34, 31, 19, 1], [34, 33, 21, 1], [34, 35, 23, 1], [34, 37, 25, 1], [34, 39, 27, 1], [34, 41, 29, 1], [34, 43, 31, 1], [34, 45, 33, 1], [36, 1, 37, 1], [36, 3, 39, 1], [36, 5, 41, 1], [36, 7, 43, 1], [36, 9, 45, 1], [36, 11, 1, 1], [36, 13, 3, 1], [36, 15, 5, 1], [36, 17, 7, 1], [36, 19, 9, 1], [36, 21, 11, 1], [36, 23, 13, 1], [36, 25, 15, 1], [36, 27, 17, 1], [36, 29, 19, 1], [36, 31, 21, 1], [36, 33, 23, 1], [36, 35, 25, 1], [36, 37, 27, 1], [36, 39, 29, 1], [36, 41, 31, 1], [36, 43, 33, 1], [36, 45, 35, 1], [38, 1, 39, 1], [38, 3, 41, 1], [38, 5, 43, 1], [38, 7, 45, 1], [38, 9, 1, 1], [38, 11, 3, 1], [38, 13, 5, 1], [38, 15, 7, 1], [38, 17, 9, 1], [38, 19, 11, 1], [38, 21, 13, 1], [38, 23, 15, 1], [38, 25, 17, 1], [38, 27, 19, 1], [38, 29, 21, 1], [38, 31, 23, 1], [38, 33, 25, 1], [38, 35, 27, 1], [38, 37, 29, 1], [38, 39, 31, 1], [38, 41, 33, 1], [38, 43, 35, 1], [38, 45, 37, 1], [40, 1, 41, 1], [40, 3, 43, 1], [40, 5, 45, 1], [40, 7, 1, 1], [40, 9, 3, 1], [40, 11, 5, 1], [40, 13, 7, 1], [40, 15, 9, 1], [40, 17, 11, 1], [40, 19, 13, 1], [40, 21, 15, 1], [40, 23, 17, 1], [40, 25, 19, 1], [40, 27, 21, 1], [40, 29, 23, 1], [40, 31, 25, 1], [40, 33, 27, 1], [40, 35, 29, 1], [40, 37, 31, 1], [40, 39, 33, 1], [40, 41, 35, 1], [40, 43, 37, 1], [40, 45, 39, 1], [42, 1, 43, 1], [42, 3, 45, 1], [42, 5, 1, 1], [42, 7, 3, 1], [42, 9, 5, 1], [42, 11, 7, 1], [42, 13, 9, 1], [42, 15, 11, 1], [42, 17, 13, 1], [42, 19, 15, 1], [42, 21, 17, 1], [42, 23, 19, 1], [42, 25, 21, 1], [42, 27, 23, 1], [42, 29, 25, 1], [42, 31, 27, 1], [42, 33, 29, 1], [42, 35, 31, 1], [42, 37, 33, 1], [42, 39, 35, 1], [42, 41, 37, 1], [42, 43, 39, 1], [42, 45, 41, 1], [44, 1, 45, 1], [44, 3, 1, 1], [44, 5, 3, 1], [44, 7, 5, 1], [44, 9, 7, 1], [44, 11, 9, 1], [44, 13, 11, 1], [44, 15, 13, 1], [44, 17, 15, 1], [44, 19, 17, 1], [44, 21, 19, 1], [44, 23, 21, 1], [44, 25, 23, 1], [44, 27, 25, 1], [44, 29, 27, 1], [44, 31, 29, 1], [44, 33, 31, 1], [44, 35, 33, 1], [44, 37, 35, 1], [44, 39, 37, 1], [44, 41, 39, 1], [44, 43, 41, 1], [44, 45, 43, 1]], [1], 是]", "relation": "implies", "rule_id": "Quotient_EVERY_coset_pair_ALL_representative_products", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [1, 0, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44]]", "after": "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], [1, 32, 33, 1], [1, 34, 35, 1], [1, 36, 37, 1], [1, 38, 39, 1], [1, 40, 41, 1], [1, 42, 43, 1], [1, 44, 45, 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, 33, 1], [3, 32, 35, 1], [3, 34, 37, 1], [3, 36, 39, 1], [3, 38, 41, 1], [3, 40, 43, 1], [3, 42, 45, 1], [3, 44, 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, 33, 1], [5, 30, 35, 1], [5, 32, 37, 1], [5, 34, 39, 1], [5, 36, 41, 1], [5, 38, 43, 1], [5, 40, 45, 1], [5, 42, 1, 1], [5, 44, 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, 33, 1], [7, 28, 35, 1], [7, 30, 37, 1], [7, 32, 39, 1], [7, 34, 41, 1], [7, 36, 43, 1], [7, 38, 45, 1], [7, 40, 1, 1], [7, 42, 3, 1], [7, 44, 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, 33, 1], [9, 26, 35, 1], [9, 28, 37, 1], [9, 30, 39, 1], [9, 32, 41, 1], [9, 34, 43, 1], [9, 36, 45, 1], [9, 38, 1, 1], [9, 40, 3, 1], [9, 42, 5, 1], [9, 44, 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, 33, 1], [11, 24, 35, 1], [11, 26, 37, 1], [11, 28, 39, 1], [11, 30, 41, 1], [11, 32, 43, 1], [11, 34, 45, 1], [11, 36, 1, 1], [11, 38, 3, 1], [11, 40, 5, 1], [11, 42, 7, 1], [11, 44, 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, 33, 1], [13, 22, 35, 1], [13, 24, 37, 1], [13, 26, 39, 1], [13, 28, 41, 1], [13, 30, 43, 1], [13, 32, 45, 1], [13, 34, 1, 1], [13, 36, 3, 1], [13, 38, 5, 1], [13, 40, 7, 1], [13, 42, 9, 1], [13, 44, 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, 33, 1], [15, 20, 35, 1], [15, 22, 37, 1], [15, 24, 39, 1], [15, 26, 41, 1], [15, 28, 43, 1], [15, 30, 45, 1], [15, 32, 1, 1], [15, 34, 3, 1], [15, 36, 5, 1], [15, 38, 7, 1], [15, 40, 9, 1], [15, 42, 11, 1], [15, 44, 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, 33, 1], [17, 18, 35, 1], [17, 20, 37, 1], [17, 22, 39, 1], [17, 24, 41, 1], [17, 26, 43, 1], [17, 28, 45, 1], [17, 30, 1, 1], [17, 32, 3, 1], [17, 34, 5, 1], [17, 36, 7, 1], [17, 38, 9, 1], [17, 40, 11, 1], [17, 42, 13, 1], [17, 44, 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, 33, 1], [19, 16, 35, 1], [19, 18, 37, 1], [19, 20, 39, 1], [19, 22, 41, 1], [19, 24, 43, 1], [19, 26, 45, 1], [19, 28, 1, 1], [19, 30, 3, 1], [19, 32, 5, 1], [19, 34, 7, 1], [19, 36, 9, 1], [19, 38, 11, 1], [19, 40, 13, 1], [19, 42, 15, 1], [19, 44, 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, 33, 1], [21, 14, 35, 1], [21, 16, 37, 1], [21, 18, 39, 1], [21, 20, 41, 1], [21, 22, 43, 1], [21, 24, 45, 1], [21, 26, 1, 1], [21, 28, 3, 1], [21, 30, 5, 1], [21, 32, 7, 1], [21, 34, 9, 1], [21, 36, 11, 1], [21, 38, 13, 1], [21, 40, 15, 1], [21, 42, 17, 1], [21, 44, 19, 1], [23, 0, 23, 1], [23, 2, 25, 1], [23, 4, 27, 1], [23, 6, 29, 1], [23, 8, 31, 1], [23, 10, 33, 1], [23, 12, 35, 1], [23, 14, 37, 1], [23, 16, 39, 1], [23, 18, 41, 1], [23, 20, 43, 1], [23, 22, 45, 1], [23, 24, 1, 1], [23, 26, 3, 1], [23, 28, 5, 1], [23, 30, 7, 1], [23, 32, 9, 1], [23, 34, 11, 1], [23, 36, 13, 1], [23, 38, 15, 1], [23, 40, 17, 1], [23, 42, 19, 1], [23, 44, 21, 1], [25, 0, 25, 1], [25, 2, 27, 1], [25, 4, 29, 1], [25, 6, 31, 1], [25, 8, 33, 1], [25, 10, 35, 1], [25, 12, 37, 1], [25, 14, 39, 1], [25, 16, 41, 1], [25, 18, 43, 1], [25, 20, 45, 1], [25, 22, 1, 1], [25, 24, 3, 1], [25, 26, 5, 1], [25, 28, 7, 1], [25, 30, 9, 1], [25, 32, 11, 1], [25, 34, 13, 1], [25, 36, 15, 1], [25, 38, 17, 1], [25, 40, 19, 1], [25, 42, 21, 1], [25, 44, 23, 1], [27, 0, 27, 1], [27, 2, 29, 1], [27, 4, 31, 1], [27, 6, 33, 1], [27, 8, 35, 1], [27, 10, 37, 1], [27, 12, 39, 1], [27, 14, 41, 1], [27, 16, 43, 1], [27, 18, 45, 1], [27, 20, 1, 1], [27, 22, 3, 1], [27, 24, 5, 1], [27, 26, 7, 1], [27, 28, 9, 1], [27, 30, 11, 1], [27, 32, 13, 1], [27, 34, 15, 1], [27, 36, 17, 1], [27, 38, 19, 1], [27, 40, 21, 1], [27, 42, 23, 1], [27, 44, 25, 1], [29, 0, 29, 1], [29, 2, 31, 1], [29, 4, 33, 1], [29, 6, 35, 1], [29, 8, 37, 1], [29, 10, 39, 1], [29, 12, 41, 1], [29, 14, 43, 1], [29, 16, 45, 1], [29, 18, 1, 1], [29, 20, 3, 1], [29, 22, 5, 1], [29, 24, 7, 1], [29, 26, 9, 1], [29, 28, 11, 1], [29, 30, 13, 1], [29, 32, 15, 1], [29, 34, 17, 1], [29, 36, 19, 1], [29, 38, 21, 1], [29, 40, 23, 1], [29, 42, 25, 1], [29, 44, 27, 1], [31, 0, 31, 1], [31, 2, 33, 1], [31, 4, 35, 1], [31, 6, 37, 1], [31, 8, 39, 1], [31, 10, 41, 1], [31, 12, 43, 1], [31, 14, 45, 1], [31, 16, 1, 1], [31, 18, 3, 1], [31, 20, 5, 1], [31, 22, 7, 1], [31, 24, 9, 1], [31, 26, 11, 1], [31, 28, 13, 1], [31, 30, 15, 1], [31, 32, 17, 1], [31, 34, 19, 1], [31, 36, 21, 1], [31, 38, 23, 1], [31, 40, 25, 1], [31, 42, 27, 1], [31, 44, 29, 1], [33, 0, 33, 1], [33, 2, 35, 1], [33, 4, 37, 1], [33, 6, 39, 1], [33, 8, 41, 1], [33, 10, 43, 1], [33, 12, 45, 1], [33, 14, 1, 1], [33, 16, 3, 1], [33, 18, 5, 1], [33, 20, 7, 1], [33, 22, 9, 1], [33, 24, 11, 1], [33, 26, 13, 1], [33, 28, 15, 1], [33, 30, 17, 1], [33, 32, 19, 1], [33, 34, 21, 1], [33, 36, 23, 1], [33, 38, 25, 1], [33, 40, 27, 1], [33, 42, 29, 1], [33, 44, 31, 1], [35, 0, 35, 1], [35, 2, 37, 1], [35, 4, 39, 1], [35, 6, 41, 1], [35, 8, 43, 1], [35, 10, 45, 1], [35, 12, 1, 1], [35, 14, 3, 1], [35, 16, 5, 1], [35, 18, 7, 1], [35, 20, 9, 1], [35, 22, 11, 1], [35, 24, 13, 1], [35, 26, 15, 1], [35, 28, 17, 1], [35, 30, 19, 1], [35, 32, 21, 1], [35, 34, 23, 1], [35, 36, 25, 1], [35, 38, 27, 1], [35, 40, 29, 1], [35, 42, 31, 1], [35, 44, 33, 1], [37, 0, 37, 1], [37, 2, 39, 1], [37, 4, 41, 1], [37, 6, 43, 1], [37, 8, 45, 1], [37, 10, 1, 1], [37, 12, 3, 1], [37, 14, 5, 1], [37, 16, 7, 1], [37, 18, 9, 1], [37, 20, 11, 1], [37, 22, 13, 1], [37, 24, 15, 1], [37, 26, 17, 1], [37, 28, 19, 1], [37, 30, 21, 1], [37, 32, 23, 1], [37, 34, 25, 1], [37, 36, 27, 1], [37, 38, 29, 1], [37, 40, 31, 1], [37, 42, 33, 1], [37, 44, 35, 1], [39, 0, 39, 1], [39, 2, 41, 1], [39, 4, 43, 1], [39, 6, 45, 1], [39, 8, 1, 1], [39, 10, 3, 1], [39, 12, 5, 1], [39, 14, 7, 1], [39, 16, 9, 1], [39, 18, 11, 1], [39, 20, 13, 1], [39, 22, 15, 1], [39, 24, 17, 1], [39, 26, 19, 1], [39, 28, 21, 1], [39, 30, 23, 1], [39, 32, 25, 1], [39, 34, 27, 1], [39, 36, 29, 1], [39, 38, 31, 1], [39, 40, 33, 1], [39, 42, 35, 1], [39, 44, 37, 1], [41, 0, 41, 1], [41, 2, 43, 1], [41, 4, 45, 1], [41, 6, 1, 1], [41, 8, 3, 1], [41, 10, 5, 1], [41, 12, 7, 1], [41, 14, 9, 1], [41, 16, 11, 1], [41, 18, 13, 1], [41, 20, 15, 1], [41, 22, 17, 1], [41, 24, 19, 1], [41, 26, 21, 1], [41, 28, 23, 1], [41, 30, 25, 1], [41, 32, 27, 1], [41, 34, 29, 1], [41, 36, 31, 1], [41, 38, 33, 1], [41, 40, 35, 1], [41, 42, 37, 1], [41, 44, 39, 1], [43, 0, 43, 1], [43, 2, 45, 1], [43, 4, 1, 1], [43, 6, 3, 1], [43, 8, 5, 1], [43, 10, 7, 1], [43, 12, 9, 1], [43, 14, 11, 1], [43, 16, 13, 1], [43, 18, 15, 1], [43, 20, 17, 1], [43, 22, 19, 1], [43, 24, 21, 1], [43, 26, 23, 1], [43, 28, 25, 1], [43, 30, 27, 1], [43, 32, 29, 1], [43, 34, 31, 1], [43, 36, 33, 1], [43, 38, 35, 1], [43, 40, 37, 1], [43, 42, 39, 1], [43, 44, 41, 1], [45, 0, 45, 1], [45, 2, 1, 1], [45, 4, 3, 1], [45, 6, 5, 1], [45, 8, 7, 1], [45, 10, 9, 1], [45, 12, 11, 1], [45, 14, 13, 1], [45, 16, 15, 1], [45, 18, 17, 1], [45, 20, 19, 1], [45, 22, 21, 1], [45, 24, 23, 1], [45, 26, 25, 1], [45, 28, 27, 1], [45, 30, 29, 1], [45, 32, 31, 1], [45, 34, 33, 1], [45, 36, 35, 1], [45, 38, 37, 1], [45, 40, 39, 1], [45, 42, 41, 1], [45, 44, 43, 1]], [1], 是]", "relation": "implies", "rule_id": "Quotient_EVERY_coset_pair_ALL_representative_products", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [1, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]]", "after": "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, 32, 0], [1, 33, 34, 0], [1, 35, 36, 0], [1, 37, 38, 0], [1, 39, 40, 0], [1, 41, 42, 0], [1, 43, 44, 0], [1, 45, 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, 32, 0], [3, 31, 34, 0], [3, 33, 36, 0], [3, 35, 38, 0], [3, 37, 40, 0], [3, 39, 42, 0], [3, 41, 44, 0], [3, 43, 0, 0], [3, 45, 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, 32, 0], [5, 29, 34, 0], [5, 31, 36, 0], [5, 33, 38, 0], [5, 35, 40, 0], [5, 37, 42, 0], [5, 39, 44, 0], [5, 41, 0, 0], [5, 43, 2, 0], [5, 45, 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, 32, 0], [7, 27, 34, 0], [7, 29, 36, 0], [7, 31, 38, 0], [7, 33, 40, 0], [7, 35, 42, 0], [7, 37, 44, 0], [7, 39, 0, 0], [7, 41, 2, 0], [7, 43, 4, 0], [7, 45, 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, 32, 0], [9, 25, 34, 0], [9, 27, 36, 0], [9, 29, 38, 0], [9, 31, 40, 0], [9, 33, 42, 0], [9, 35, 44, 0], [9, 37, 0, 0], [9, 39, 2, 0], [9, 41, 4, 0], [9, 43, 6, 0], [9, 45, 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, 32, 0], [11, 23, 34, 0], [11, 25, 36, 0], [11, 27, 38, 0], [11, 29, 40, 0], [11, 31, 42, 0], [11, 33, 44, 0], [11, 35, 0, 0], [11, 37, 2, 0], [11, 39, 4, 0], [11, 41, 6, 0], [11, 43, 8, 0], [11, 45, 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, 32, 0], [13, 21, 34, 0], [13, 23, 36, 0], [13, 25, 38, 0], [13, 27, 40, 0], [13, 29, 42, 0], [13, 31, 44, 0], [13, 33, 0, 0], [13, 35, 2, 0], [13, 37, 4, 0], [13, 39, 6, 0], [13, 41, 8, 0], [13, 43, 10, 0], [13, 45, 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, 32, 0], [15, 19, 34, 0], [15, 21, 36, 0], [15, 23, 38, 0], [15, 25, 40, 0], [15, 27, 42, 0], [15, 29, 44, 0], [15, 31, 0, 0], [15, 33, 2, 0], [15, 35, 4, 0], [15, 37, 6, 0], [15, 39, 8, 0], [15, 41, 10, 0], [15, 43, 12, 0], [15, 45, 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, 32, 0], [17, 17, 34, 0], [17, 19, 36, 0], [17, 21, 38, 0], [17, 23, 40, 0], [17, 25, 42, 0], [17, 27, 44, 0], [17, 29, 0, 0], [17, 31, 2, 0], [17, 33, 4, 0], [17, 35, 6, 0], [17, 37, 8, 0], [17, 39, 10, 0], [17, 41, 12, 0], [17, 43, 14, 0], [17, 45, 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, 32, 0], [19, 15, 34, 0], [19, 17, 36, 0], [19, 19, 38, 0], [19, 21, 40, 0], [19, 23, 42, 0], [19, 25, 44, 0], [19, 27, 0, 0], [19, 29, 2, 0], [19, 31, 4, 0], [19, 33, 6, 0], [19, 35, 8, 0], [19, 37, 10, 0], [19, 39, 12, 0], [19, 41, 14, 0], [19, 43, 16, 0], [19, 45, 18, 0], [21, 1, 22, 0], [21, 3, 24, 0], [21, 5, 26, 0], [21, 7, 28, 0], [21, 9, 30, 0], [21, 11, 32, 0], [21, 13, 34, 0], [21, 15, 36, 0], [21, 17, 38, 0], [21, 19, 40, 0], [21, 21, 42, 0], [21, 23, 44, 0], [21, 25, 0, 0], [21, 27, 2, 0], [21, 29, 4, 0], [21, 31, 6, 0], [21, 33, 8, 0], [21, 35, 10, 0], [21, 37, 12, 0], [21, 39, 14, 0], [21, 41, 16, 0], [21, 43, 18, 0], [21, 45, 20, 0], [23, 1, 24, 0], [23, 3, 26, 0], [23, 5, 28, 0], [23, 7, 30, 0], [23, 9, 32, 0], [23, 11, 34, 0], [23, 13, 36, 0], [23, 15, 38, 0], [23, 17, 40, 0], [23, 19, 42, 0], [23, 21, 44, 0], [23, 23, 0, 0], [23, 25, 2, 0], [23, 27, 4, 0], [23, 29, 6, 0], [23, 31, 8, 0], [23, 33, 10, 0], [23, 35, 12, 0], [23, 37, 14, 0], [23, 39, 16, 0], [23, 41, 18, 0], [23, 43, 20, 0], [23, 45, 22, 0], [25, 1, 26, 0], [25, 3, 28, 0], [25, 5, 30, 0], [25, 7, 32, 0], [25, 9, 34, 0], [25, 11, 36, 0], [25, 13, 38, 0], [25, 15, 40, 0], [25, 17, 42, 0], [25, 19, 44, 0], [25, 21, 0, 0], [25, 23, 2, 0], [25, 25, 4, 0], [25, 27, 6, 0], [25, 29, 8, 0], [25, 31, 10, 0], [25, 33, 12, 0], [25, 35, 14, 0], [25, 37, 16, 0], [25, 39, 18, 0], [25, 41, 20, 0], [25, 43, 22, 0], [25, 45, 24, 0], [27, 1, 28, 0], [27, 3, 30, 0], [27, 5, 32, 0], [27, 7, 34, 0], [27, 9, 36, 0], [27, 11, 38, 0], [27, 13, 40, 0], [27, 15, 42, 0], [27, 17, 44, 0], [27, 19, 0, 0], [27, 21, 2, 0], [27, 23, 4, 0], [27, 25, 6, 0], [27, 27, 8, 0], [27, 29, 10, 0], [27, 31, 12, 0], [27, 33, 14, 0], [27, 35, 16, 0], [27, 37, 18, 0], [27, 39, 20, 0], [27, 41, 22, 0], [27, 43, 24, 0], [27, 45, 26, 0], [29, 1, 30, 0], [29, 3, 32, 0], [29, 5, 34, 0], [29, 7, 36, 0], [29, 9, 38, 0], [29, 11, 40, 0], [29, 13, 42, 0], [29, 15, 44, 0], [29, 17, 0, 0], [29, 19, 2, 0], [29, 21, 4, 0], [29, 23, 6, 0], [29, 25, 8, 0], [29, 27, 10, 0], [29, 29, 12, 0], [29, 31, 14, 0], [29, 33, 16, 0], [29, 35, 18, 0], [29, 37, 20, 0], [29, 39, 22, 0], [29, 41, 24, 0], [29, 43, 26, 0], [29, 45, 28, 0], [31, 1, 32, 0], [31, 3, 34, 0], [31, 5, 36, 0], [31, 7, 38, 0], [31, 9, 40, 0], [31, 11, 42, 0], [31, 13, 44, 0], [31, 15, 0, 0], [31, 17, 2, 0], [31, 19, 4, 0], [31, 21, 6, 0], [31, 23, 8, 0], [31, 25, 10, 0], [31, 27, 12, 0], [31, 29, 14, 0], [31, 31, 16, 0], [31, 33, 18, 0], [31, 35, 20, 0], [31, 37, 22, 0], [31, 39, 24, 0], [31, 41, 26, 0], [31, 43, 28, 0], [31, 45, 30, 0], [33, 1, 34, 0], [33, 3, 36, 0], [33, 5, 38, 0], [33, 7, 40, 0], [33, 9, 42, 0], [33, 11, 44, 0], [33, 13, 0, 0], [33, 15, 2, 0], [33, 17, 4, 0], [33, 19, 6, 0], [33, 21, 8, 0], [33, 23, 10, 0], [33, 25, 12, 0], [33, 27, 14, 0], [33, 29, 16, 0], [33, 31, 18, 0], [33, 33, 20, 0], [33, 35, 22, 0], [33, 37, 24, 0], [33, 39, 26, 0], [33, 41, 28, 0], [33, 43, 30, 0], [33, 45, 32, 0], [35, 1, 36, 0], [35, 3, 38, 0], [35, 5, 40, 0], [35, 7, 42, 0], [35, 9, 44, 0], [35, 11, 0, 0], [35, 13, 2, 0], [35, 15, 4, 0], [35, 17, 6, 0], [35, 19, 8, 0], [35, 21, 10, 0], [35, 23, 12, 0], [35, 25, 14, 0], [35, 27, 16, 0], [35, 29, 18, 0], [35, 31, 20, 0], [35, 33, 22, 0], [35, 35, 24, 0], [35, 37, 26, 0], [35, 39, 28, 0], [35, 41, 30, 0], [35, 43, 32, 0], [35, 45, 34, 0], [37, 1, 38, 0], [37, 3, 40, 0], [37, 5, 42, 0], [37, 7, 44, 0], [37, 9, 0, 0], [37, 11, 2, 0], [37, 13, 4, 0], [37, 15, 6, 0], [37, 17, 8, 0], [37, 19, 10, 0], [37, 21, 12, 0], [37, 23, 14, 0], [37, 25, 16, 0], [37, 27, 18, 0], [37, 29, 20, 0], [37, 31, 22, 0], [37, 33, 24, 0], [37, 35, 26, 0], [37, 37, 28, 0], [37, 39, 30, 0], [37, 41, 32, 0], [37, 43, 34, 0], [37, 45, 36, 0], [39, 1, 40, 0], [39, 3, 42, 0], [39, 5, 44, 0], [39, 7, 0, 0], [39, 9, 2, 0], [39, 11, 4, 0], [39, 13, 6, 0], [39, 15, 8, 0], [39, 17, 10, 0], [39, 19, 12, 0], [39, 21, 14, 0], [39, 23, 16, 0], [39, 25, 18, 0], [39, 27, 20, 0], [39, 29, 22, 0], [39, 31, 24, 0], [39, 33, 26, 0], [39, 35, 28, 0], [39, 37, 30, 0], [39, 39, 32, 0], [39, 41, 34, 0], [39, 43, 36, 0], [39, 45, 38, 0], [41, 1, 42, 0], [41, 3, 44, 0], [41, 5, 0, 0], [41, 7, 2, 0], [41, 9, 4, 0], [41, 11, 6, 0], [41, 13, 8, 0], [41, 15, 10, 0], [41, 17, 12, 0], [41, 19, 14, 0], [41, 21, 16, 0], [41, 23, 18, 0], [41, 25, 20, 0], [41, 27, 22, 0], [41, 29, 24, 0], [41, 31, 26, 0], [41, 33, 28, 0], [41, 35, 30, 0], [41, 37, 32, 0], [41, 39, 34, 0], [41, 41, 36, 0], [41, 43, 38, 0], [41, 45, 40, 0], [43, 1, 44, 0], [43, 3, 0, 0], [43, 5, 2, 0], [43, 7, 4, 0], [43, 9, 6, 0], [43, 11, 8, 0], [43, 13, 10, 0], [43, 15, 12, 0], [43, 17, 14, 0], [43, 19, 16, 0], [43, 21, 18, 0], [43, 23, 20, 0], [43, 25, 22, 0], [43, 27, 24, 0], [43, 29, 26, 0], [43, 31, 28, 0], [43, 33, 30, 0], [43, 35, 32, 0], [43, 37, 34, 0], [43, 39, 36, 0], [43, 41, 38, 0], [43, 43, 40, 0], [43, 45, 42, 0], [45, 1, 0, 0], [45, 3, 2, 0], [45, 5, 4, 0], [45, 7, 6, 0], [45, 9, 8, 0], [45, 11, 10, 0], [45, 13, 12, 0], [45, 15, 14, 0], [45, 17, 16, 0], [45, 19, 18, 0], [45, 21, 20, 0], [45, 23, 22, 0], [45, 25, 24, 0], [45, 27, 26, 0], [45, 29, 28, 0], [45, 31, 30, 0], [45, 33, 32, 0], [45, 35, 34, 0], [45, 37, 36, 0], [45, 39, 38, 0], [45, 41, 40, 0], [45, 43, 42, 0], [45, 45, 44, 0]], [0], 是]", "relation": "implies", "rule_id": "Quotient_EVERY_coset_pair_ALL_representative_products", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], 是, [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]]]", "after": "derived = [[[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], 是, 否]", "relation": "implies", "rule_id": "Quotient_GENERAL_normality_representative_replacement_and_group_laws", "conditions": [], "explanation": { "zh-Hant": "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。", "en": "Replacing a by ah and b by bk yields(ah)(bk)=ab(b^-1hb)k in abH,using normality for representative independence.Associativity descends from G,eH is identity and a^-1H inverse.The fixed No withholds automatic commutativity;it does not call a particular cyclic quotient noncommutative or invoke first isomorphism." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ [ [ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44 ], [ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45 ] ], [ [ 0, 0, [ 0 ], true ], [ 0, 1, [ 1 ], true ], [ 1, 0, [ 1 ], true ], [ 1, 1, [ 0 ], true ] ], true, "every_representative_pair_lands_in_one_declared_product_coset" ] }, "conclusion": { "canonical_value": "([[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]], [[0, 0, [0], 是], [0, 1, [1], 是], [1, 0, [1], 是], [1, 1, [0], 是]], 是, every_representative_pair_lands_in_one_declared_product_coset)", "explanation": { "zh-Hant": "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。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "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。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261114-r02-finite-quotient-group-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261114-r02-finite-quotient-group-foundations/questions.jsonl", "batch_sha256": "0398c3196e77c77c239fa72091680c1fa5e961a520dc298756d2d4591c79a2ff", "original_line": 72, "original_line_sha256": "40816dbc4bd222de4413fb2b174a54a51629ca16d93ff64937ae3ac06fd3a039", "question_compact_sha256": "e90cfd20c5b0e62ee87d48c51645aabec11ee5f0dc940992d7e875973f8a59a2", "english_row_compact_sha256": "5c750b8bbd2364dfbf0f67ebcda16739a3837fdbd6c9f362a4b48ac6071454e0", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261114-r02-finite-quotient-group-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261114-r02-finite-quotient-group-foundations.en.jsonl" }, "mko_bindings": { "mko-finite-quotient-group": { "sha256": "f3e42a5b51685f805af8727447418b3b158bf4c512da0e78fd2804f7a04bba24", "english_sha256": "036ebd3e0c66d661a3e151b97f378a2d60aae4fc019e98ba7d711292436a7f89", "original_path": "public/data/mko/finite-quotient-group.json", "english_path": "public/data/i18n/editions/mko-finite-quotient-group.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "991c73df066352bfcfb1142a2492e104cd1d20dafe7e9ecea8cff26a4f9d8e50", "original_candidate_sha256": "f52cc1c1dc75c42e284c861feaa6986cfa4c1f65ed9d016af5b28f326d52a3e4", "original_candidate_line_sha256": "f52cc1c1dc75c42e284c861feaa6986cfa4c1f65ed9d016af5b28f326d52a3e4" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-010-p50", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3", "question_first_witness_sha256": "f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 10, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "1962a2b5c29e57971ad083df9e85d34c4f70e051f1f689241e4f70d2d37a75e4", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }