OCME中英數學知識與推導English

正規語言Pumping引理基礎 · 脈衝 R07

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations · 2026-10-15 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

欄序與來源綁定資料

1. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":1,"word":["a","b"],"x_length":0,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(1, [a, b], 2, 0, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(1, [a, b], 2, 0, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

2. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":2,"word":["a","a","b"],"x_length":1,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(2, [a, a, b], 3, 1, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(2, [a, a, b], 3, 1, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

3. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":3,"word":["a","a","a","b"],"x_length":2,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(3, [a, a, a, b], 4, 2, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(3, [a, a, a, b], 4, 2, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

4. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":4,"word":["a","a","a","a","b"],"x_length":3,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(4, [a, a, a, a, b], 5, 3, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(4, [a, a, a, a, b], 5, 3, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

5. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":5,"word":["a","a","a","a","a","b"],"x_length":4,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(5, [a, a, a, a, a, b], 6, 4, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(5, [a, a, a, a, a, b], 6, 4, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

6. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":6,"word":["a","a","a","a","a","a","b"],"x_length":5,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(6, [a, a, a, a, a, a, b], 7, 5, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(6, [a, a, a, a, a, a, b], 7, 5, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

7. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":7,"word":["a","a","a","a","a","a","a","b"],"x_length":6,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(7, [a, a, a, a, a, a, a, b], 8, 6, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(7, [a, a, a, a, a, a, a, b], 8, 6, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

8. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":8,"word":["a","a","a","a","a","a","a","a","b"],"x_length":7,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(8, [a, a, a, a, a, a, a, a, b], 9, 7, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(8, [a, a, a, a, a, a, a, a, b], 9, 7, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

9. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":9,"word":["a","a","a","a","a","a","a","a","a","b"],"x_length":8,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(9, [a, a, a, a, a, a, a, a, a, b], 10, 8, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(9, [a, a, a, a, a, a, a, a, a, b], 10, 8, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

10. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":10,"word":["a","a","a","a","a","a","a","a","a","a","b"],"x_length":9,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(10, [a, a, a, a, a, a, a, a, a, a, b], 11, 9, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(10, [a, a, a, a, a, a, a, a, a, a, b], 11, 9, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

11. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":11,"word":["a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":10,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(11, [a, a, a, a, a, a, a, a, a, a, a, b], 12, 10, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(11, [a, a, a, a, a, a, a, a, a, a, a, b], 12, 10, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

12. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":12,"word":["a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":11,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(12, [a, a, a, a, a, a, a, a, a, a, a, a, b], 13, 11, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(12, [a, a, a, a, a, a, a, a, a, a, a, a, b], 13, 11, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

13. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":13,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":12,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(13, [a, a, a, a, a, a, a, a, a, a, a, a, a, b], 14, 12, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(13, [a, a, a, a, a, a, a, a, a, a, a, a, a, b], 14, 12, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

14. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":14,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":13,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(14, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 15, 13, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(14, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 15, 13, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

15. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":15,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":14,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(15, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 16, 14, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(15, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 16, 14, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

16. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":16,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":15,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(16, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 17, 15, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(16, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 17, 15, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

17. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":17,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":16,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(17, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 18, 16, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(17, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 18, 16, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

18. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":18,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":17,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(18, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 19, 17, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(18, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 19, 17, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

19. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":19,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":18,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(19, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 20, 18, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(19, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 20, 18, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

20. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":20,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":19,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 21, 19, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 21, 19, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

21. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":21,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":20,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(21, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 22, 20, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(21, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 22, 20, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

22. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":22,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":21,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(22, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 23, 21, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(22, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 23, 21, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

23. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":23,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":22,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(23, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 24, 22, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(23, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 24, 22, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

24. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":24,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":23,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(24, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 25, 23, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(24, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 25, 23, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

25. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={"p":25,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":24,"y_length":1}。請依 [p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order] 欄序回答。

查看答案與解釋

(25, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 26, 24, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)

Wave6第十二層核對=(25, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 26, 24, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-quantifier-audit · candidate

26. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":1,"word_length":1}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(1, 1, [0, 0], [0, 1, 0], 0, 1, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(1, 1, [0, 0], [0, 1, 0], 0, 1, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

27. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":2,"word_length":4}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(2, 4, [0, 1, 0], [0, 2, 0], 0, 2, 2, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(2, 4, [0, 1, 0], [0, 2, 0], 0, 2, 2, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

28. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":3,"word_length":9}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(3, 9, [0, 1, 2, 0], [0, 3, 0], 0, 3, 6, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(3, 9, [0, 1, 2, 0], [0, 3, 0], 0, 3, 6, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

29. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":4,"word_length":16}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(4, 16, [0, 1, 2, 3, 0], [0, 4, 0], 0, 4, 12, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(4, 16, [0, 1, 2, 3, 0], [0, 4, 0], 0, 4, 12, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

30. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":5,"word_length":5}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(5, 5, [0, 1, 2, 3, 4, 0], [0, 5, 0], 0, 5, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(5, 5, [0, 1, 2, 3, 4, 0], [0, 5, 0], 0, 5, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

31. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":6,"word_length":12}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(6, 12, [0, 1, 2, 3, 4, 5, 0], [0, 6, 0], 0, 6, 6, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(6, 12, [0, 1, 2, 3, 4, 5, 0], [0, 6, 0], 0, 6, 6, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

32. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":7,"word_length":21}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(7, 21, [0, 1, 2, 3, 4, 5, 6, 0], [0, 7, 0], 0, 7, 14, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(7, 21, [0, 1, 2, 3, 4, 5, 6, 0], [0, 7, 0], 0, 7, 14, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

33. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":8,"word_length":32}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(8, 32, [0, 1, 2, 3, 4, 5, 6, 7, 0], [0, 8, 0], 0, 8, 24, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(8, 32, [0, 1, 2, 3, 4, 5, 6, 7, 0], [0, 8, 0], 0, 8, 24, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

34. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":9,"word_length":9}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(9, 9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 0], [0, 9, 0], 0, 9, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(9, 9, [0, 1, 2, 3, 4, 5, 6, 7, 8, 0], [0, 9, 0], 0, 9, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

35. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":10,"word_length":20}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(10, 20, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [0, 10, 0], 0, 10, 10, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(10, 20, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [0, 10, 0], 0, 10, 10, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

36. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":11,"word_length":33}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(11, 33, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [0, 11, 0], 0, 11, 22, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(11, 33, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [0, 11, 0], 0, 11, 22, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

37. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":12,"word_length":48}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(12, 48, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [0, 12, 0], 0, 12, 36, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(12, 48, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [0, 12, 0], 0, 12, 36, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

38. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":13,"word_length":13}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(13, 13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 13, 0], 0, 13, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(13, 13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 13, 0], 0, 13, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

39. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":14,"word_length":28}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(14, 28, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [0, 14, 0], 0, 14, 14, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(14, 28, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [0, 14, 0], 0, 14, 14, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

40. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":15,"word_length":45}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(15, 45, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [0, 15, 0], 0, 15, 30, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(15, 45, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [0, 15, 0], 0, 15, 30, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

41. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":16,"word_length":64}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(16, 64, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [0, 16, 0], 0, 16, 48, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(16, 64, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [0, 16, 0], 0, 16, 48, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

42. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":17,"word_length":17}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(17, 17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [0, 17, 0], 0, 17, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(17, 17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [0, 17, 0], 0, 17, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

43. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":18,"word_length":36}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(18, 36, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [0, 18, 0], 0, 18, 18, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(18, 36, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [0, 18, 0], 0, 18, 18, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

44. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":19,"word_length":57}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(19, 57, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [0, 19, 0], 0, 19, 38, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(19, 57, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [0, 19, 0], 0, 19, 38, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

45. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":20,"word_length":80}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(20, 80, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [0, 20, 0], 0, 20, 60, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(20, 80, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [0, 20, 0], 0, 20, 60, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

46. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":21,"word_length":21}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(21, 21, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [0, 21, 0], 0, 21, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(21, 21, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [0, 21, 0], 0, 21, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

47. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":22,"word_length":44}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(22, 44, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [0, 22, 0], 0, 22, 22, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(22, 44, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [0, 22, 0], 0, 22, 22, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

48. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":23,"word_length":69}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(23, 69, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [0, 23, 0], 0, 23, 46, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(23, 69, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [0, 23, 0], 0, 23, 46, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

49. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":24,"word_length":96}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(24, 96, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [0, 24, 0], 0, 24, 72, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(24, 96, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [0, 24, 0], 0, 24, 72, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

50. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-dfa-loop-witness；資料={"modulus":25,"word_length":25}。請依 [modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified] 欄序回答。

查看答案與解釋

(25, 25, [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, 0], [0, 25, 0], 0, 25, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)

Wave6第十二層核對=(25, 25, [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, 0], [0, 25, 0], 0, 25, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-dfa-loop-witness · candidate

51. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":1}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(1, a^1b^1, [[0, 1, 0, a^0b^1, 否]], 1, 1, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(1, a^1b^1, [[0, 1, 0, a^0b^1, 否]], 1, 1, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

52. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":2}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(2, a^2b^2, [[0, 1, 1, a^1b^2, 否], [0, 2, 0, a^0b^2, 否], [1, 1, 0, a^1b^2, 否]], 3, 3, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(2, a^2b^2, [[0, 1, 1, a^1b^2, 否], [0, 2, 0, a^0b^2, 否], [1, 1, 0, a^1b^2, 否]], 3, 3, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

53. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":3}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

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(3, a^3b^3, [[0, 1, 2, a^2b^3, 否], [0, 2, 1, a^1b^3, 否], [0, 3, 0, a^0b^3, 否], [1, 1, 1, a^2b^3, 否], [1, 2, 0, a^1b^3, 否], [2, 1, 0, a^2b^3, 否]], 6, 6, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(3, a^3b^3, [[0, 1, 2, a^2b^3, 否], [0, 2, 1, a^1b^3, 否], [0, 3, 0, a^0b^3, 否], [1, 1, 1, a^2b^3, 否], [1, 2, 0, a^1b^3, 否], [2, 1, 0, a^2b^3, 否]], 6, 6, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

54. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":4}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

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(4, a^4b^4, [[0, 1, 3, a^3b^4, 否], [0, 2, 2, a^2b^4, 否], [0, 3, 1, a^1b^4, 否], [0, 4, 0, a^0b^4, 否], [1, 1, 2, a^3b^4, 否], [1, 2, 1, a^2b^4, 否], [1, 3, 0, a^1b^4, 否], [2, 1, 1, a^3b^4, 否], [2, 2, 0, a^2b^4, 否], [3, 1, 0, a^3b^4, 否]], 10, 10, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(4, a^4b^4, [[0, 1, 3, a^3b^4, 否], [0, 2, 2, a^2b^4, 否], [0, 3, 1, a^1b^4, 否], [0, 4, 0, a^0b^4, 否], [1, 1, 2, a^3b^4, 否], [1, 2, 1, a^2b^4, 否], [1, 3, 0, a^1b^4, 否], [2, 1, 1, a^3b^4, 否], [2, 2, 0, a^2b^4, 否], [3, 1, 0, a^3b^4, 否]], 10, 10, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

55. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":5}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

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(5, a^5b^5, [[0, 1, 4, a^4b^5, 否], [0, 2, 3, a^3b^5, 否], [0, 3, 2, a^2b^5, 否], [0, 4, 1, a^1b^5, 否], [0, 5, 0, a^0b^5, 否], [1, 1, 3, a^4b^5, 否], [1, 2, 2, a^3b^5, 否], [1, 3, 1, a^2b^5, 否], [1, 4, 0, a^1b^5, 否], [2, 1, 2, a^4b^5, 否], [2, 2, 1, a^3b^5, 否], [2, 3, 0, a^2b^5, 否], [3, 1, 1, a^4b^5, 否], [3, 2, 0, a^3b^5, 否], [4, 1, 0, a^4b^5, 否]], 15, 15, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(5, a^5b^5, [[0, 1, 4, a^4b^5, 否], [0, 2, 3, a^3b^5, 否], [0, 3, 2, a^2b^5, 否], [0, 4, 1, a^1b^5, 否], [0, 5, 0, a^0b^5, 否], [1, 1, 3, a^4b^5, 否], [1, 2, 2, a^3b^5, 否], [1, 3, 1, a^2b^5, 否], [1, 4, 0, a^1b^5, 否], [2, 1, 2, a^4b^5, 否], [2, 2, 1, a^3b^5, 否], [2, 3, 0, a^2b^5, 否], [3, 1, 1, a^4b^5, 否], [3, 2, 0, a^3b^5, 否], [4, 1, 0, a^4b^5, 否]], 15, 15, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

56. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":6}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

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(6, a^6b^6, [[0, 1, 5, a^5b^6, 否], [0, 2, 4, a^4b^6, 否], [0, 3, 3, a^3b^6, 否], [0, 4, 2, a^2b^6, 否], [0, 5, 1, a^1b^6, 否], [0, 6, 0, a^0b^6, 否], [1, 1, 4, a^5b^6, 否], [1, 2, 3, a^4b^6, 否], [1, 3, 2, a^3b^6, 否], [1, 4, 1, a^2b^6, 否], [1, 5, 0, a^1b^6, 否], [2, 1, 3, a^5b^6, 否], [2, 2, 2, a^4b^6, 否], [2, 3, 1, a^3b^6, 否], [2, 4, 0, a^2b^6, 否], [3, 1, 2, a^5b^6, 否], [3, 2, 1, a^4b^6, 否], [3, 3, 0, a^3b^6, 否], [4, 1, 1, a^5b^6, 否], [4, 2, 0, a^4b^6, 否], [5, 1, 0, a^5b^6, 否]], 21, 21, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(6, a^6b^6, [[0, 1, 5, a^5b^6, 否], [0, 2, 4, a^4b^6, 否], [0, 3, 3, a^3b^6, 否], [0, 4, 2, a^2b^6, 否], [0, 5, 1, a^1b^6, 否], [0, 6, 0, a^0b^6, 否], [1, 1, 4, a^5b^6, 否], [1, 2, 3, a^4b^6, 否], [1, 3, 2, a^3b^6, 否], [1, 4, 1, a^2b^6, 否], [1, 5, 0, a^1b^6, 否], [2, 1, 3, a^5b^6, 否], [2, 2, 2, a^4b^6, 否], [2, 3, 1, a^3b^6, 否], [2, 4, 0, a^2b^6, 否], [3, 1, 2, a^5b^6, 否], [3, 2, 1, a^4b^6, 否], [3, 3, 0, a^3b^6, 否], [4, 1, 1, a^5b^6, 否], [4, 2, 0, a^4b^6, 否], [5, 1, 0, a^5b^6, 否]], 21, 21, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

57. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":7}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

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(7, a^7b^7, [[0, 1, 6, a^6b^7, 否], [0, 2, 5, a^5b^7, 否], [0, 3, 4, a^4b^7, 否], [0, 4, 3, a^3b^7, 否], [0, 5, 2, a^2b^7, 否], [0, 6, 1, a^1b^7, 否], [0, 7, 0, a^0b^7, 否], [1, 1, 5, a^6b^7, 否], [1, 2, 4, a^5b^7, 否], [1, 3, 3, a^4b^7, 否], [1, 4, 2, a^3b^7, 否], [1, 5, 1, a^2b^7, 否], [1, 6, 0, a^1b^7, 否], [2, 1, 4, a^6b^7, 否], [2, 2, 3, a^5b^7, 否], [2, 3, 2, a^4b^7, 否], [2, 4, 1, a^3b^7, 否], [2, 5, 0, a^2b^7, 否], [3, 1, 3, a^6b^7, 否], [3, 2, 2, a^5b^7, 否], [3, 3, 1, a^4b^7, 否], [3, 4, 0, a^3b^7, 否], [4, 1, 2, a^6b^7, 否], [4, 2, 1, a^5b^7, 否], [4, 3, 0, a^4b^7, 否], [5, 1, 1, a^6b^7, 否], [5, 2, 0, a^5b^7, 否], [6, 1, 0, a^6b^7, 否]], 28, 28, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(7, a^7b^7, [[0, 1, 6, a^6b^7, 否], [0, 2, 5, a^5b^7, 否], [0, 3, 4, a^4b^7, 否], [0, 4, 3, a^3b^7, 否], [0, 5, 2, a^2b^7, 否], [0, 6, 1, a^1b^7, 否], [0, 7, 0, a^0b^7, 否], [1, 1, 5, a^6b^7, 否], [1, 2, 4, a^5b^7, 否], [1, 3, 3, a^4b^7, 否], [1, 4, 2, a^3b^7, 否], [1, 5, 1, a^2b^7, 否], [1, 6, 0, a^1b^7, 否], [2, 1, 4, a^6b^7, 否], [2, 2, 3, a^5b^7, 否], [2, 3, 2, a^4b^7, 否], [2, 4, 1, a^3b^7, 否], [2, 5, 0, a^2b^7, 否], [3, 1, 3, a^6b^7, 否], [3, 2, 2, a^5b^7, 否], [3, 3, 1, a^4b^7, 否], [3, 4, 0, a^3b^7, 否], [4, 1, 2, a^6b^7, 否], [4, 2, 1, a^5b^7, 否], [4, 3, 0, a^4b^7, 否], [5, 1, 1, a^6b^7, 否], [5, 2, 0, a^5b^7, 否], [6, 1, 0, a^6b^7, 否]], 28, 28, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

58. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":8}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

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(8, a^8b^8, [[0, 1, 7, a^7b^8, 否], [0, 2, 6, a^6b^8, 否], [0, 3, 5, a^5b^8, 否], [0, 4, 4, a^4b^8, 否], [0, 5, 3, a^3b^8, 否], [0, 6, 2, a^2b^8, 否], [0, 7, 1, a^1b^8, 否], [0, 8, 0, a^0b^8, 否], [1, 1, 6, a^7b^8, 否], [1, 2, 5, a^6b^8, 否], [1, 3, 4, a^5b^8, 否], [1, 4, 3, a^4b^8, 否], [1, 5, 2, a^3b^8, 否], [1, 6, 1, a^2b^8, 否], [1, 7, 0, a^1b^8, 否], [2, 1, 5, a^7b^8, 否], [2, 2, 4, a^6b^8, 否], [2, 3, 3, a^5b^8, 否], [2, 4, 2, a^4b^8, 否], [2, 5, 1, a^3b^8, 否], [2, 6, 0, a^2b^8, 否], [3, 1, 4, a^7b^8, 否], [3, 2, 3, a^6b^8, 否], [3, 3, 2, a^5b^8, 否], [3, 4, 1, a^4b^8, 否], [3, 5, 0, a^3b^8, 否], [4, 1, 3, a^7b^8, 否], [4, 2, 2, a^6b^8, 否], [4, 3, 1, a^5b^8, 否], [4, 4, 0, a^4b^8, 否], [5, 1, 2, a^7b^8, 否], [5, 2, 1, a^6b^8, 否], [5, 3, 0, a^5b^8, 否], [6, 1, 1, a^7b^8, 否], [6, 2, 0, a^6b^8, 否], [7, 1, 0, a^7b^8, 否]], 36, 36, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(8, a^8b^8, [[0, 1, 7, a^7b^8, 否], [0, 2, 6, a^6b^8, 否], [0, 3, 5, a^5b^8, 否], [0, 4, 4, a^4b^8, 否], [0, 5, 3, a^3b^8, 否], [0, 6, 2, a^2b^8, 否], [0, 7, 1, a^1b^8, 否], [0, 8, 0, a^0b^8, 否], [1, 1, 6, a^7b^8, 否], [1, 2, 5, a^6b^8, 否], [1, 3, 4, a^5b^8, 否], [1, 4, 3, a^4b^8, 否], [1, 5, 2, a^3b^8, 否], [1, 6, 1, a^2b^8, 否], [1, 7, 0, a^1b^8, 否], [2, 1, 5, a^7b^8, 否], [2, 2, 4, a^6b^8, 否], [2, 3, 3, a^5b^8, 否], [2, 4, 2, a^4b^8, 否], [2, 5, 1, a^3b^8, 否], [2, 6, 0, a^2b^8, 否], [3, 1, 4, a^7b^8, 否], [3, 2, 3, a^6b^8, 否], [3, 3, 2, a^5b^8, 否], [3, 4, 1, a^4b^8, 否], [3, 5, 0, a^3b^8, 否], [4, 1, 3, a^7b^8, 否], [4, 2, 2, a^6b^8, 否], [4, 3, 1, a^5b^8, 否], [4, 4, 0, a^4b^8, 否], [5, 1, 2, a^7b^8, 否], [5, 2, 1, a^6b^8, 否], [5, 3, 0, a^5b^8, 否], [6, 1, 1, a^7b^8, 否], [6, 2, 0, a^6b^8, 否], [7, 1, 0, a^7b^8, 否]], 36, 36, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

59. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":9}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

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(9, a^9b^9, [[0, 1, 8, a^8b^9, 否], [0, 2, 7, a^7b^9, 否], [0, 3, 6, a^6b^9, 否], [0, 4, 5, a^5b^9, 否], [0, 5, 4, a^4b^9, 否], [0, 6, 3, a^3b^9, 否], [0, 7, 2, a^2b^9, 否], [0, 8, 1, a^1b^9, 否], [0, 9, 0, a^0b^9, 否], [1, 1, 7, a^8b^9, 否], [1, 2, 6, a^7b^9, 否], [1, 3, 5, a^6b^9, 否], [1, 4, 4, a^5b^9, 否], [1, 5, 3, a^4b^9, 否], [1, 6, 2, a^3b^9, 否], [1, 7, 1, a^2b^9, 否], [1, 8, 0, a^1b^9, 否], [2, 1, 6, a^8b^9, 否], [2, 2, 5, a^7b^9, 否], [2, 3, 4, a^6b^9, 否], [2, 4, 3, a^5b^9, 否], [2, 5, 2, a^4b^9, 否], [2, 6, 1, a^3b^9, 否], [2, 7, 0, a^2b^9, 否], [3, 1, 5, a^8b^9, 否], [3, 2, 4, a^7b^9, 否], [3, 3, 3, a^6b^9, 否], [3, 4, 2, a^5b^9, 否], [3, 5, 1, a^4b^9, 否], [3, 6, 0, a^3b^9, 否], [4, 1, 4, a^8b^9, 否], [4, 2, 3, a^7b^9, 否], [4, 3, 2, a^6b^9, 否], [4, 4, 1, a^5b^9, 否], [4, 5, 0, a^4b^9, 否], [5, 1, 3, a^8b^9, 否], [5, 2, 2, a^7b^9, 否], [5, 3, 1, a^6b^9, 否], [5, 4, 0, a^5b^9, 否], [6, 1, 2, a^8b^9, 否], [6, 2, 1, a^7b^9, 否], [6, 3, 0, a^6b^9, 否], [7, 1, 1, a^8b^9, 否], [7, 2, 0, a^7b^9, 否], [8, 1, 0, a^8b^9, 否]], 45, 45, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(9, a^9b^9, [[0, 1, 8, a^8b^9, 否], [0, 2, 7, a^7b^9, 否], [0, 3, 6, a^6b^9, 否], [0, 4, 5, a^5b^9, 否], [0, 5, 4, a^4b^9, 否], [0, 6, 3, a^3b^9, 否], [0, 7, 2, a^2b^9, 否], [0, 8, 1, a^1b^9, 否], [0, 9, 0, a^0b^9, 否], [1, 1, 7, a^8b^9, 否], [1, 2, 6, a^7b^9, 否], [1, 3, 5, a^6b^9, 否], [1, 4, 4, a^5b^9, 否], [1, 5, 3, a^4b^9, 否], [1, 6, 2, a^3b^9, 否], [1, 7, 1, a^2b^9, 否], [1, 8, 0, a^1b^9, 否], [2, 1, 6, a^8b^9, 否], [2, 2, 5, a^7b^9, 否], [2, 3, 4, a^6b^9, 否], [2, 4, 3, a^5b^9, 否], [2, 5, 2, a^4b^9, 否], [2, 6, 1, a^3b^9, 否], [2, 7, 0, a^2b^9, 否], [3, 1, 5, a^8b^9, 否], [3, 2, 4, a^7b^9, 否], [3, 3, 3, a^6b^9, 否], [3, 4, 2, a^5b^9, 否], [3, 5, 1, a^4b^9, 否], [3, 6, 0, a^3b^9, 否], [4, 1, 4, a^8b^9, 否], [4, 2, 3, a^7b^9, 否], [4, 3, 2, a^6b^9, 否], [4, 4, 1, a^5b^9, 否], [4, 5, 0, a^4b^9, 否], [5, 1, 3, a^8b^9, 否], [5, 2, 2, a^7b^9, 否], [5, 3, 1, a^6b^9, 否], [5, 4, 0, a^5b^9, 否], [6, 1, 2, a^8b^9, 否], [6, 2, 1, a^7b^9, 否], [6, 3, 0, a^6b^9, 否], [7, 1, 1, a^8b^9, 否], [7, 2, 0, a^7b^9, 否], [8, 1, 0, a^8b^9, 否]], 45, 45, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

60. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":10}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(10, a^10b^10, [[0, 1, 9, a^9b^10, 否], [0, 2, 8, a^8b^10, 否], [0, 3, 7, a^7b^10, 否], [0, 4, 6, a^6b^10, 否], [0, 5, 5, a^5b^10, 否], [0, 6, 4, a^4b^10, 否], [0, 7, 3, a^3b^10, 否], [0, 8, 2, a^2b^10, 否], [0, 9, 1, a^1b^10, 否], [0, 10, 0, a^0b^10, 否], [1, 1, 8, a^9b^10, 否], [1, 2, 7, a^8b^10, 否], [1, 3, 6, a^7b^10, 否], [1, 4, 5, a^6b^10, 否], [1, 5, 4, a^5b^10, 否], [1, 6, 3, a^4b^10, 否], [1, 7, 2, a^3b^10, 否], [1, 8, 1, a^2b^10, 否], [1, 9, 0, a^1b^10, 否], [2, 1, 7, a^9b^10, 否], [2, 2, 6, a^8b^10, 否], [2, 3, 5, a^7b^10, 否], [2, 4, 4, a^6b^10, 否], [2, 5, 3, a^5b^10, 否], [2, 6, 2, a^4b^10, 否], [2, 7, 1, a^3b^10, 否], [2, 8, 0, a^2b^10, 否], [3, 1, 6, a^9b^10, 否], [3, 2, 5, a^8b^10, 否], [3, 3, 4, a^7b^10, 否], [3, 4, 3, a^6b^10, 否], [3, 5, 2, a^5b^10, 否], [3, 6, 1, a^4b^10, 否], [3, 7, 0, a^3b^10, 否], [4, 1, 5, a^9b^10, 否], [4, 2, 4, a^8b^10, 否], [4, 3, 3, a^7b^10, 否], [4, 4, 2, a^6b^10, 否], [4, 5, 1, a^5b^10, 否], [4, 6, 0, a^4b^10, 否], [5, 1, 4, a^9b^10, 否], [5, 2, 3, a^8b^10, 否], [5, 3, 2, a^7b^10, 否], [5, 4, 1, a^6b^10, 否], [5, 5, 0, a^5b^10, 否], [6, 1, 3, a^9b^10, 否], [6, 2, 2, a^8b^10, 否], [6, 3, 1, a^7b^10, 否], [6, 4, 0, a^6b^10, 否], [7, 1, 2, a^9b^10, 否], [7, 2, 1, a^8b^10, 否], [7, 3, 0, a^7b^10, 否], [8, 1, 1, a^9b^10, 否], [8, 2, 0, a^8b^10, 否], [9, 1, 0, a^9b^10, 否]], 55, 55, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(10, a^10b^10, [[0, 1, 9, a^9b^10, 否], [0, 2, 8, a^8b^10, 否], [0, 3, 7, a^7b^10, 否], [0, 4, 6, a^6b^10, 否], [0, 5, 5, a^5b^10, 否], [0, 6, 4, a^4b^10, 否], [0, 7, 3, a^3b^10, 否], [0, 8, 2, a^2b^10, 否], [0, 9, 1, a^1b^10, 否], [0, 10, 0, a^0b^10, 否], [1, 1, 8, a^9b^10, 否], [1, 2, 7, a^8b^10, 否], [1, 3, 6, a^7b^10, 否], [1, 4, 5, a^6b^10, 否], [1, 5, 4, a^5b^10, 否], [1, 6, 3, a^4b^10, 否], [1, 7, 2, a^3b^10, 否], [1, 8, 1, a^2b^10, 否], [1, 9, 0, a^1b^10, 否], [2, 1, 7, a^9b^10, 否], [2, 2, 6, a^8b^10, 否], [2, 3, 5, a^7b^10, 否], [2, 4, 4, a^6b^10, 否], [2, 5, 3, a^5b^10, 否], [2, 6, 2, a^4b^10, 否], [2, 7, 1, a^3b^10, 否], [2, 8, 0, a^2b^10, 否], [3, 1, 6, a^9b^10, 否], [3, 2, 5, a^8b^10, 否], [3, 3, 4, a^7b^10, 否], [3, 4, 3, a^6b^10, 否], [3, 5, 2, a^5b^10, 否], [3, 6, 1, a^4b^10, 否], [3, 7, 0, a^3b^10, 否], [4, 1, 5, a^9b^10, 否], [4, 2, 4, a^8b^10, 否], [4, 3, 3, a^7b^10, 否], [4, 4, 2, a^6b^10, 否], [4, 5, 1, a^5b^10, 否], [4, 6, 0, a^4b^10, 否], [5, 1, 4, a^9b^10, 否], [5, 2, 3, a^8b^10, 否], [5, 3, 2, a^7b^10, 否], [5, 4, 1, a^6b^10, 否], [5, 5, 0, a^5b^10, 否], [6, 1, 3, a^9b^10, 否], [6, 2, 2, a^8b^10, 否], [6, 3, 1, a^7b^10, 否], [6, 4, 0, a^6b^10, 否], [7, 1, 2, a^9b^10, 否], [7, 2, 1, a^8b^10, 否], [7, 3, 0, a^7b^10, 否], [8, 1, 1, a^9b^10, 否], [8, 2, 0, a^8b^10, 否], [9, 1, 0, a^9b^10, 否]], 55, 55, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

61. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":11}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(11, a^11b^11, [[0, 1, 10, a^10b^11, 否], [0, 2, 9, a^9b^11, 否], [0, 3, 8, a^8b^11, 否], [0, 4, 7, a^7b^11, 否], [0, 5, 6, a^6b^11, 否], [0, 6, 5, a^5b^11, 否], [0, 7, 4, a^4b^11, 否], [0, 8, 3, a^3b^11, 否], [0, 9, 2, a^2b^11, 否], [0, 10, 1, a^1b^11, 否], [0, 11, 0, a^0b^11, 否], [1, 1, 9, a^10b^11, 否], [1, 2, 8, a^9b^11, 否], [1, 3, 7, a^8b^11, 否], [1, 4, 6, a^7b^11, 否], [1, 5, 5, a^6b^11, 否], [1, 6, 4, a^5b^11, 否], [1, 7, 3, a^4b^11, 否], [1, 8, 2, a^3b^11, 否], [1, 9, 1, a^2b^11, 否], [1, 10, 0, a^1b^11, 否], [2, 1, 8, a^10b^11, 否], [2, 2, 7, a^9b^11, 否], [2, 3, 6, a^8b^11, 否], [2, 4, 5, a^7b^11, 否], [2, 5, 4, a^6b^11, 否], [2, 6, 3, a^5b^11, 否], [2, 7, 2, a^4b^11, 否], [2, 8, 1, a^3b^11, 否], [2, 9, 0, a^2b^11, 否], [3, 1, 7, a^10b^11, 否], [3, 2, 6, a^9b^11, 否], [3, 3, 5, a^8b^11, 否], [3, 4, 4, a^7b^11, 否], [3, 5, 3, a^6b^11, 否], [3, 6, 2, a^5b^11, 否], [3, 7, 1, a^4b^11, 否], [3, 8, 0, a^3b^11, 否], [4, 1, 6, a^10b^11, 否], [4, 2, 5, a^9b^11, 否], [4, 3, 4, a^8b^11, 否], [4, 4, 3, a^7b^11, 否], [4, 5, 2, a^6b^11, 否], [4, 6, 1, a^5b^11, 否], [4, 7, 0, a^4b^11, 否], [5, 1, 5, a^10b^11, 否], [5, 2, 4, a^9b^11, 否], [5, 3, 3, a^8b^11, 否], [5, 4, 2, a^7b^11, 否], [5, 5, 1, a^6b^11, 否], [5, 6, 0, a^5b^11, 否], [6, 1, 4, a^10b^11, 否], [6, 2, 3, a^9b^11, 否], [6, 3, 2, a^8b^11, 否], [6, 4, 1, a^7b^11, 否], [6, 5, 0, a^6b^11, 否], [7, 1, 3, a^10b^11, 否], [7, 2, 2, a^9b^11, 否], [7, 3, 1, a^8b^11, 否], [7, 4, 0, a^7b^11, 否], [8, 1, 2, a^10b^11, 否], [8, 2, 1, a^9b^11, 否], [8, 3, 0, a^8b^11, 否], [9, 1, 1, a^10b^11, 否], [9, 2, 0, a^9b^11, 否], [10, 1, 0, a^10b^11, 否]], 66, 66, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(11, a^11b^11, [[0, 1, 10, a^10b^11, 否], [0, 2, 9, a^9b^11, 否], [0, 3, 8, a^8b^11, 否], [0, 4, 7, a^7b^11, 否], [0, 5, 6, a^6b^11, 否], [0, 6, 5, a^5b^11, 否], [0, 7, 4, a^4b^11, 否], [0, 8, 3, a^3b^11, 否], [0, 9, 2, a^2b^11, 否], [0, 10, 1, a^1b^11, 否], [0, 11, 0, a^0b^11, 否], [1, 1, 9, a^10b^11, 否], [1, 2, 8, a^9b^11, 否], [1, 3, 7, a^8b^11, 否], [1, 4, 6, a^7b^11, 否], [1, 5, 5, a^6b^11, 否], [1, 6, 4, a^5b^11, 否], [1, 7, 3, a^4b^11, 否], [1, 8, 2, a^3b^11, 否], [1, 9, 1, a^2b^11, 否], [1, 10, 0, a^1b^11, 否], [2, 1, 8, a^10b^11, 否], [2, 2, 7, a^9b^11, 否], [2, 3, 6, a^8b^11, 否], [2, 4, 5, a^7b^11, 否], [2, 5, 4, a^6b^11, 否], [2, 6, 3, a^5b^11, 否], [2, 7, 2, a^4b^11, 否], [2, 8, 1, a^3b^11, 否], [2, 9, 0, a^2b^11, 否], [3, 1, 7, a^10b^11, 否], [3, 2, 6, a^9b^11, 否], [3, 3, 5, a^8b^11, 否], [3, 4, 4, a^7b^11, 否], [3, 5, 3, a^6b^11, 否], [3, 6, 2, a^5b^11, 否], [3, 7, 1, a^4b^11, 否], [3, 8, 0, a^3b^11, 否], [4, 1, 6, a^10b^11, 否], [4, 2, 5, a^9b^11, 否], [4, 3, 4, a^8b^11, 否], [4, 4, 3, a^7b^11, 否], [4, 5, 2, a^6b^11, 否], [4, 6, 1, a^5b^11, 否], [4, 7, 0, a^4b^11, 否], [5, 1, 5, a^10b^11, 否], [5, 2, 4, a^9b^11, 否], [5, 3, 3, a^8b^11, 否], [5, 4, 2, a^7b^11, 否], [5, 5, 1, a^6b^11, 否], [5, 6, 0, a^5b^11, 否], [6, 1, 4, a^10b^11, 否], [6, 2, 3, a^9b^11, 否], [6, 3, 2, a^8b^11, 否], [6, 4, 1, a^7b^11, 否], [6, 5, 0, a^6b^11, 否], [7, 1, 3, a^10b^11, 否], [7, 2, 2, a^9b^11, 否], [7, 3, 1, a^8b^11, 否], [7, 4, 0, a^7b^11, 否], [8, 1, 2, a^10b^11, 否], [8, 2, 1, a^9b^11, 否], [8, 3, 0, a^8b^11, 否], [9, 1, 1, a^10b^11, 否], [9, 2, 0, a^9b^11, 否], [10, 1, 0, a^10b^11, 否]], 66, 66, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

62. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":12}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(12, a^12b^12, [[0, 1, 11, a^11b^12, 否], [0, 2, 10, a^10b^12, 否], [0, 3, 9, a^9b^12, 否], [0, 4, 8, a^8b^12, 否], [0, 5, 7, a^7b^12, 否], [0, 6, 6, a^6b^12, 否], [0, 7, 5, a^5b^12, 否], [0, 8, 4, a^4b^12, 否], [0, 9, 3, a^3b^12, 否], [0, 10, 2, a^2b^12, 否], [0, 11, 1, a^1b^12, 否], [0, 12, 0, a^0b^12, 否], [1, 1, 10, a^11b^12, 否], [1, 2, 9, a^10b^12, 否], [1, 3, 8, a^9b^12, 否], [1, 4, 7, a^8b^12, 否], [1, 5, 6, a^7b^12, 否], [1, 6, 5, a^6b^12, 否], [1, 7, 4, a^5b^12, 否], [1, 8, 3, a^4b^12, 否], [1, 9, 2, a^3b^12, 否], [1, 10, 1, a^2b^12, 否], [1, 11, 0, a^1b^12, 否], [2, 1, 9, a^11b^12, 否], [2, 2, 8, a^10b^12, 否], [2, 3, 7, a^9b^12, 否], [2, 4, 6, a^8b^12, 否], [2, 5, 5, a^7b^12, 否], [2, 6, 4, a^6b^12, 否], [2, 7, 3, a^5b^12, 否], [2, 8, 2, a^4b^12, 否], [2, 9, 1, a^3b^12, 否], [2, 10, 0, a^2b^12, 否], [3, 1, 8, a^11b^12, 否], [3, 2, 7, a^10b^12, 否], [3, 3, 6, a^9b^12, 否], [3, 4, 5, a^8b^12, 否], [3, 5, 4, a^7b^12, 否], [3, 6, 3, a^6b^12, 否], [3, 7, 2, a^5b^12, 否], [3, 8, 1, a^4b^12, 否], [3, 9, 0, a^3b^12, 否], [4, 1, 7, a^11b^12, 否], [4, 2, 6, a^10b^12, 否], [4, 3, 5, a^9b^12, 否], [4, 4, 4, a^8b^12, 否], [4, 5, 3, a^7b^12, 否], [4, 6, 2, a^6b^12, 否], [4, 7, 1, a^5b^12, 否], [4, 8, 0, a^4b^12, 否], [5, 1, 6, a^11b^12, 否], [5, 2, 5, a^10b^12, 否], [5, 3, 4, a^9b^12, 否], [5, 4, 3, a^8b^12, 否], [5, 5, 2, a^7b^12, 否], [5, 6, 1, a^6b^12, 否], [5, 7, 0, a^5b^12, 否], [6, 1, 5, a^11b^12, 否], [6, 2, 4, a^10b^12, 否], [6, 3, 3, a^9b^12, 否], [6, 4, 2, a^8b^12, 否], [6, 5, 1, a^7b^12, 否], [6, 6, 0, a^6b^12, 否], [7, 1, 4, a^11b^12, 否], [7, 2, 3, a^10b^12, 否], [7, 3, 2, a^9b^12, 否], [7, 4, 1, a^8b^12, 否], [7, 5, 0, a^7b^12, 否], [8, 1, 3, a^11b^12, 否], [8, 2, 2, a^10b^12, 否], [8, 3, 1, a^9b^12, 否], [8, 4, 0, a^8b^12, 否], [9, 1, 2, a^11b^12, 否], [9, 2, 1, a^10b^12, 否], [9, 3, 0, a^9b^12, 否], [10, 1, 1, a^11b^12, 否], [10, 2, 0, a^10b^12, 否], [11, 1, 0, a^11b^12, 否]], 78, 78, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(12, a^12b^12, [[0, 1, 11, a^11b^12, 否], [0, 2, 10, a^10b^12, 否], [0, 3, 9, a^9b^12, 否], [0, 4, 8, a^8b^12, 否], [0, 5, 7, a^7b^12, 否], [0, 6, 6, a^6b^12, 否], [0, 7, 5, a^5b^12, 否], [0, 8, 4, a^4b^12, 否], [0, 9, 3, a^3b^12, 否], [0, 10, 2, a^2b^12, 否], [0, 11, 1, a^1b^12, 否], [0, 12, 0, a^0b^12, 否], [1, 1, 10, a^11b^12, 否], [1, 2, 9, a^10b^12, 否], [1, 3, 8, a^9b^12, 否], [1, 4, 7, a^8b^12, 否], [1, 5, 6, a^7b^12, 否], [1, 6, 5, a^6b^12, 否], [1, 7, 4, a^5b^12, 否], [1, 8, 3, a^4b^12, 否], [1, 9, 2, a^3b^12, 否], [1, 10, 1, a^2b^12, 否], [1, 11, 0, a^1b^12, 否], [2, 1, 9, a^11b^12, 否], [2, 2, 8, a^10b^12, 否], [2, 3, 7, a^9b^12, 否], [2, 4, 6, a^8b^12, 否], [2, 5, 5, a^7b^12, 否], [2, 6, 4, a^6b^12, 否], [2, 7, 3, a^5b^12, 否], [2, 8, 2, a^4b^12, 否], [2, 9, 1, a^3b^12, 否], [2, 10, 0, a^2b^12, 否], [3, 1, 8, a^11b^12, 否], [3, 2, 7, a^10b^12, 否], [3, 3, 6, a^9b^12, 否], [3, 4, 5, a^8b^12, 否], [3, 5, 4, a^7b^12, 否], [3, 6, 3, a^6b^12, 否], [3, 7, 2, a^5b^12, 否], [3, 8, 1, a^4b^12, 否], [3, 9, 0, a^3b^12, 否], [4, 1, 7, a^11b^12, 否], [4, 2, 6, a^10b^12, 否], [4, 3, 5, a^9b^12, 否], [4, 4, 4, a^8b^12, 否], [4, 5, 3, a^7b^12, 否], [4, 6, 2, a^6b^12, 否], [4, 7, 1, a^5b^12, 否], [4, 8, 0, a^4b^12, 否], [5, 1, 6, a^11b^12, 否], [5, 2, 5, a^10b^12, 否], [5, 3, 4, a^9b^12, 否], [5, 4, 3, a^8b^12, 否], [5, 5, 2, a^7b^12, 否], [5, 6, 1, a^6b^12, 否], [5, 7, 0, a^5b^12, 否], [6, 1, 5, a^11b^12, 否], [6, 2, 4, a^10b^12, 否], [6, 3, 3, a^9b^12, 否], [6, 4, 2, a^8b^12, 否], [6, 5, 1, a^7b^12, 否], [6, 6, 0, a^6b^12, 否], [7, 1, 4, a^11b^12, 否], [7, 2, 3, a^10b^12, 否], [7, 3, 2, a^9b^12, 否], [7, 4, 1, a^8b^12, 否], [7, 5, 0, a^7b^12, 否], [8, 1, 3, a^11b^12, 否], [8, 2, 2, a^10b^12, 否], [8, 3, 1, a^9b^12, 否], [8, 4, 0, a^8b^12, 否], [9, 1, 2, a^11b^12, 否], [9, 2, 1, a^10b^12, 否], [9, 3, 0, a^9b^12, 否], [10, 1, 1, a^11b^12, 否], [10, 2, 0, a^10b^12, 否], [11, 1, 0, a^11b^12, 否]], 78, 78, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

63. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":13}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(13, a^13b^13, [[0, 1, 12, a^12b^13, 否], [0, 2, 11, a^11b^13, 否], [0, 3, 10, a^10b^13, 否], [0, 4, 9, a^9b^13, 否], [0, 5, 8, a^8b^13, 否], [0, 6, 7, a^7b^13, 否], [0, 7, 6, a^6b^13, 否], [0, 8, 5, a^5b^13, 否], [0, 9, 4, a^4b^13, 否], [0, 10, 3, a^3b^13, 否], [0, 11, 2, a^2b^13, 否], [0, 12, 1, a^1b^13, 否], [0, 13, 0, a^0b^13, 否], [1, 1, 11, a^12b^13, 否], [1, 2, 10, a^11b^13, 否], [1, 3, 9, a^10b^13, 否], [1, 4, 8, a^9b^13, 否], [1, 5, 7, a^8b^13, 否], [1, 6, 6, a^7b^13, 否], [1, 7, 5, a^6b^13, 否], [1, 8, 4, a^5b^13, 否], [1, 9, 3, a^4b^13, 否], [1, 10, 2, a^3b^13, 否], [1, 11, 1, a^2b^13, 否], [1, 12, 0, a^1b^13, 否], [2, 1, 10, a^12b^13, 否], [2, 2, 9, a^11b^13, 否], [2, 3, 8, a^10b^13, 否], [2, 4, 7, a^9b^13, 否], [2, 5, 6, a^8b^13, 否], [2, 6, 5, a^7b^13, 否], [2, 7, 4, a^6b^13, 否], [2, 8, 3, a^5b^13, 否], [2, 9, 2, a^4b^13, 否], [2, 10, 1, a^3b^13, 否], [2, 11, 0, a^2b^13, 否], [3, 1, 9, a^12b^13, 否], [3, 2, 8, a^11b^13, 否], [3, 3, 7, a^10b^13, 否], [3, 4, 6, a^9b^13, 否], [3, 5, 5, a^8b^13, 否], [3, 6, 4, a^7b^13, 否], [3, 7, 3, a^6b^13, 否], [3, 8, 2, a^5b^13, 否], [3, 9, 1, a^4b^13, 否], [3, 10, 0, a^3b^13, 否], [4, 1, 8, a^12b^13, 否], [4, 2, 7, a^11b^13, 否], [4, 3, 6, a^10b^13, 否], [4, 4, 5, a^9b^13, 否], [4, 5, 4, a^8b^13, 否], [4, 6, 3, a^7b^13, 否], [4, 7, 2, a^6b^13, 否], [4, 8, 1, a^5b^13, 否], [4, 9, 0, a^4b^13, 否], [5, 1, 7, a^12b^13, 否], [5, 2, 6, a^11b^13, 否], [5, 3, 5, a^10b^13, 否], [5, 4, 4, a^9b^13, 否], [5, 5, 3, a^8b^13, 否], [5, 6, 2, a^7b^13, 否], [5, 7, 1, a^6b^13, 否], [5, 8, 0, a^5b^13, 否], [6, 1, 6, a^12b^13, 否], [6, 2, 5, a^11b^13, 否], [6, 3, 4, a^10b^13, 否], [6, 4, 3, a^9b^13, 否], [6, 5, 2, a^8b^13, 否], [6, 6, 1, a^7b^13, 否], [6, 7, 0, a^6b^13, 否], [7, 1, 5, a^12b^13, 否], [7, 2, 4, a^11b^13, 否], [7, 3, 3, a^10b^13, 否], [7, 4, 2, a^9b^13, 否], [7, 5, 1, a^8b^13, 否], [7, 6, 0, a^7b^13, 否], [8, 1, 4, a^12b^13, 否], [8, 2, 3, a^11b^13, 否], [8, 3, 2, a^10b^13, 否], [8, 4, 1, a^9b^13, 否], [8, 5, 0, a^8b^13, 否], [9, 1, 3, a^12b^13, 否], [9, 2, 2, a^11b^13, 否], [9, 3, 1, a^10b^13, 否], [9, 4, 0, a^9b^13, 否], [10, 1, 2, a^12b^13, 否], [10, 2, 1, a^11b^13, 否], [10, 3, 0, a^10b^13, 否], [11, 1, 1, a^12b^13, 否], [11, 2, 0, a^11b^13, 否], [12, 1, 0, a^12b^13, 否]], 91, 91, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(13, a^13b^13, [[0, 1, 12, a^12b^13, 否], [0, 2, 11, a^11b^13, 否], [0, 3, 10, a^10b^13, 否], [0, 4, 9, a^9b^13, 否], [0, 5, 8, a^8b^13, 否], [0, 6, 7, a^7b^13, 否], [0, 7, 6, a^6b^13, 否], [0, 8, 5, a^5b^13, 否], [0, 9, 4, a^4b^13, 否], [0, 10, 3, a^3b^13, 否], [0, 11, 2, a^2b^13, 否], [0, 12, 1, a^1b^13, 否], [0, 13, 0, a^0b^13, 否], [1, 1, 11, a^12b^13, 否], [1, 2, 10, a^11b^13, 否], [1, 3, 9, a^10b^13, 否], [1, 4, 8, a^9b^13, 否], [1, 5, 7, a^8b^13, 否], [1, 6, 6, a^7b^13, 否], [1, 7, 5, a^6b^13, 否], [1, 8, 4, a^5b^13, 否], [1, 9, 3, a^4b^13, 否], [1, 10, 2, a^3b^13, 否], [1, 11, 1, a^2b^13, 否], [1, 12, 0, a^1b^13, 否], [2, 1, 10, a^12b^13, 否], [2, 2, 9, a^11b^13, 否], [2, 3, 8, a^10b^13, 否], [2, 4, 7, a^9b^13, 否], [2, 5, 6, a^8b^13, 否], [2, 6, 5, a^7b^13, 否], [2, 7, 4, a^6b^13, 否], [2, 8, 3, a^5b^13, 否], [2, 9, 2, a^4b^13, 否], [2, 10, 1, a^3b^13, 否], [2, 11, 0, a^2b^13, 否], [3, 1, 9, a^12b^13, 否], [3, 2, 8, a^11b^13, 否], [3, 3, 7, a^10b^13, 否], [3, 4, 6, a^9b^13, 否], [3, 5, 5, a^8b^13, 否], [3, 6, 4, a^7b^13, 否], [3, 7, 3, a^6b^13, 否], [3, 8, 2, a^5b^13, 否], [3, 9, 1, a^4b^13, 否], [3, 10, 0, a^3b^13, 否], [4, 1, 8, a^12b^13, 否], [4, 2, 7, a^11b^13, 否], [4, 3, 6, a^10b^13, 否], [4, 4, 5, a^9b^13, 否], [4, 5, 4, a^8b^13, 否], [4, 6, 3, a^7b^13, 否], [4, 7, 2, a^6b^13, 否], [4, 8, 1, a^5b^13, 否], [4, 9, 0, a^4b^13, 否], [5, 1, 7, a^12b^13, 否], [5, 2, 6, a^11b^13, 否], [5, 3, 5, a^10b^13, 否], [5, 4, 4, a^9b^13, 否], [5, 5, 3, a^8b^13, 否], [5, 6, 2, a^7b^13, 否], [5, 7, 1, a^6b^13, 否], [5, 8, 0, a^5b^13, 否], [6, 1, 6, a^12b^13, 否], [6, 2, 5, a^11b^13, 否], [6, 3, 4, a^10b^13, 否], [6, 4, 3, a^9b^13, 否], [6, 5, 2, a^8b^13, 否], [6, 6, 1, a^7b^13, 否], [6, 7, 0, a^6b^13, 否], [7, 1, 5, a^12b^13, 否], [7, 2, 4, a^11b^13, 否], [7, 3, 3, a^10b^13, 否], [7, 4, 2, a^9b^13, 否], [7, 5, 1, a^8b^13, 否], [7, 6, 0, a^7b^13, 否], [8, 1, 4, a^12b^13, 否], [8, 2, 3, a^11b^13, 否], [8, 3, 2, a^10b^13, 否], [8, 4, 1, a^9b^13, 否], [8, 5, 0, a^8b^13, 否], [9, 1, 3, a^12b^13, 否], [9, 2, 2, a^11b^13, 否], [9, 3, 1, a^10b^13, 否], [9, 4, 0, a^9b^13, 否], [10, 1, 2, a^12b^13, 否], [10, 2, 1, a^11b^13, 否], [10, 3, 0, a^10b^13, 否], [11, 1, 1, a^12b^13, 否], [11, 2, 0, a^11b^13, 否], [12, 1, 0, a^12b^13, 否]], 91, 91, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

64. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":14}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(14, a^14b^14, [[0, 1, 13, a^13b^14, 否], [0, 2, 12, a^12b^14, 否], [0, 3, 11, a^11b^14, 否], [0, 4, 10, a^10b^14, 否], [0, 5, 9, a^9b^14, 否], [0, 6, 8, a^8b^14, 否], [0, 7, 7, a^7b^14, 否], [0, 8, 6, a^6b^14, 否], [0, 9, 5, a^5b^14, 否], [0, 10, 4, a^4b^14, 否], [0, 11, 3, a^3b^14, 否], [0, 12, 2, a^2b^14, 否], [0, 13, 1, a^1b^14, 否], [0, 14, 0, a^0b^14, 否], [1, 1, 12, a^13b^14, 否], [1, 2, 11, a^12b^14, 否], [1, 3, 10, a^11b^14, 否], [1, 4, 9, a^10b^14, 否], [1, 5, 8, a^9b^14, 否], [1, 6, 7, a^8b^14, 否], [1, 7, 6, a^7b^14, 否], [1, 8, 5, a^6b^14, 否], [1, 9, 4, a^5b^14, 否], [1, 10, 3, a^4b^14, 否], [1, 11, 2, a^3b^14, 否], [1, 12, 1, a^2b^14, 否], [1, 13, 0, a^1b^14, 否], [2, 1, 11, a^13b^14, 否], [2, 2, 10, a^12b^14, 否], [2, 3, 9, a^11b^14, 否], [2, 4, 8, a^10b^14, 否], [2, 5, 7, a^9b^14, 否], [2, 6, 6, a^8b^14, 否], [2, 7, 5, a^7b^14, 否], [2, 8, 4, a^6b^14, 否], [2, 9, 3, a^5b^14, 否], [2, 10, 2, a^4b^14, 否], [2, 11, 1, a^3b^14, 否], [2, 12, 0, a^2b^14, 否], [3, 1, 10, a^13b^14, 否], [3, 2, 9, a^12b^14, 否], [3, 3, 8, a^11b^14, 否], [3, 4, 7, a^10b^14, 否], [3, 5, 6, a^9b^14, 否], [3, 6, 5, a^8b^14, 否], [3, 7, 4, a^7b^14, 否], [3, 8, 3, a^6b^14, 否], [3, 9, 2, a^5b^14, 否], [3, 10, 1, a^4b^14, 否], [3, 11, 0, a^3b^14, 否], [4, 1, 9, a^13b^14, 否], [4, 2, 8, a^12b^14, 否], [4, 3, 7, a^11b^14, 否], [4, 4, 6, a^10b^14, 否], [4, 5, 5, a^9b^14, 否], [4, 6, 4, a^8b^14, 否], [4, 7, 3, a^7b^14, 否], [4, 8, 2, a^6b^14, 否], [4, 9, 1, a^5b^14, 否], [4, 10, 0, a^4b^14, 否], [5, 1, 8, a^13b^14, 否], [5, 2, 7, a^12b^14, 否], [5, 3, 6, a^11b^14, 否], [5, 4, 5, a^10b^14, 否], [5, 5, 4, a^9b^14, 否], [5, 6, 3, a^8b^14, 否], [5, 7, 2, a^7b^14, 否], [5, 8, 1, a^6b^14, 否], [5, 9, 0, a^5b^14, 否], [6, 1, 7, a^13b^14, 否], [6, 2, 6, a^12b^14, 否], [6, 3, 5, a^11b^14, 否], [6, 4, 4, a^10b^14, 否], [6, 5, 3, a^9b^14, 否], [6, 6, 2, a^8b^14, 否], [6, 7, 1, a^7b^14, 否], [6, 8, 0, a^6b^14, 否], [7, 1, 6, a^13b^14, 否], [7, 2, 5, a^12b^14, 否], [7, 3, 4, a^11b^14, 否], [7, 4, 3, a^10b^14, 否], [7, 5, 2, a^9b^14, 否], [7, 6, 1, a^8b^14, 否], [7, 7, 0, a^7b^14, 否], [8, 1, 5, a^13b^14, 否], [8, 2, 4, a^12b^14, 否], [8, 3, 3, a^11b^14, 否], [8, 4, 2, a^10b^14, 否], [8, 5, 1, a^9b^14, 否], [8, 6, 0, a^8b^14, 否], [9, 1, 4, a^13b^14, 否], [9, 2, 3, a^12b^14, 否], [9, 3, 2, a^11b^14, 否], [9, 4, 1, a^10b^14, 否], [9, 5, 0, a^9b^14, 否], [10, 1, 3, a^13b^14, 否], [10, 2, 2, a^12b^14, 否], [10, 3, 1, a^11b^14, 否], [10, 4, 0, a^10b^14, 否], [11, 1, 2, a^13b^14, 否], [11, 2, 1, a^12b^14, 否], [11, 3, 0, a^11b^14, 否], [12, 1, 1, a^13b^14, 否], [12, 2, 0, a^12b^14, 否], [13, 1, 0, a^13b^14, 否]], 105, 105, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(14, a^14b^14, [[0, 1, 13, a^13b^14, 否], [0, 2, 12, a^12b^14, 否], [0, 3, 11, a^11b^14, 否], [0, 4, 10, a^10b^14, 否], [0, 5, 9, a^9b^14, 否], [0, 6, 8, a^8b^14, 否], [0, 7, 7, a^7b^14, 否], [0, 8, 6, a^6b^14, 否], [0, 9, 5, a^5b^14, 否], [0, 10, 4, a^4b^14, 否], [0, 11, 3, a^3b^14, 否], [0, 12, 2, a^2b^14, 否], [0, 13, 1, a^1b^14, 否], [0, 14, 0, a^0b^14, 否], [1, 1, 12, a^13b^14, 否], [1, 2, 11, a^12b^14, 否], [1, 3, 10, a^11b^14, 否], [1, 4, 9, a^10b^14, 否], [1, 5, 8, a^9b^14, 否], [1, 6, 7, a^8b^14, 否], [1, 7, 6, a^7b^14, 否], [1, 8, 5, a^6b^14, 否], [1, 9, 4, a^5b^14, 否], [1, 10, 3, a^4b^14, 否], [1, 11, 2, a^3b^14, 否], [1, 12, 1, a^2b^14, 否], [1, 13, 0, a^1b^14, 否], [2, 1, 11, a^13b^14, 否], [2, 2, 10, a^12b^14, 否], [2, 3, 9, a^11b^14, 否], [2, 4, 8, a^10b^14, 否], [2, 5, 7, a^9b^14, 否], [2, 6, 6, a^8b^14, 否], [2, 7, 5, a^7b^14, 否], [2, 8, 4, a^6b^14, 否], [2, 9, 3, a^5b^14, 否], [2, 10, 2, a^4b^14, 否], [2, 11, 1, a^3b^14, 否], [2, 12, 0, a^2b^14, 否], [3, 1, 10, a^13b^14, 否], [3, 2, 9, a^12b^14, 否], [3, 3, 8, a^11b^14, 否], [3, 4, 7, a^10b^14, 否], [3, 5, 6, a^9b^14, 否], [3, 6, 5, a^8b^14, 否], [3, 7, 4, a^7b^14, 否], [3, 8, 3, a^6b^14, 否], [3, 9, 2, a^5b^14, 否], [3, 10, 1, a^4b^14, 否], [3, 11, 0, a^3b^14, 否], [4, 1, 9, a^13b^14, 否], [4, 2, 8, a^12b^14, 否], [4, 3, 7, a^11b^14, 否], [4, 4, 6, a^10b^14, 否], [4, 5, 5, a^9b^14, 否], [4, 6, 4, a^8b^14, 否], [4, 7, 3, a^7b^14, 否], [4, 8, 2, a^6b^14, 否], [4, 9, 1, a^5b^14, 否], [4, 10, 0, a^4b^14, 否], [5, 1, 8, a^13b^14, 否], [5, 2, 7, a^12b^14, 否], [5, 3, 6, a^11b^14, 否], [5, 4, 5, a^10b^14, 否], [5, 5, 4, a^9b^14, 否], [5, 6, 3, a^8b^14, 否], [5, 7, 2, a^7b^14, 否], [5, 8, 1, a^6b^14, 否], [5, 9, 0, a^5b^14, 否], [6, 1, 7, a^13b^14, 否], [6, 2, 6, a^12b^14, 否], [6, 3, 5, a^11b^14, 否], [6, 4, 4, a^10b^14, 否], [6, 5, 3, a^9b^14, 否], [6, 6, 2, a^8b^14, 否], [6, 7, 1, a^7b^14, 否], [6, 8, 0, a^6b^14, 否], [7, 1, 6, a^13b^14, 否], [7, 2, 5, a^12b^14, 否], [7, 3, 4, a^11b^14, 否], [7, 4, 3, a^10b^14, 否], [7, 5, 2, a^9b^14, 否], [7, 6, 1, a^8b^14, 否], [7, 7, 0, a^7b^14, 否], [8, 1, 5, a^13b^14, 否], [8, 2, 4, a^12b^14, 否], [8, 3, 3, a^11b^14, 否], [8, 4, 2, a^10b^14, 否], [8, 5, 1, a^9b^14, 否], [8, 6, 0, a^8b^14, 否], [9, 1, 4, a^13b^14, 否], [9, 2, 3, a^12b^14, 否], [9, 3, 2, a^11b^14, 否], [9, 4, 1, a^10b^14, 否], [9, 5, 0, a^9b^14, 否], [10, 1, 3, a^13b^14, 否], [10, 2, 2, a^12b^14, 否], [10, 3, 1, a^11b^14, 否], [10, 4, 0, a^10b^14, 否], [11, 1, 2, a^13b^14, 否], [11, 2, 1, a^12b^14, 否], [11, 3, 0, a^11b^14, 否], [12, 1, 1, a^13b^14, 否], [12, 2, 0, a^12b^14, 否], [13, 1, 0, a^13b^14, 否]], 105, 105, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

65. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":15}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(15, a^15b^15, [[0, 1, 14, a^14b^15, 否], [0, 2, 13, a^13b^15, 否], [0, 3, 12, a^12b^15, 否], [0, 4, 11, a^11b^15, 否], [0, 5, 10, a^10b^15, 否], [0, 6, 9, a^9b^15, 否], [0, 7, 8, a^8b^15, 否], [0, 8, 7, a^7b^15, 否], [0, 9, 6, a^6b^15, 否], [0, 10, 5, a^5b^15, 否], [0, 11, 4, a^4b^15, 否], [0, 12, 3, a^3b^15, 否], [0, 13, 2, a^2b^15, 否], [0, 14, 1, a^1b^15, 否], [0, 15, 0, a^0b^15, 否], [1, 1, 13, a^14b^15, 否], [1, 2, 12, a^13b^15, 否], [1, 3, 11, a^12b^15, 否], [1, 4, 10, a^11b^15, 否], [1, 5, 9, a^10b^15, 否], [1, 6, 8, a^9b^15, 否], [1, 7, 7, a^8b^15, 否], [1, 8, 6, a^7b^15, 否], [1, 9, 5, a^6b^15, 否], [1, 10, 4, a^5b^15, 否], [1, 11, 3, a^4b^15, 否], [1, 12, 2, a^3b^15, 否], [1, 13, 1, a^2b^15, 否], [1, 14, 0, a^1b^15, 否], [2, 1, 12, a^14b^15, 否], [2, 2, 11, a^13b^15, 否], [2, 3, 10, a^12b^15, 否], [2, 4, 9, a^11b^15, 否], [2, 5, 8, a^10b^15, 否], [2, 6, 7, a^9b^15, 否], [2, 7, 6, a^8b^15, 否], [2, 8, 5, a^7b^15, 否], [2, 9, 4, a^6b^15, 否], [2, 10, 3, a^5b^15, 否], [2, 11, 2, a^4b^15, 否], [2, 12, 1, a^3b^15, 否], [2, 13, 0, a^2b^15, 否], [3, 1, 11, a^14b^15, 否], [3, 2, 10, a^13b^15, 否], [3, 3, 9, a^12b^15, 否], [3, 4, 8, a^11b^15, 否], [3, 5, 7, a^10b^15, 否], [3, 6, 6, a^9b^15, 否], [3, 7, 5, a^8b^15, 否], [3, 8, 4, a^7b^15, 否], [3, 9, 3, a^6b^15, 否], [3, 10, 2, a^5b^15, 否], [3, 11, 1, a^4b^15, 否], [3, 12, 0, a^3b^15, 否], [4, 1, 10, a^14b^15, 否], [4, 2, 9, a^13b^15, 否], [4, 3, 8, a^12b^15, 否], [4, 4, 7, a^11b^15, 否], [4, 5, 6, a^10b^15, 否], [4, 6, 5, a^9b^15, 否], [4, 7, 4, a^8b^15, 否], [4, 8, 3, a^7b^15, 否], [4, 9, 2, a^6b^15, 否], [4, 10, 1, a^5b^15, 否], [4, 11, 0, a^4b^15, 否], [5, 1, 9, a^14b^15, 否], [5, 2, 8, a^13b^15, 否], [5, 3, 7, a^12b^15, 否], [5, 4, 6, a^11b^15, 否], [5, 5, 5, a^10b^15, 否], [5, 6, 4, a^9b^15, 否], [5, 7, 3, a^8b^15, 否], [5, 8, 2, a^7b^15, 否], [5, 9, 1, a^6b^15, 否], [5, 10, 0, a^5b^15, 否], [6, 1, 8, a^14b^15, 否], [6, 2, 7, a^13b^15, 否], [6, 3, 6, a^12b^15, 否], [6, 4, 5, a^11b^15, 否], [6, 5, 4, a^10b^15, 否], [6, 6, 3, a^9b^15, 否], [6, 7, 2, a^8b^15, 否], [6, 8, 1, a^7b^15, 否], [6, 9, 0, a^6b^15, 否], [7, 1, 7, a^14b^15, 否], [7, 2, 6, a^13b^15, 否], [7, 3, 5, a^12b^15, 否], [7, 4, 4, a^11b^15, 否], [7, 5, 3, a^10b^15, 否], [7, 6, 2, a^9b^15, 否], [7, 7, 1, a^8b^15, 否], [7, 8, 0, a^7b^15, 否], [8, 1, 6, a^14b^15, 否], [8, 2, 5, a^13b^15, 否], [8, 3, 4, a^12b^15, 否], [8, 4, 3, a^11b^15, 否], [8, 5, 2, a^10b^15, 否], [8, 6, 1, a^9b^15, 否], [8, 7, 0, a^8b^15, 否], [9, 1, 5, a^14b^15, 否], [9, 2, 4, a^13b^15, 否], [9, 3, 3, a^12b^15, 否], [9, 4, 2, a^11b^15, 否], [9, 5, 1, a^10b^15, 否], [9, 6, 0, a^9b^15, 否], [10, 1, 4, a^14b^15, 否], [10, 2, 3, a^13b^15, 否], [10, 3, 2, a^12b^15, 否], [10, 4, 1, a^11b^15, 否], [10, 5, 0, a^10b^15, 否], [11, 1, 3, a^14b^15, 否], [11, 2, 2, a^13b^15, 否], [11, 3, 1, a^12b^15, 否], [11, 4, 0, a^11b^15, 否], [12, 1, 2, a^14b^15, 否], [12, 2, 1, a^13b^15, 否], [12, 3, 0, a^12b^15, 否], [13, 1, 1, a^14b^15, 否], [13, 2, 0, a^13b^15, 否], [14, 1, 0, a^14b^15, 否]], 120, 120, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(15, a^15b^15, [[0, 1, 14, a^14b^15, 否], [0, 2, 13, a^13b^15, 否], [0, 3, 12, a^12b^15, 否], [0, 4, 11, a^11b^15, 否], [0, 5, 10, a^10b^15, 否], [0, 6, 9, a^9b^15, 否], [0, 7, 8, a^8b^15, 否], [0, 8, 7, a^7b^15, 否], [0, 9, 6, a^6b^15, 否], [0, 10, 5, a^5b^15, 否], [0, 11, 4, a^4b^15, 否], [0, 12, 3, a^3b^15, 否], [0, 13, 2, a^2b^15, 否], [0, 14, 1, a^1b^15, 否], [0, 15, 0, a^0b^15, 否], [1, 1, 13, a^14b^15, 否], [1, 2, 12, a^13b^15, 否], [1, 3, 11, a^12b^15, 否], [1, 4, 10, a^11b^15, 否], [1, 5, 9, a^10b^15, 否], [1, 6, 8, a^9b^15, 否], [1, 7, 7, a^8b^15, 否], [1, 8, 6, a^7b^15, 否], [1, 9, 5, a^6b^15, 否], [1, 10, 4, a^5b^15, 否], [1, 11, 3, a^4b^15, 否], [1, 12, 2, a^3b^15, 否], [1, 13, 1, a^2b^15, 否], [1, 14, 0, a^1b^15, 否], [2, 1, 12, a^14b^15, 否], [2, 2, 11, a^13b^15, 否], [2, 3, 10, a^12b^15, 否], [2, 4, 9, a^11b^15, 否], [2, 5, 8, a^10b^15, 否], [2, 6, 7, a^9b^15, 否], [2, 7, 6, a^8b^15, 否], [2, 8, 5, a^7b^15, 否], [2, 9, 4, a^6b^15, 否], [2, 10, 3, a^5b^15, 否], [2, 11, 2, a^4b^15, 否], [2, 12, 1, a^3b^15, 否], [2, 13, 0, a^2b^15, 否], [3, 1, 11, a^14b^15, 否], [3, 2, 10, a^13b^15, 否], [3, 3, 9, a^12b^15, 否], [3, 4, 8, a^11b^15, 否], [3, 5, 7, a^10b^15, 否], [3, 6, 6, a^9b^15, 否], [3, 7, 5, a^8b^15, 否], [3, 8, 4, a^7b^15, 否], [3, 9, 3, a^6b^15, 否], [3, 10, 2, a^5b^15, 否], [3, 11, 1, a^4b^15, 否], [3, 12, 0, a^3b^15, 否], [4, 1, 10, a^14b^15, 否], [4, 2, 9, a^13b^15, 否], [4, 3, 8, a^12b^15, 否], [4, 4, 7, a^11b^15, 否], [4, 5, 6, a^10b^15, 否], [4, 6, 5, a^9b^15, 否], [4, 7, 4, a^8b^15, 否], [4, 8, 3, a^7b^15, 否], [4, 9, 2, a^6b^15, 否], [4, 10, 1, a^5b^15, 否], [4, 11, 0, a^4b^15, 否], [5, 1, 9, a^14b^15, 否], [5, 2, 8, a^13b^15, 否], [5, 3, 7, a^12b^15, 否], [5, 4, 6, a^11b^15, 否], [5, 5, 5, a^10b^15, 否], [5, 6, 4, a^9b^15, 否], [5, 7, 3, a^8b^15, 否], [5, 8, 2, a^7b^15, 否], [5, 9, 1, a^6b^15, 否], [5, 10, 0, a^5b^15, 否], [6, 1, 8, a^14b^15, 否], [6, 2, 7, a^13b^15, 否], [6, 3, 6, a^12b^15, 否], [6, 4, 5, a^11b^15, 否], [6, 5, 4, a^10b^15, 否], [6, 6, 3, a^9b^15, 否], [6, 7, 2, a^8b^15, 否], [6, 8, 1, a^7b^15, 否], [6, 9, 0, a^6b^15, 否], [7, 1, 7, a^14b^15, 否], [7, 2, 6, a^13b^15, 否], [7, 3, 5, a^12b^15, 否], [7, 4, 4, a^11b^15, 否], [7, 5, 3, a^10b^15, 否], [7, 6, 2, a^9b^15, 否], [7, 7, 1, a^8b^15, 否], [7, 8, 0, a^7b^15, 否], [8, 1, 6, a^14b^15, 否], [8, 2, 5, a^13b^15, 否], [8, 3, 4, a^12b^15, 否], [8, 4, 3, a^11b^15, 否], [8, 5, 2, a^10b^15, 否], [8, 6, 1, a^9b^15, 否], [8, 7, 0, a^8b^15, 否], [9, 1, 5, a^14b^15, 否], [9, 2, 4, a^13b^15, 否], [9, 3, 3, a^12b^15, 否], [9, 4, 2, a^11b^15, 否], [9, 5, 1, a^10b^15, 否], [9, 6, 0, a^9b^15, 否], [10, 1, 4, a^14b^15, 否], [10, 2, 3, a^13b^15, 否], [10, 3, 2, a^12b^15, 否], [10, 4, 1, a^11b^15, 否], [10, 5, 0, a^10b^15, 否], [11, 1, 3, a^14b^15, 否], [11, 2, 2, a^13b^15, 否], [11, 3, 1, a^12b^15, 否], [11, 4, 0, a^11b^15, 否], [12, 1, 2, a^14b^15, 否], [12, 2, 1, a^13b^15, 否], [12, 3, 0, a^12b^15, 否], [13, 1, 1, a^14b^15, 否], [13, 2, 0, a^13b^15, 否], [14, 1, 0, a^14b^15, 否]], 120, 120, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

66. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":16}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(16, a^16b^16, [[0, 1, 15, a^15b^16, 否], [0, 2, 14, a^14b^16, 否], [0, 3, 13, a^13b^16, 否], [0, 4, 12, a^12b^16, 否], [0, 5, 11, a^11b^16, 否], [0, 6, 10, a^10b^16, 否], [0, 7, 9, a^9b^16, 否], [0, 8, 8, a^8b^16, 否], [0, 9, 7, a^7b^16, 否], [0, 10, 6, a^6b^16, 否], [0, 11, 5, a^5b^16, 否], [0, 12, 4, a^4b^16, 否], [0, 13, 3, a^3b^16, 否], [0, 14, 2, a^2b^16, 否], [0, 15, 1, a^1b^16, 否], [0, 16, 0, a^0b^16, 否], [1, 1, 14, a^15b^16, 否], [1, 2, 13, a^14b^16, 否], [1, 3, 12, a^13b^16, 否], [1, 4, 11, a^12b^16, 否], [1, 5, 10, a^11b^16, 否], [1, 6, 9, a^10b^16, 否], [1, 7, 8, a^9b^16, 否], [1, 8, 7, a^8b^16, 否], [1, 9, 6, a^7b^16, 否], [1, 10, 5, a^6b^16, 否], [1, 11, 4, a^5b^16, 否], [1, 12, 3, a^4b^16, 否], [1, 13, 2, a^3b^16, 否], [1, 14, 1, a^2b^16, 否], [1, 15, 0, a^1b^16, 否], [2, 1, 13, a^15b^16, 否], [2, 2, 12, a^14b^16, 否], [2, 3, 11, a^13b^16, 否], [2, 4, 10, a^12b^16, 否], [2, 5, 9, a^11b^16, 否], [2, 6, 8, a^10b^16, 否], [2, 7, 7, a^9b^16, 否], [2, 8, 6, a^8b^16, 否], [2, 9, 5, a^7b^16, 否], [2, 10, 4, a^6b^16, 否], [2, 11, 3, a^5b^16, 否], [2, 12, 2, a^4b^16, 否], [2, 13, 1, a^3b^16, 否], [2, 14, 0, a^2b^16, 否], [3, 1, 12, a^15b^16, 否], [3, 2, 11, a^14b^16, 否], [3, 3, 10, a^13b^16, 否], [3, 4, 9, a^12b^16, 否], [3, 5, 8, a^11b^16, 否], [3, 6, 7, a^10b^16, 否], [3, 7, 6, a^9b^16, 否], [3, 8, 5, a^8b^16, 否], [3, 9, 4, a^7b^16, 否], [3, 10, 3, a^6b^16, 否], [3, 11, 2, a^5b^16, 否], [3, 12, 1, a^4b^16, 否], [3, 13, 0, a^3b^16, 否], [4, 1, 11, a^15b^16, 否], [4, 2, 10, a^14b^16, 否], [4, 3, 9, a^13b^16, 否], [4, 4, 8, a^12b^16, 否], [4, 5, 7, a^11b^16, 否], [4, 6, 6, a^10b^16, 否], [4, 7, 5, a^9b^16, 否], [4, 8, 4, a^8b^16, 否], [4, 9, 3, a^7b^16, 否], [4, 10, 2, a^6b^16, 否], [4, 11, 1, a^5b^16, 否], [4, 12, 0, a^4b^16, 否], [5, 1, 10, a^15b^16, 否], [5, 2, 9, a^14b^16, 否], [5, 3, 8, a^13b^16, 否], [5, 4, 7, a^12b^16, 否], [5, 5, 6, a^11b^16, 否], [5, 6, 5, a^10b^16, 否], [5, 7, 4, a^9b^16, 否], [5, 8, 3, a^8b^16, 否], [5, 9, 2, a^7b^16, 否], [5, 10, 1, a^6b^16, 否], [5, 11, 0, a^5b^16, 否], [6, 1, 9, a^15b^16, 否], [6, 2, 8, a^14b^16, 否], [6, 3, 7, a^13b^16, 否], [6, 4, 6, a^12b^16, 否], [6, 5, 5, a^11b^16, 否], [6, 6, 4, a^10b^16, 否], [6, 7, 3, a^9b^16, 否], [6, 8, 2, a^8b^16, 否], [6, 9, 1, a^7b^16, 否], [6, 10, 0, a^6b^16, 否], [7, 1, 8, a^15b^16, 否], [7, 2, 7, a^14b^16, 否], [7, 3, 6, a^13b^16, 否], [7, 4, 5, a^12b^16, 否], [7, 5, 4, a^11b^16, 否], [7, 6, 3, a^10b^16, 否], [7, 7, 2, a^9b^16, 否], [7, 8, 1, a^8b^16, 否], [7, 9, 0, a^7b^16, 否], [8, 1, 7, a^15b^16, 否], [8, 2, 6, a^14b^16, 否], [8, 3, 5, a^13b^16, 否], [8, 4, 4, a^12b^16, 否], [8, 5, 3, a^11b^16, 否], [8, 6, 2, a^10b^16, 否], [8, 7, 1, a^9b^16, 否], [8, 8, 0, a^8b^16, 否], [9, 1, 6, a^15b^16, 否], [9, 2, 5, a^14b^16, 否], [9, 3, 4, a^13b^16, 否], [9, 4, 3, a^12b^16, 否], [9, 5, 2, a^11b^16, 否], [9, 6, 1, a^10b^16, 否], [9, 7, 0, a^9b^16, 否], [10, 1, 5, a^15b^16, 否], [10, 2, 4, a^14b^16, 否], [10, 3, 3, a^13b^16, 否], [10, 4, 2, a^12b^16, 否], [10, 5, 1, a^11b^16, 否], [10, 6, 0, a^10b^16, 否], [11, 1, 4, a^15b^16, 否], [11, 2, 3, a^14b^16, 否], [11, 3, 2, a^13b^16, 否], [11, 4, 1, a^12b^16, 否], [11, 5, 0, a^11b^16, 否], [12, 1, 3, a^15b^16, 否], [12, 2, 2, a^14b^16, 否], [12, 3, 1, a^13b^16, 否], [12, 4, 0, a^12b^16, 否], [13, 1, 2, a^15b^16, 否], [13, 2, 1, a^14b^16, 否], [13, 3, 0, a^13b^16, 否], [14, 1, 1, a^15b^16, 否], [14, 2, 0, a^14b^16, 否], [15, 1, 0, a^15b^16, 否]], 136, 136, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(16, a^16b^16, [[0, 1, 15, a^15b^16, 否], [0, 2, 14, a^14b^16, 否], [0, 3, 13, a^13b^16, 否], [0, 4, 12, a^12b^16, 否], [0, 5, 11, a^11b^16, 否], [0, 6, 10, a^10b^16, 否], [0, 7, 9, a^9b^16, 否], [0, 8, 8, a^8b^16, 否], [0, 9, 7, a^7b^16, 否], [0, 10, 6, a^6b^16, 否], [0, 11, 5, a^5b^16, 否], [0, 12, 4, a^4b^16, 否], [0, 13, 3, a^3b^16, 否], [0, 14, 2, a^2b^16, 否], [0, 15, 1, a^1b^16, 否], [0, 16, 0, a^0b^16, 否], [1, 1, 14, a^15b^16, 否], [1, 2, 13, a^14b^16, 否], [1, 3, 12, a^13b^16, 否], [1, 4, 11, a^12b^16, 否], [1, 5, 10, a^11b^16, 否], [1, 6, 9, a^10b^16, 否], [1, 7, 8, a^9b^16, 否], [1, 8, 7, a^8b^16, 否], [1, 9, 6, a^7b^16, 否], [1, 10, 5, a^6b^16, 否], [1, 11, 4, a^5b^16, 否], [1, 12, 3, a^4b^16, 否], [1, 13, 2, a^3b^16, 否], [1, 14, 1, a^2b^16, 否], [1, 15, 0, a^1b^16, 否], [2, 1, 13, a^15b^16, 否], [2, 2, 12, a^14b^16, 否], [2, 3, 11, a^13b^16, 否], [2, 4, 10, a^12b^16, 否], [2, 5, 9, a^11b^16, 否], [2, 6, 8, a^10b^16, 否], [2, 7, 7, a^9b^16, 否], [2, 8, 6, a^8b^16, 否], [2, 9, 5, a^7b^16, 否], [2, 10, 4, a^6b^16, 否], [2, 11, 3, a^5b^16, 否], [2, 12, 2, a^4b^16, 否], [2, 13, 1, a^3b^16, 否], [2, 14, 0, a^2b^16, 否], [3, 1, 12, a^15b^16, 否], [3, 2, 11, a^14b^16, 否], [3, 3, 10, a^13b^16, 否], [3, 4, 9, a^12b^16, 否], [3, 5, 8, a^11b^16, 否], [3, 6, 7, a^10b^16, 否], [3, 7, 6, a^9b^16, 否], [3, 8, 5, a^8b^16, 否], [3, 9, 4, a^7b^16, 否], [3, 10, 3, a^6b^16, 否], [3, 11, 2, a^5b^16, 否], [3, 12, 1, a^4b^16, 否], [3, 13, 0, a^3b^16, 否], [4, 1, 11, a^15b^16, 否], [4, 2, 10, a^14b^16, 否], [4, 3, 9, a^13b^16, 否], [4, 4, 8, a^12b^16, 否], [4, 5, 7, a^11b^16, 否], [4, 6, 6, a^10b^16, 否], [4, 7, 5, a^9b^16, 否], [4, 8, 4, a^8b^16, 否], [4, 9, 3, a^7b^16, 否], [4, 10, 2, a^6b^16, 否], [4, 11, 1, a^5b^16, 否], [4, 12, 0, a^4b^16, 否], [5, 1, 10, a^15b^16, 否], [5, 2, 9, a^14b^16, 否], [5, 3, 8, a^13b^16, 否], [5, 4, 7, a^12b^16, 否], [5, 5, 6, a^11b^16, 否], [5, 6, 5, a^10b^16, 否], [5, 7, 4, a^9b^16, 否], [5, 8, 3, a^8b^16, 否], [5, 9, 2, a^7b^16, 否], [5, 10, 1, a^6b^16, 否], [5, 11, 0, a^5b^16, 否], [6, 1, 9, a^15b^16, 否], [6, 2, 8, a^14b^16, 否], [6, 3, 7, a^13b^16, 否], [6, 4, 6, a^12b^16, 否], [6, 5, 5, a^11b^16, 否], [6, 6, 4, a^10b^16, 否], [6, 7, 3, a^9b^16, 否], [6, 8, 2, a^8b^16, 否], [6, 9, 1, a^7b^16, 否], [6, 10, 0, a^6b^16, 否], [7, 1, 8, a^15b^16, 否], [7, 2, 7, a^14b^16, 否], [7, 3, 6, a^13b^16, 否], [7, 4, 5, a^12b^16, 否], [7, 5, 4, a^11b^16, 否], [7, 6, 3, a^10b^16, 否], [7, 7, 2, a^9b^16, 否], [7, 8, 1, a^8b^16, 否], [7, 9, 0, a^7b^16, 否], [8, 1, 7, a^15b^16, 否], [8, 2, 6, a^14b^16, 否], [8, 3, 5, a^13b^16, 否], [8, 4, 4, a^12b^16, 否], [8, 5, 3, a^11b^16, 否], [8, 6, 2, a^10b^16, 否], [8, 7, 1, a^9b^16, 否], [8, 8, 0, a^8b^16, 否], [9, 1, 6, a^15b^16, 否], [9, 2, 5, a^14b^16, 否], [9, 3, 4, a^13b^16, 否], [9, 4, 3, a^12b^16, 否], [9, 5, 2, a^11b^16, 否], [9, 6, 1, a^10b^16, 否], [9, 7, 0, a^9b^16, 否], [10, 1, 5, a^15b^16, 否], [10, 2, 4, a^14b^16, 否], [10, 3, 3, a^13b^16, 否], [10, 4, 2, a^12b^16, 否], [10, 5, 1, a^11b^16, 否], [10, 6, 0, a^10b^16, 否], [11, 1, 4, a^15b^16, 否], [11, 2, 3, a^14b^16, 否], [11, 3, 2, a^13b^16, 否], [11, 4, 1, a^12b^16, 否], [11, 5, 0, a^11b^16, 否], [12, 1, 3, a^15b^16, 否], [12, 2, 2, a^14b^16, 否], [12, 3, 1, a^13b^16, 否], [12, 4, 0, a^12b^16, 否], [13, 1, 2, a^15b^16, 否], [13, 2, 1, a^14b^16, 否], [13, 3, 0, a^13b^16, 否], [14, 1, 1, a^15b^16, 否], [14, 2, 0, a^14b^16, 否], [15, 1, 0, a^15b^16, 否]], 136, 136, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

67. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":17}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(17, a^17b^17, [[0, 1, 16, a^16b^17, 否], [0, 2, 15, a^15b^17, 否], [0, 3, 14, a^14b^17, 否], [0, 4, 13, a^13b^17, 否], [0, 5, 12, a^12b^17, 否], [0, 6, 11, a^11b^17, 否], [0, 7, 10, a^10b^17, 否], [0, 8, 9, a^9b^17, 否], [0, 9, 8, a^8b^17, 否], [0, 10, 7, a^7b^17, 否], [0, 11, 6, a^6b^17, 否], [0, 12, 5, a^5b^17, 否], [0, 13, 4, a^4b^17, 否], [0, 14, 3, a^3b^17, 否], [0, 15, 2, a^2b^17, 否], [0, 16, 1, a^1b^17, 否], [0, 17, 0, a^0b^17, 否], [1, 1, 15, a^16b^17, 否], [1, 2, 14, a^15b^17, 否], [1, 3, 13, a^14b^17, 否], [1, 4, 12, a^13b^17, 否], [1, 5, 11, a^12b^17, 否], [1, 6, 10, a^11b^17, 否], [1, 7, 9, a^10b^17, 否], [1, 8, 8, a^9b^17, 否], [1, 9, 7, a^8b^17, 否], [1, 10, 6, a^7b^17, 否], [1, 11, 5, a^6b^17, 否], [1, 12, 4, a^5b^17, 否], [1, 13, 3, a^4b^17, 否], [1, 14, 2, a^3b^17, 否], [1, 15, 1, a^2b^17, 否], [1, 16, 0, a^1b^17, 否], [2, 1, 14, a^16b^17, 否], [2, 2, 13, a^15b^17, 否], [2, 3, 12, a^14b^17, 否], [2, 4, 11, a^13b^17, 否], [2, 5, 10, a^12b^17, 否], [2, 6, 9, a^11b^17, 否], [2, 7, 8, a^10b^17, 否], [2, 8, 7, a^9b^17, 否], [2, 9, 6, a^8b^17, 否], [2, 10, 5, a^7b^17, 否], [2, 11, 4, a^6b^17, 否], [2, 12, 3, a^5b^17, 否], [2, 13, 2, a^4b^17, 否], [2, 14, 1, a^3b^17, 否], [2, 15, 0, a^2b^17, 否], [3, 1, 13, a^16b^17, 否], [3, 2, 12, a^15b^17, 否], [3, 3, 11, a^14b^17, 否], [3, 4, 10, a^13b^17, 否], [3, 5, 9, a^12b^17, 否], [3, 6, 8, a^11b^17, 否], [3, 7, 7, a^10b^17, 否], [3, 8, 6, a^9b^17, 否], [3, 9, 5, a^8b^17, 否], [3, 10, 4, a^7b^17, 否], [3, 11, 3, a^6b^17, 否], [3, 12, 2, a^5b^17, 否], [3, 13, 1, a^4b^17, 否], [3, 14, 0, a^3b^17, 否], [4, 1, 12, a^16b^17, 否], [4, 2, 11, a^15b^17, 否], [4, 3, 10, a^14b^17, 否], [4, 4, 9, a^13b^17, 否], [4, 5, 8, a^12b^17, 否], [4, 6, 7, a^11b^17, 否], [4, 7, 6, a^10b^17, 否], [4, 8, 5, a^9b^17, 否], [4, 9, 4, a^8b^17, 否], [4, 10, 3, a^7b^17, 否], [4, 11, 2, a^6b^17, 否], [4, 12, 1, a^5b^17, 否], [4, 13, 0, a^4b^17, 否], [5, 1, 11, a^16b^17, 否], [5, 2, 10, a^15b^17, 否], [5, 3, 9, a^14b^17, 否], [5, 4, 8, a^13b^17, 否], [5, 5, 7, a^12b^17, 否], [5, 6, 6, a^11b^17, 否], [5, 7, 5, a^10b^17, 否], [5, 8, 4, a^9b^17, 否], [5, 9, 3, a^8b^17, 否], [5, 10, 2, a^7b^17, 否], [5, 11, 1, a^6b^17, 否], [5, 12, 0, a^5b^17, 否], [6, 1, 10, a^16b^17, 否], [6, 2, 9, a^15b^17, 否], [6, 3, 8, a^14b^17, 否], [6, 4, 7, a^13b^17, 否], [6, 5, 6, a^12b^17, 否], [6, 6, 5, a^11b^17, 否], [6, 7, 4, a^10b^17, 否], [6, 8, 3, a^9b^17, 否], [6, 9, 2, a^8b^17, 否], [6, 10, 1, a^7b^17, 否], [6, 11, 0, a^6b^17, 否], [7, 1, 9, a^16b^17, 否], [7, 2, 8, a^15b^17, 否], [7, 3, 7, a^14b^17, 否], [7, 4, 6, a^13b^17, 否], [7, 5, 5, a^12b^17, 否], [7, 6, 4, a^11b^17, 否], [7, 7, 3, a^10b^17, 否], [7, 8, 2, a^9b^17, 否], [7, 9, 1, a^8b^17, 否], [7, 10, 0, a^7b^17, 否], [8, 1, 8, a^16b^17, 否], [8, 2, 7, a^15b^17, 否], [8, 3, 6, a^14b^17, 否], [8, 4, 5, a^13b^17, 否], [8, 5, 4, a^12b^17, 否], [8, 6, 3, a^11b^17, 否], [8, 7, 2, a^10b^17, 否], [8, 8, 1, a^9b^17, 否], [8, 9, 0, a^8b^17, 否], [9, 1, 7, a^16b^17, 否], [9, 2, 6, a^15b^17, 否], [9, 3, 5, a^14b^17, 否], [9, 4, 4, a^13b^17, 否], [9, 5, 3, a^12b^17, 否], [9, 6, 2, a^11b^17, 否], [9, 7, 1, a^10b^17, 否], [9, 8, 0, a^9b^17, 否], [10, 1, 6, a^16b^17, 否], [10, 2, 5, a^15b^17, 否], [10, 3, 4, a^14b^17, 否], [10, 4, 3, a^13b^17, 否], [10, 5, 2, a^12b^17, 否], [10, 6, 1, a^11b^17, 否], [10, 7, 0, a^10b^17, 否], [11, 1, 5, a^16b^17, 否], [11, 2, 4, a^15b^17, 否], [11, 3, 3, a^14b^17, 否], [11, 4, 2, a^13b^17, 否], [11, 5, 1, a^12b^17, 否], [11, 6, 0, a^11b^17, 否], [12, 1, 4, a^16b^17, 否], [12, 2, 3, a^15b^17, 否], [12, 3, 2, a^14b^17, 否], [12, 4, 1, a^13b^17, 否], [12, 5, 0, a^12b^17, 否], [13, 1, 3, a^16b^17, 否], [13, 2, 2, a^15b^17, 否], [13, 3, 1, a^14b^17, 否], [13, 4, 0, a^13b^17, 否], [14, 1, 2, a^16b^17, 否], [14, 2, 1, a^15b^17, 否], [14, 3, 0, a^14b^17, 否], [15, 1, 1, a^16b^17, 否], [15, 2, 0, a^15b^17, 否], [16, 1, 0, a^16b^17, 否]], 153, 153, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(17, a^17b^17, [[0, 1, 16, a^16b^17, 否], [0, 2, 15, a^15b^17, 否], [0, 3, 14, a^14b^17, 否], [0, 4, 13, a^13b^17, 否], [0, 5, 12, a^12b^17, 否], [0, 6, 11, a^11b^17, 否], [0, 7, 10, a^10b^17, 否], [0, 8, 9, a^9b^17, 否], [0, 9, 8, a^8b^17, 否], [0, 10, 7, a^7b^17, 否], [0, 11, 6, a^6b^17, 否], [0, 12, 5, a^5b^17, 否], [0, 13, 4, a^4b^17, 否], [0, 14, 3, a^3b^17, 否], [0, 15, 2, a^2b^17, 否], [0, 16, 1, a^1b^17, 否], [0, 17, 0, a^0b^17, 否], [1, 1, 15, a^16b^17, 否], [1, 2, 14, a^15b^17, 否], [1, 3, 13, a^14b^17, 否], [1, 4, 12, a^13b^17, 否], [1, 5, 11, a^12b^17, 否], [1, 6, 10, a^11b^17, 否], [1, 7, 9, a^10b^17, 否], [1, 8, 8, a^9b^17, 否], [1, 9, 7, a^8b^17, 否], [1, 10, 6, a^7b^17, 否], [1, 11, 5, a^6b^17, 否], [1, 12, 4, a^5b^17, 否], [1, 13, 3, a^4b^17, 否], [1, 14, 2, a^3b^17, 否], [1, 15, 1, a^2b^17, 否], [1, 16, 0, a^1b^17, 否], [2, 1, 14, a^16b^17, 否], [2, 2, 13, a^15b^17, 否], [2, 3, 12, a^14b^17, 否], [2, 4, 11, a^13b^17, 否], [2, 5, 10, a^12b^17, 否], [2, 6, 9, a^11b^17, 否], [2, 7, 8, a^10b^17, 否], [2, 8, 7, a^9b^17, 否], [2, 9, 6, a^8b^17, 否], [2, 10, 5, a^7b^17, 否], [2, 11, 4, a^6b^17, 否], [2, 12, 3, a^5b^17, 否], [2, 13, 2, a^4b^17, 否], [2, 14, 1, a^3b^17, 否], [2, 15, 0, a^2b^17, 否], [3, 1, 13, a^16b^17, 否], [3, 2, 12, a^15b^17, 否], [3, 3, 11, a^14b^17, 否], [3, 4, 10, a^13b^17, 否], [3, 5, 9, a^12b^17, 否], [3, 6, 8, a^11b^17, 否], [3, 7, 7, a^10b^17, 否], [3, 8, 6, a^9b^17, 否], [3, 9, 5, a^8b^17, 否], [3, 10, 4, a^7b^17, 否], [3, 11, 3, a^6b^17, 否], [3, 12, 2, a^5b^17, 否], [3, 13, 1, a^4b^17, 否], [3, 14, 0, a^3b^17, 否], [4, 1, 12, a^16b^17, 否], [4, 2, 11, a^15b^17, 否], [4, 3, 10, a^14b^17, 否], [4, 4, 9, a^13b^17, 否], [4, 5, 8, a^12b^17, 否], [4, 6, 7, a^11b^17, 否], [4, 7, 6, a^10b^17, 否], [4, 8, 5, a^9b^17, 否], [4, 9, 4, a^8b^17, 否], [4, 10, 3, a^7b^17, 否], [4, 11, 2, a^6b^17, 否], [4, 12, 1, a^5b^17, 否], [4, 13, 0, a^4b^17, 否], [5, 1, 11, a^16b^17, 否], [5, 2, 10, a^15b^17, 否], [5, 3, 9, a^14b^17, 否], [5, 4, 8, a^13b^17, 否], [5, 5, 7, a^12b^17, 否], [5, 6, 6, a^11b^17, 否], [5, 7, 5, a^10b^17, 否], [5, 8, 4, a^9b^17, 否], [5, 9, 3, a^8b^17, 否], [5, 10, 2, a^7b^17, 否], [5, 11, 1, a^6b^17, 否], [5, 12, 0, a^5b^17, 否], [6, 1, 10, a^16b^17, 否], [6, 2, 9, a^15b^17, 否], [6, 3, 8, a^14b^17, 否], [6, 4, 7, a^13b^17, 否], [6, 5, 6, a^12b^17, 否], [6, 6, 5, a^11b^17, 否], [6, 7, 4, a^10b^17, 否], [6, 8, 3, a^9b^17, 否], [6, 9, 2, a^8b^17, 否], [6, 10, 1, a^7b^17, 否], [6, 11, 0, a^6b^17, 否], [7, 1, 9, a^16b^17, 否], [7, 2, 8, a^15b^17, 否], [7, 3, 7, a^14b^17, 否], [7, 4, 6, a^13b^17, 否], [7, 5, 5, a^12b^17, 否], [7, 6, 4, a^11b^17, 否], [7, 7, 3, a^10b^17, 否], [7, 8, 2, a^9b^17, 否], [7, 9, 1, a^8b^17, 否], [7, 10, 0, a^7b^17, 否], [8, 1, 8, a^16b^17, 否], [8, 2, 7, a^15b^17, 否], [8, 3, 6, a^14b^17, 否], [8, 4, 5, a^13b^17, 否], [8, 5, 4, a^12b^17, 否], [8, 6, 3, a^11b^17, 否], [8, 7, 2, a^10b^17, 否], [8, 8, 1, a^9b^17, 否], [8, 9, 0, a^8b^17, 否], [9, 1, 7, a^16b^17, 否], [9, 2, 6, a^15b^17, 否], [9, 3, 5, a^14b^17, 否], [9, 4, 4, a^13b^17, 否], [9, 5, 3, a^12b^17, 否], [9, 6, 2, a^11b^17, 否], [9, 7, 1, a^10b^17, 否], [9, 8, 0, a^9b^17, 否], [10, 1, 6, a^16b^17, 否], [10, 2, 5, a^15b^17, 否], [10, 3, 4, a^14b^17, 否], [10, 4, 3, a^13b^17, 否], [10, 5, 2, a^12b^17, 否], [10, 6, 1, a^11b^17, 否], [10, 7, 0, a^10b^17, 否], [11, 1, 5, a^16b^17, 否], [11, 2, 4, a^15b^17, 否], [11, 3, 3, a^14b^17, 否], [11, 4, 2, a^13b^17, 否], [11, 5, 1, a^12b^17, 否], [11, 6, 0, a^11b^17, 否], [12, 1, 4, a^16b^17, 否], [12, 2, 3, a^15b^17, 否], [12, 3, 2, a^14b^17, 否], [12, 4, 1, a^13b^17, 否], [12, 5, 0, a^12b^17, 否], [13, 1, 3, a^16b^17, 否], [13, 2, 2, a^15b^17, 否], [13, 3, 1, a^14b^17, 否], [13, 4, 0, a^13b^17, 否], [14, 1, 2, a^16b^17, 否], [14, 2, 1, a^15b^17, 否], [14, 3, 0, a^14b^17, 否], [15, 1, 1, a^16b^17, 否], [15, 2, 0, a^15b^17, 否], [16, 1, 0, a^16b^17, 否]], 153, 153, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

68. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":18}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

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(18, a^18b^18, [[0, 1, 17, a^17b^18, 否], [0, 2, 16, a^16b^18, 否], [0, 3, 15, a^15b^18, 否], [0, 4, 14, a^14b^18, 否], [0, 5, 13, a^13b^18, 否], [0, 6, 12, a^12b^18, 否], [0, 7, 11, a^11b^18, 否], [0, 8, 10, a^10b^18, 否], [0, 9, 9, a^9b^18, 否], [0, 10, 8, a^8b^18, 否], [0, 11, 7, a^7b^18, 否], [0, 12, 6, a^6b^18, 否], [0, 13, 5, a^5b^18, 否], [0, 14, 4, a^4b^18, 否], [0, 15, 3, a^3b^18, 否], [0, 16, 2, a^2b^18, 否], [0, 17, 1, a^1b^18, 否], [0, 18, 0, a^0b^18, 否], [1, 1, 16, a^17b^18, 否], [1, 2, 15, a^16b^18, 否], [1, 3, 14, a^15b^18, 否], [1, 4, 13, a^14b^18, 否], [1, 5, 12, a^13b^18, 否], [1, 6, 11, a^12b^18, 否], [1, 7, 10, a^11b^18, 否], [1, 8, 9, a^10b^18, 否], [1, 9, 8, a^9b^18, 否], [1, 10, 7, a^8b^18, 否], [1, 11, 6, a^7b^18, 否], [1, 12, 5, a^6b^18, 否], [1, 13, 4, a^5b^18, 否], [1, 14, 3, a^4b^18, 否], [1, 15, 2, a^3b^18, 否], [1, 16, 1, a^2b^18, 否], [1, 17, 0, a^1b^18, 否], [2, 1, 15, a^17b^18, 否], [2, 2, 14, a^16b^18, 否], [2, 3, 13, a^15b^18, 否], [2, 4, 12, a^14b^18, 否], [2, 5, 11, a^13b^18, 否], [2, 6, 10, a^12b^18, 否], [2, 7, 9, a^11b^18, 否], [2, 8, 8, a^10b^18, 否], [2, 9, 7, a^9b^18, 否], [2, 10, 6, a^8b^18, 否], [2, 11, 5, a^7b^18, 否], [2, 12, 4, a^6b^18, 否], [2, 13, 3, a^5b^18, 否], [2, 14, 2, a^4b^18, 否], [2, 15, 1, a^3b^18, 否], [2, 16, 0, a^2b^18, 否], [3, 1, 14, a^17b^18, 否], [3, 2, 13, a^16b^18, 否], [3, 3, 12, a^15b^18, 否], [3, 4, 11, a^14b^18, 否], [3, 5, 10, a^13b^18, 否], [3, 6, 9, a^12b^18, 否], [3, 7, 8, a^11b^18, 否], [3, 8, 7, a^10b^18, 否], [3, 9, 6, a^9b^18, 否], [3, 10, 5, a^8b^18, 否], [3, 11, 4, a^7b^18, 否], [3, 12, 3, a^6b^18, 否], [3, 13, 2, a^5b^18, 否], [3, 14, 1, a^4b^18, 否], [3, 15, 0, a^3b^18, 否], [4, 1, 13, a^17b^18, 否], [4, 2, 12, a^16b^18, 否], [4, 3, 11, a^15b^18, 否], [4, 4, 10, a^14b^18, 否], [4, 5, 9, a^13b^18, 否], [4, 6, 8, a^12b^18, 否], [4, 7, 7, a^11b^18, 否], [4, 8, 6, a^10b^18, 否], [4, 9, 5, a^9b^18, 否], [4, 10, 4, a^8b^18, 否], [4, 11, 3, a^7b^18, 否], [4, 12, 2, a^6b^18, 否], [4, 13, 1, a^5b^18, 否], [4, 14, 0, a^4b^18, 否], [5, 1, 12, a^17b^18, 否], [5, 2, 11, a^16b^18, 否], [5, 3, 10, a^15b^18, 否], [5, 4, 9, a^14b^18, 否], [5, 5, 8, a^13b^18, 否], [5, 6, 7, a^12b^18, 否], [5, 7, 6, a^11b^18, 否], [5, 8, 5, a^10b^18, 否], [5, 9, 4, a^9b^18, 否], [5, 10, 3, a^8b^18, 否], [5, 11, 2, a^7b^18, 否], [5, 12, 1, a^6b^18, 否], [5, 13, 0, a^5b^18, 否], [6, 1, 11, a^17b^18, 否], [6, 2, 10, a^16b^18, 否], [6, 3, 9, a^15b^18, 否], [6, 4, 8, a^14b^18, 否], [6, 5, 7, a^13b^18, 否], [6, 6, 6, a^12b^18, 否], [6, 7, 5, a^11b^18, 否], [6, 8, 4, a^10b^18, 否], [6, 9, 3, a^9b^18, 否], [6, 10, 2, a^8b^18, 否], [6, 11, 1, a^7b^18, 否], [6, 12, 0, a^6b^18, 否], [7, 1, 10, a^17b^18, 否], [7, 2, 9, a^16b^18, 否], [7, 3, 8, a^15b^18, 否], [7, 4, 7, a^14b^18, 否], [7, 5, 6, a^13b^18, 否], [7, 6, 5, a^12b^18, 否], [7, 7, 4, a^11b^18, 否], [7, 8, 3, a^10b^18, 否], [7, 9, 2, a^9b^18, 否], [7, 10, 1, a^8b^18, 否], [7, 11, 0, a^7b^18, 否], [8, 1, 9, a^17b^18, 否], [8, 2, 8, a^16b^18, 否], [8, 3, 7, a^15b^18, 否], [8, 4, 6, a^14b^18, 否], [8, 5, 5, a^13b^18, 否], [8, 6, 4, a^12b^18, 否], [8, 7, 3, a^11b^18, 否], [8, 8, 2, a^10b^18, 否], [8, 9, 1, a^9b^18, 否], [8, 10, 0, a^8b^18, 否], [9, 1, 8, a^17b^18, 否], [9, 2, 7, a^16b^18, 否], [9, 3, 6, a^15b^18, 否], [9, 4, 5, a^14b^18, 否], [9, 5, 4, a^13b^18, 否], [9, 6, 3, a^12b^18, 否], [9, 7, 2, a^11b^18, 否], [9, 8, 1, a^10b^18, 否], [9, 9, 0, a^9b^18, 否], [10, 1, 7, a^17b^18, 否], [10, 2, 6, a^16b^18, 否], [10, 3, 5, a^15b^18, 否], [10, 4, 4, a^14b^18, 否], [10, 5, 3, a^13b^18, 否], [10, 6, 2, a^12b^18, 否], [10, 7, 1, a^11b^18, 否], [10, 8, 0, a^10b^18, 否], [11, 1, 6, a^17b^18, 否], [11, 2, 5, a^16b^18, 否], [11, 3, 4, a^15b^18, 否], [11, 4, 3, a^14b^18, 否], [11, 5, 2, a^13b^18, 否], [11, 6, 1, a^12b^18, 否], [11, 7, 0, a^11b^18, 否], [12, 1, 5, a^17b^18, 否], [12, 2, 4, a^16b^18, 否], [12, 3, 3, a^15b^18, 否], [12, 4, 2, a^14b^18, 否], [12, 5, 1, a^13b^18, 否], [12, 6, 0, a^12b^18, 否], [13, 1, 4, a^17b^18, 否], [13, 2, 3, a^16b^18, 否], [13, 3, 2, a^15b^18, 否], [13, 4, 1, a^14b^18, 否], [13, 5, 0, a^13b^18, 否], [14, 1, 3, a^17b^18, 否], [14, 2, 2, a^16b^18, 否], [14, 3, 1, a^15b^18, 否], [14, 4, 0, a^14b^18, 否], [15, 1, 2, a^17b^18, 否], [15, 2, 1, a^16b^18, 否], [15, 3, 0, a^15b^18, 否], [16, 1, 1, a^17b^18, 否], [16, 2, 0, a^16b^18, 否], [17, 1, 0, a^17b^18, 否]], 171, 171, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(18, a^18b^18, [[0, 1, 17, a^17b^18, 否], [0, 2, 16, a^16b^18, 否], [0, 3, 15, a^15b^18, 否], [0, 4, 14, a^14b^18, 否], [0, 5, 13, a^13b^18, 否], [0, 6, 12, a^12b^18, 否], [0, 7, 11, a^11b^18, 否], [0, 8, 10, a^10b^18, 否], [0, 9, 9, a^9b^18, 否], [0, 10, 8, a^8b^18, 否], [0, 11, 7, a^7b^18, 否], [0, 12, 6, a^6b^18, 否], [0, 13, 5, a^5b^18, 否], [0, 14, 4, a^4b^18, 否], [0, 15, 3, a^3b^18, 否], [0, 16, 2, a^2b^18, 否], [0, 17, 1, a^1b^18, 否], [0, 18, 0, a^0b^18, 否], [1, 1, 16, a^17b^18, 否], [1, 2, 15, a^16b^18, 否], [1, 3, 14, a^15b^18, 否], [1, 4, 13, a^14b^18, 否], [1, 5, 12, a^13b^18, 否], [1, 6, 11, a^12b^18, 否], [1, 7, 10, a^11b^18, 否], [1, 8, 9, a^10b^18, 否], [1, 9, 8, a^9b^18, 否], [1, 10, 7, a^8b^18, 否], [1, 11, 6, a^7b^18, 否], [1, 12, 5, a^6b^18, 否], [1, 13, 4, a^5b^18, 否], [1, 14, 3, a^4b^18, 否], [1, 15, 2, a^3b^18, 否], [1, 16, 1, a^2b^18, 否], [1, 17, 0, a^1b^18, 否], [2, 1, 15, a^17b^18, 否], [2, 2, 14, a^16b^18, 否], [2, 3, 13, a^15b^18, 否], [2, 4, 12, a^14b^18, 否], [2, 5, 11, a^13b^18, 否], [2, 6, 10, a^12b^18, 否], [2, 7, 9, a^11b^18, 否], [2, 8, 8, a^10b^18, 否], [2, 9, 7, a^9b^18, 否], [2, 10, 6, a^8b^18, 否], [2, 11, 5, a^7b^18, 否], [2, 12, 4, a^6b^18, 否], [2, 13, 3, a^5b^18, 否], [2, 14, 2, a^4b^18, 否], [2, 15, 1, a^3b^18, 否], [2, 16, 0, a^2b^18, 否], [3, 1, 14, a^17b^18, 否], [3, 2, 13, a^16b^18, 否], [3, 3, 12, a^15b^18, 否], [3, 4, 11, a^14b^18, 否], [3, 5, 10, a^13b^18, 否], [3, 6, 9, a^12b^18, 否], [3, 7, 8, a^11b^18, 否], [3, 8, 7, a^10b^18, 否], [3, 9, 6, a^9b^18, 否], [3, 10, 5, a^8b^18, 否], [3, 11, 4, a^7b^18, 否], [3, 12, 3, a^6b^18, 否], [3, 13, 2, a^5b^18, 否], [3, 14, 1, a^4b^18, 否], [3, 15, 0, a^3b^18, 否], [4, 1, 13, a^17b^18, 否], [4, 2, 12, a^16b^18, 否], [4, 3, 11, a^15b^18, 否], [4, 4, 10, a^14b^18, 否], [4, 5, 9, a^13b^18, 否], [4, 6, 8, a^12b^18, 否], [4, 7, 7, a^11b^18, 否], [4, 8, 6, a^10b^18, 否], [4, 9, 5, a^9b^18, 否], [4, 10, 4, a^8b^18, 否], [4, 11, 3, a^7b^18, 否], [4, 12, 2, a^6b^18, 否], [4, 13, 1, a^5b^18, 否], [4, 14, 0, a^4b^18, 否], [5, 1, 12, a^17b^18, 否], [5, 2, 11, a^16b^18, 否], [5, 3, 10, a^15b^18, 否], [5, 4, 9, a^14b^18, 否], [5, 5, 8, a^13b^18, 否], [5, 6, 7, a^12b^18, 否], [5, 7, 6, a^11b^18, 否], [5, 8, 5, a^10b^18, 否], [5, 9, 4, a^9b^18, 否], [5, 10, 3, a^8b^18, 否], [5, 11, 2, a^7b^18, 否], [5, 12, 1, a^6b^18, 否], [5, 13, 0, a^5b^18, 否], [6, 1, 11, a^17b^18, 否], [6, 2, 10, a^16b^18, 否], [6, 3, 9, a^15b^18, 否], [6, 4, 8, a^14b^18, 否], [6, 5, 7, a^13b^18, 否], [6, 6, 6, a^12b^18, 否], [6, 7, 5, a^11b^18, 否], [6, 8, 4, a^10b^18, 否], [6, 9, 3, a^9b^18, 否], [6, 10, 2, a^8b^18, 否], [6, 11, 1, a^7b^18, 否], [6, 12, 0, a^6b^18, 否], [7, 1, 10, a^17b^18, 否], [7, 2, 9, a^16b^18, 否], [7, 3, 8, a^15b^18, 否], [7, 4, 7, a^14b^18, 否], [7, 5, 6, a^13b^18, 否], [7, 6, 5, a^12b^18, 否], [7, 7, 4, a^11b^18, 否], [7, 8, 3, a^10b^18, 否], [7, 9, 2, a^9b^18, 否], [7, 10, 1, a^8b^18, 否], [7, 11, 0, a^7b^18, 否], [8, 1, 9, a^17b^18, 否], [8, 2, 8, a^16b^18, 否], [8, 3, 7, a^15b^18, 否], [8, 4, 6, a^14b^18, 否], [8, 5, 5, a^13b^18, 否], [8, 6, 4, a^12b^18, 否], [8, 7, 3, a^11b^18, 否], [8, 8, 2, a^10b^18, 否], [8, 9, 1, a^9b^18, 否], [8, 10, 0, a^8b^18, 否], [9, 1, 8, a^17b^18, 否], [9, 2, 7, a^16b^18, 否], [9, 3, 6, a^15b^18, 否], [9, 4, 5, a^14b^18, 否], [9, 5, 4, a^13b^18, 否], [9, 6, 3, a^12b^18, 否], [9, 7, 2, a^11b^18, 否], [9, 8, 1, a^10b^18, 否], [9, 9, 0, a^9b^18, 否], [10, 1, 7, a^17b^18, 否], [10, 2, 6, a^16b^18, 否], [10, 3, 5, a^15b^18, 否], [10, 4, 4, a^14b^18, 否], [10, 5, 3, a^13b^18, 否], [10, 6, 2, a^12b^18, 否], [10, 7, 1, a^11b^18, 否], [10, 8, 0, a^10b^18, 否], [11, 1, 6, a^17b^18, 否], [11, 2, 5, a^16b^18, 否], [11, 3, 4, a^15b^18, 否], [11, 4, 3, a^14b^18, 否], [11, 5, 2, a^13b^18, 否], [11, 6, 1, a^12b^18, 否], [11, 7, 0, a^11b^18, 否], [12, 1, 5, a^17b^18, 否], [12, 2, 4, a^16b^18, 否], [12, 3, 3, a^15b^18, 否], [12, 4, 2, a^14b^18, 否], [12, 5, 1, a^13b^18, 否], [12, 6, 0, a^12b^18, 否], [13, 1, 4, a^17b^18, 否], [13, 2, 3, a^16b^18, 否], [13, 3, 2, a^15b^18, 否], [13, 4, 1, a^14b^18, 否], [13, 5, 0, a^13b^18, 否], [14, 1, 3, a^17b^18, 否], [14, 2, 2, a^16b^18, 否], [14, 3, 1, a^15b^18, 否], [14, 4, 0, a^14b^18, 否], [15, 1, 2, a^17b^18, 否], [15, 2, 1, a^16b^18, 否], [15, 3, 0, a^15b^18, 否], [16, 1, 1, a^17b^18, 否], [16, 2, 0, a^16b^18, 否], [17, 1, 0, a^17b^18, 否]], 171, 171, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

69. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":19}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(19, a^19b^19, [[0, 1, 18, a^18b^19, 否], [0, 2, 17, a^17b^19, 否], [0, 3, 16, a^16b^19, 否], [0, 4, 15, a^15b^19, 否], [0, 5, 14, a^14b^19, 否], [0, 6, 13, a^13b^19, 否], [0, 7, 12, a^12b^19, 否], [0, 8, 11, a^11b^19, 否], [0, 9, 10, a^10b^19, 否], [0, 10, 9, a^9b^19, 否], [0, 11, 8, a^8b^19, 否], [0, 12, 7, a^7b^19, 否], [0, 13, 6, a^6b^19, 否], [0, 14, 5, a^5b^19, 否], [0, 15, 4, a^4b^19, 否], [0, 16, 3, a^3b^19, 否], [0, 17, 2, a^2b^19, 否], [0, 18, 1, a^1b^19, 否], [0, 19, 0, a^0b^19, 否], [1, 1, 17, a^18b^19, 否], [1, 2, 16, a^17b^19, 否], [1, 3, 15, a^16b^19, 否], [1, 4, 14, a^15b^19, 否], [1, 5, 13, a^14b^19, 否], [1, 6, 12, a^13b^19, 否], [1, 7, 11, a^12b^19, 否], [1, 8, 10, a^11b^19, 否], [1, 9, 9, a^10b^19, 否], [1, 10, 8, a^9b^19, 否], [1, 11, 7, a^8b^19, 否], [1, 12, 6, a^7b^19, 否], [1, 13, 5, a^6b^19, 否], [1, 14, 4, a^5b^19, 否], [1, 15, 3, a^4b^19, 否], [1, 16, 2, a^3b^19, 否], [1, 17, 1, a^2b^19, 否], [1, 18, 0, a^1b^19, 否], [2, 1, 16, a^18b^19, 否], [2, 2, 15, a^17b^19, 否], [2, 3, 14, a^16b^19, 否], [2, 4, 13, a^15b^19, 否], [2, 5, 12, a^14b^19, 否], [2, 6, 11, a^13b^19, 否], [2, 7, 10, a^12b^19, 否], [2, 8, 9, a^11b^19, 否], [2, 9, 8, a^10b^19, 否], [2, 10, 7, a^9b^19, 否], [2, 11, 6, a^8b^19, 否], [2, 12, 5, a^7b^19, 否], [2, 13, 4, a^6b^19, 否], [2, 14, 3, a^5b^19, 否], [2, 15, 2, a^4b^19, 否], [2, 16, 1, a^3b^19, 否], [2, 17, 0, a^2b^19, 否], [3, 1, 15, a^18b^19, 否], [3, 2, 14, a^17b^19, 否], [3, 3, 13, a^16b^19, 否], [3, 4, 12, a^15b^19, 否], [3, 5, 11, a^14b^19, 否], [3, 6, 10, a^13b^19, 否], [3, 7, 9, a^12b^19, 否], [3, 8, 8, a^11b^19, 否], [3, 9, 7, a^10b^19, 否], [3, 10, 6, a^9b^19, 否], [3, 11, 5, a^8b^19, 否], [3, 12, 4, a^7b^19, 否], [3, 13, 3, a^6b^19, 否], [3, 14, 2, a^5b^19, 否], [3, 15, 1, a^4b^19, 否], [3, 16, 0, a^3b^19, 否], [4, 1, 14, a^18b^19, 否], [4, 2, 13, a^17b^19, 否], [4, 3, 12, a^16b^19, 否], [4, 4, 11, a^15b^19, 否], [4, 5, 10, a^14b^19, 否], [4, 6, 9, a^13b^19, 否], [4, 7, 8, a^12b^19, 否], [4, 8, 7, a^11b^19, 否], [4, 9, 6, a^10b^19, 否], [4, 10, 5, a^9b^19, 否], [4, 11, 4, a^8b^19, 否], [4, 12, 3, a^7b^19, 否], [4, 13, 2, a^6b^19, 否], [4, 14, 1, a^5b^19, 否], [4, 15, 0, a^4b^19, 否], [5, 1, 13, a^18b^19, 否], [5, 2, 12, a^17b^19, 否], [5, 3, 11, a^16b^19, 否], [5, 4, 10, a^15b^19, 否], [5, 5, 9, a^14b^19, 否], [5, 6, 8, a^13b^19, 否], [5, 7, 7, a^12b^19, 否], [5, 8, 6, a^11b^19, 否], [5, 9, 5, a^10b^19, 否], [5, 10, 4, a^9b^19, 否], [5, 11, 3, a^8b^19, 否], [5, 12, 2, a^7b^19, 否], [5, 13, 1, a^6b^19, 否], [5, 14, 0, a^5b^19, 否], [6, 1, 12, a^18b^19, 否], [6, 2, 11, a^17b^19, 否], [6, 3, 10, a^16b^19, 否], [6, 4, 9, a^15b^19, 否], [6, 5, 8, a^14b^19, 否], [6, 6, 7, a^13b^19, 否], [6, 7, 6, a^12b^19, 否], [6, 8, 5, a^11b^19, 否], [6, 9, 4, a^10b^19, 否], [6, 10, 3, a^9b^19, 否], [6, 11, 2, a^8b^19, 否], [6, 12, 1, a^7b^19, 否], [6, 13, 0, a^6b^19, 否], [7, 1, 11, a^18b^19, 否], [7, 2, 10, a^17b^19, 否], [7, 3, 9, a^16b^19, 否], [7, 4, 8, a^15b^19, 否], [7, 5, 7, a^14b^19, 否], [7, 6, 6, a^13b^19, 否], [7, 7, 5, a^12b^19, 否], [7, 8, 4, a^11b^19, 否], [7, 9, 3, a^10b^19, 否], [7, 10, 2, a^9b^19, 否], [7, 11, 1, a^8b^19, 否], [7, 12, 0, a^7b^19, 否], [8, 1, 10, a^18b^19, 否], [8, 2, 9, a^17b^19, 否], [8, 3, 8, a^16b^19, 否], [8, 4, 7, a^15b^19, 否], [8, 5, 6, a^14b^19, 否], [8, 6, 5, a^13b^19, 否], [8, 7, 4, a^12b^19, 否], [8, 8, 3, a^11b^19, 否], [8, 9, 2, a^10b^19, 否], [8, 10, 1, a^9b^19, 否], [8, 11, 0, a^8b^19, 否], [9, 1, 9, a^18b^19, 否], [9, 2, 8, a^17b^19, 否], [9, 3, 7, a^16b^19, 否], [9, 4, 6, a^15b^19, 否], [9, 5, 5, a^14b^19, 否], [9, 6, 4, a^13b^19, 否], [9, 7, 3, a^12b^19, 否], [9, 8, 2, a^11b^19, 否], [9, 9, 1, a^10b^19, 否], [9, 10, 0, a^9b^19, 否], [10, 1, 8, a^18b^19, 否], [10, 2, 7, a^17b^19, 否], [10, 3, 6, a^16b^19, 否], [10, 4, 5, a^15b^19, 否], [10, 5, 4, a^14b^19, 否], [10, 6, 3, a^13b^19, 否], [10, 7, 2, a^12b^19, 否], [10, 8, 1, a^11b^19, 否], [10, 9, 0, a^10b^19, 否], [11, 1, 7, a^18b^19, 否], [11, 2, 6, a^17b^19, 否], [11, 3, 5, a^16b^19, 否], [11, 4, 4, a^15b^19, 否], [11, 5, 3, a^14b^19, 否], [11, 6, 2, a^13b^19, 否], [11, 7, 1, a^12b^19, 否], [11, 8, 0, a^11b^19, 否], [12, 1, 6, a^18b^19, 否], [12, 2, 5, a^17b^19, 否], [12, 3, 4, a^16b^19, 否], [12, 4, 3, a^15b^19, 否], [12, 5, 2, a^14b^19, 否], [12, 6, 1, a^13b^19, 否], [12, 7, 0, a^12b^19, 否], [13, 1, 5, a^18b^19, 否], [13, 2, 4, a^17b^19, 否], [13, 3, 3, a^16b^19, 否], [13, 4, 2, a^15b^19, 否], [13, 5, 1, a^14b^19, 否], [13, 6, 0, a^13b^19, 否], [14, 1, 4, a^18b^19, 否], [14, 2, 3, a^17b^19, 否], [14, 3, 2, a^16b^19, 否], [14, 4, 1, a^15b^19, 否], [14, 5, 0, a^14b^19, 否], [15, 1, 3, a^18b^19, 否], [15, 2, 2, a^17b^19, 否], [15, 3, 1, a^16b^19, 否], [15, 4, 0, a^15b^19, 否], [16, 1, 2, a^18b^19, 否], [16, 2, 1, a^17b^19, 否], [16, 3, 0, a^16b^19, 否], [17, 1, 1, a^18b^19, 否], [17, 2, 0, a^17b^19, 否], [18, 1, 0, a^18b^19, 否]], 190, 190, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(19, a^19b^19, [[0, 1, 18, a^18b^19, 否], [0, 2, 17, a^17b^19, 否], [0, 3, 16, a^16b^19, 否], [0, 4, 15, a^15b^19, 否], [0, 5, 14, a^14b^19, 否], [0, 6, 13, a^13b^19, 否], [0, 7, 12, a^12b^19, 否], [0, 8, 11, a^11b^19, 否], [0, 9, 10, a^10b^19, 否], [0, 10, 9, a^9b^19, 否], [0, 11, 8, a^8b^19, 否], [0, 12, 7, a^7b^19, 否], [0, 13, 6, a^6b^19, 否], [0, 14, 5, a^5b^19, 否], [0, 15, 4, a^4b^19, 否], [0, 16, 3, a^3b^19, 否], [0, 17, 2, a^2b^19, 否], [0, 18, 1, a^1b^19, 否], [0, 19, 0, a^0b^19, 否], [1, 1, 17, a^18b^19, 否], [1, 2, 16, a^17b^19, 否], [1, 3, 15, a^16b^19, 否], [1, 4, 14, a^15b^19, 否], [1, 5, 13, a^14b^19, 否], [1, 6, 12, a^13b^19, 否], [1, 7, 11, a^12b^19, 否], [1, 8, 10, a^11b^19, 否], [1, 9, 9, a^10b^19, 否], [1, 10, 8, a^9b^19, 否], [1, 11, 7, a^8b^19, 否], [1, 12, 6, a^7b^19, 否], [1, 13, 5, a^6b^19, 否], [1, 14, 4, a^5b^19, 否], [1, 15, 3, a^4b^19, 否], [1, 16, 2, a^3b^19, 否], [1, 17, 1, a^2b^19, 否], [1, 18, 0, a^1b^19, 否], [2, 1, 16, a^18b^19, 否], [2, 2, 15, a^17b^19, 否], [2, 3, 14, a^16b^19, 否], [2, 4, 13, a^15b^19, 否], [2, 5, 12, a^14b^19, 否], [2, 6, 11, a^13b^19, 否], [2, 7, 10, a^12b^19, 否], [2, 8, 9, a^11b^19, 否], [2, 9, 8, a^10b^19, 否], [2, 10, 7, a^9b^19, 否], [2, 11, 6, a^8b^19, 否], [2, 12, 5, a^7b^19, 否], [2, 13, 4, a^6b^19, 否], [2, 14, 3, a^5b^19, 否], [2, 15, 2, a^4b^19, 否], [2, 16, 1, a^3b^19, 否], [2, 17, 0, a^2b^19, 否], [3, 1, 15, a^18b^19, 否], [3, 2, 14, a^17b^19, 否], [3, 3, 13, a^16b^19, 否], [3, 4, 12, a^15b^19, 否], [3, 5, 11, a^14b^19, 否], [3, 6, 10, a^13b^19, 否], [3, 7, 9, a^12b^19, 否], [3, 8, 8, a^11b^19, 否], [3, 9, 7, a^10b^19, 否], [3, 10, 6, a^9b^19, 否], [3, 11, 5, a^8b^19, 否], [3, 12, 4, a^7b^19, 否], [3, 13, 3, a^6b^19, 否], [3, 14, 2, a^5b^19, 否], [3, 15, 1, a^4b^19, 否], [3, 16, 0, a^3b^19, 否], [4, 1, 14, a^18b^19, 否], [4, 2, 13, a^17b^19, 否], [4, 3, 12, a^16b^19, 否], [4, 4, 11, a^15b^19, 否], [4, 5, 10, a^14b^19, 否], [4, 6, 9, a^13b^19, 否], [4, 7, 8, a^12b^19, 否], [4, 8, 7, a^11b^19, 否], [4, 9, 6, a^10b^19, 否], [4, 10, 5, a^9b^19, 否], [4, 11, 4, a^8b^19, 否], [4, 12, 3, a^7b^19, 否], [4, 13, 2, a^6b^19, 否], [4, 14, 1, a^5b^19, 否], [4, 15, 0, a^4b^19, 否], [5, 1, 13, a^18b^19, 否], [5, 2, 12, a^17b^19, 否], [5, 3, 11, a^16b^19, 否], [5, 4, 10, a^15b^19, 否], [5, 5, 9, a^14b^19, 否], [5, 6, 8, a^13b^19, 否], [5, 7, 7, a^12b^19, 否], [5, 8, 6, a^11b^19, 否], [5, 9, 5, a^10b^19, 否], [5, 10, 4, a^9b^19, 否], [5, 11, 3, a^8b^19, 否], [5, 12, 2, a^7b^19, 否], [5, 13, 1, a^6b^19, 否], [5, 14, 0, a^5b^19, 否], [6, 1, 12, a^18b^19, 否], [6, 2, 11, a^17b^19, 否], [6, 3, 10, a^16b^19, 否], [6, 4, 9, a^15b^19, 否], [6, 5, 8, a^14b^19, 否], [6, 6, 7, a^13b^19, 否], [6, 7, 6, a^12b^19, 否], [6, 8, 5, a^11b^19, 否], [6, 9, 4, a^10b^19, 否], [6, 10, 3, a^9b^19, 否], [6, 11, 2, a^8b^19, 否], [6, 12, 1, a^7b^19, 否], [6, 13, 0, a^6b^19, 否], [7, 1, 11, a^18b^19, 否], [7, 2, 10, a^17b^19, 否], [7, 3, 9, a^16b^19, 否], [7, 4, 8, a^15b^19, 否], [7, 5, 7, a^14b^19, 否], [7, 6, 6, a^13b^19, 否], [7, 7, 5, a^12b^19, 否], [7, 8, 4, a^11b^19, 否], [7, 9, 3, a^10b^19, 否], [7, 10, 2, a^9b^19, 否], [7, 11, 1, a^8b^19, 否], [7, 12, 0, a^7b^19, 否], [8, 1, 10, a^18b^19, 否], [8, 2, 9, a^17b^19, 否], [8, 3, 8, a^16b^19, 否], [8, 4, 7, a^15b^19, 否], [8, 5, 6, a^14b^19, 否], [8, 6, 5, a^13b^19, 否], [8, 7, 4, a^12b^19, 否], [8, 8, 3, a^11b^19, 否], [8, 9, 2, a^10b^19, 否], [8, 10, 1, a^9b^19, 否], [8, 11, 0, a^8b^19, 否], [9, 1, 9, a^18b^19, 否], [9, 2, 8, a^17b^19, 否], [9, 3, 7, a^16b^19, 否], [9, 4, 6, a^15b^19, 否], [9, 5, 5, a^14b^19, 否], [9, 6, 4, a^13b^19, 否], [9, 7, 3, a^12b^19, 否], [9, 8, 2, a^11b^19, 否], [9, 9, 1, a^10b^19, 否], [9, 10, 0, a^9b^19, 否], [10, 1, 8, a^18b^19, 否], [10, 2, 7, a^17b^19, 否], [10, 3, 6, a^16b^19, 否], [10, 4, 5, a^15b^19, 否], [10, 5, 4, a^14b^19, 否], [10, 6, 3, a^13b^19, 否], [10, 7, 2, a^12b^19, 否], [10, 8, 1, a^11b^19, 否], [10, 9, 0, a^10b^19, 否], [11, 1, 7, a^18b^19, 否], [11, 2, 6, a^17b^19, 否], [11, 3, 5, a^16b^19, 否], [11, 4, 4, a^15b^19, 否], [11, 5, 3, a^14b^19, 否], [11, 6, 2, a^13b^19, 否], [11, 7, 1, a^12b^19, 否], [11, 8, 0, a^11b^19, 否], [12, 1, 6, a^18b^19, 否], [12, 2, 5, a^17b^19, 否], [12, 3, 4, a^16b^19, 否], [12, 4, 3, a^15b^19, 否], [12, 5, 2, a^14b^19, 否], [12, 6, 1, a^13b^19, 否], [12, 7, 0, a^12b^19, 否], [13, 1, 5, a^18b^19, 否], [13, 2, 4, a^17b^19, 否], [13, 3, 3, a^16b^19, 否], [13, 4, 2, a^15b^19, 否], [13, 5, 1, a^14b^19, 否], [13, 6, 0, a^13b^19, 否], [14, 1, 4, a^18b^19, 否], [14, 2, 3, a^17b^19, 否], [14, 3, 2, a^16b^19, 否], [14, 4, 1, a^15b^19, 否], [14, 5, 0, a^14b^19, 否], [15, 1, 3, a^18b^19, 否], [15, 2, 2, a^17b^19, 否], [15, 3, 1, a^16b^19, 否], [15, 4, 0, a^15b^19, 否], [16, 1, 2, a^18b^19, 否], [16, 2, 1, a^17b^19, 否], [16, 3, 0, a^16b^19, 否], [17, 1, 1, a^18b^19, 否], [17, 2, 0, a^17b^19, 否], [18, 1, 0, a^18b^19, 否]], 190, 190, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

70. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":20}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(20, a^20b^20, [[0, 1, 19, a^19b^20, 否], [0, 2, 18, a^18b^20, 否], [0, 3, 17, a^17b^20, 否], [0, 4, 16, a^16b^20, 否], [0, 5, 15, a^15b^20, 否], [0, 6, 14, a^14b^20, 否], [0, 7, 13, a^13b^20, 否], [0, 8, 12, a^12b^20, 否], [0, 9, 11, a^11b^20, 否], [0, 10, 10, a^10b^20, 否], [0, 11, 9, a^9b^20, 否], [0, 12, 8, a^8b^20, 否], [0, 13, 7, a^7b^20, 否], [0, 14, 6, a^6b^20, 否], [0, 15, 5, a^5b^20, 否], [0, 16, 4, a^4b^20, 否], [0, 17, 3, a^3b^20, 否], [0, 18, 2, a^2b^20, 否], [0, 19, 1, a^1b^20, 否], [0, 20, 0, a^0b^20, 否], [1, 1, 18, a^19b^20, 否], [1, 2, 17, a^18b^20, 否], [1, 3, 16, a^17b^20, 否], [1, 4, 15, a^16b^20, 否], [1, 5, 14, a^15b^20, 否], [1, 6, 13, a^14b^20, 否], [1, 7, 12, a^13b^20, 否], [1, 8, 11, a^12b^20, 否], [1, 9, 10, a^11b^20, 否], [1, 10, 9, a^10b^20, 否], [1, 11, 8, a^9b^20, 否], [1, 12, 7, a^8b^20, 否], [1, 13, 6, a^7b^20, 否], [1, 14, 5, a^6b^20, 否], [1, 15, 4, a^5b^20, 否], [1, 16, 3, a^4b^20, 否], [1, 17, 2, a^3b^20, 否], [1, 18, 1, a^2b^20, 否], [1, 19, 0, a^1b^20, 否], [2, 1, 17, a^19b^20, 否], [2, 2, 16, a^18b^20, 否], [2, 3, 15, a^17b^20, 否], [2, 4, 14, a^16b^20, 否], [2, 5, 13, a^15b^20, 否], [2, 6, 12, a^14b^20, 否], [2, 7, 11, a^13b^20, 否], [2, 8, 10, a^12b^20, 否], [2, 9, 9, a^11b^20, 否], [2, 10, 8, a^10b^20, 否], [2, 11, 7, a^9b^20, 否], [2, 12, 6, a^8b^20, 否], [2, 13, 5, a^7b^20, 否], [2, 14, 4, a^6b^20, 否], [2, 15, 3, a^5b^20, 否], [2, 16, 2, a^4b^20, 否], [2, 17, 1, a^3b^20, 否], [2, 18, 0, a^2b^20, 否], [3, 1, 16, a^19b^20, 否], [3, 2, 15, a^18b^20, 否], [3, 3, 14, a^17b^20, 否], [3, 4, 13, a^16b^20, 否], [3, 5, 12, a^15b^20, 否], [3, 6, 11, a^14b^20, 否], [3, 7, 10, a^13b^20, 否], [3, 8, 9, a^12b^20, 否], [3, 9, 8, a^11b^20, 否], [3, 10, 7, a^10b^20, 否], [3, 11, 6, a^9b^20, 否], [3, 12, 5, a^8b^20, 否], [3, 13, 4, a^7b^20, 否], [3, 14, 3, a^6b^20, 否], [3, 15, 2, a^5b^20, 否], [3, 16, 1, a^4b^20, 否], [3, 17, 0, a^3b^20, 否], [4, 1, 15, a^19b^20, 否], [4, 2, 14, a^18b^20, 否], [4, 3, 13, a^17b^20, 否], [4, 4, 12, a^16b^20, 否], [4, 5, 11, a^15b^20, 否], [4, 6, 10, a^14b^20, 否], [4, 7, 9, a^13b^20, 否], [4, 8, 8, a^12b^20, 否], [4, 9, 7, a^11b^20, 否], [4, 10, 6, a^10b^20, 否], [4, 11, 5, a^9b^20, 否], [4, 12, 4, a^8b^20, 否], [4, 13, 3, a^7b^20, 否], [4, 14, 2, a^6b^20, 否], [4, 15, 1, a^5b^20, 否], [4, 16, 0, a^4b^20, 否], [5, 1, 14, a^19b^20, 否], [5, 2, 13, a^18b^20, 否], [5, 3, 12, a^17b^20, 否], [5, 4, 11, a^16b^20, 否], [5, 5, 10, a^15b^20, 否], [5, 6, 9, a^14b^20, 否], [5, 7, 8, a^13b^20, 否], [5, 8, 7, a^12b^20, 否], [5, 9, 6, a^11b^20, 否], [5, 10, 5, a^10b^20, 否], [5, 11, 4, a^9b^20, 否], [5, 12, 3, a^8b^20, 否], [5, 13, 2, a^7b^20, 否], [5, 14, 1, a^6b^20, 否], [5, 15, 0, a^5b^20, 否], [6, 1, 13, a^19b^20, 否], [6, 2, 12, a^18b^20, 否], [6, 3, 11, a^17b^20, 否], [6, 4, 10, a^16b^20, 否], [6, 5, 9, a^15b^20, 否], [6, 6, 8, a^14b^20, 否], [6, 7, 7, a^13b^20, 否], [6, 8, 6, a^12b^20, 否], [6, 9, 5, a^11b^20, 否], [6, 10, 4, a^10b^20, 否], [6, 11, 3, a^9b^20, 否], [6, 12, 2, a^8b^20, 否], [6, 13, 1, a^7b^20, 否], [6, 14, 0, a^6b^20, 否], [7, 1, 12, a^19b^20, 否], [7, 2, 11, a^18b^20, 否], [7, 3, 10, a^17b^20, 否], [7, 4, 9, a^16b^20, 否], [7, 5, 8, a^15b^20, 否], [7, 6, 7, a^14b^20, 否], [7, 7, 6, a^13b^20, 否], [7, 8, 5, a^12b^20, 否], [7, 9, 4, a^11b^20, 否], [7, 10, 3, a^10b^20, 否], [7, 11, 2, a^9b^20, 否], [7, 12, 1, a^8b^20, 否], [7, 13, 0, a^7b^20, 否], [8, 1, 11, a^19b^20, 否], [8, 2, 10, a^18b^20, 否], [8, 3, 9, a^17b^20, 否], [8, 4, 8, a^16b^20, 否], [8, 5, 7, a^15b^20, 否], [8, 6, 6, a^14b^20, 否], [8, 7, 5, a^13b^20, 否], [8, 8, 4, a^12b^20, 否], [8, 9, 3, a^11b^20, 否], [8, 10, 2, a^10b^20, 否], [8, 11, 1, a^9b^20, 否], [8, 12, 0, a^8b^20, 否], [9, 1, 10, a^19b^20, 否], [9, 2, 9, a^18b^20, 否], [9, 3, 8, a^17b^20, 否], [9, 4, 7, a^16b^20, 否], [9, 5, 6, a^15b^20, 否], [9, 6, 5, a^14b^20, 否], [9, 7, 4, a^13b^20, 否], [9, 8, 3, a^12b^20, 否], [9, 9, 2, a^11b^20, 否], [9, 10, 1, a^10b^20, 否], [9, 11, 0, a^9b^20, 否], [10, 1, 9, a^19b^20, 否], [10, 2, 8, a^18b^20, 否], [10, 3, 7, a^17b^20, 否], [10, 4, 6, a^16b^20, 否], [10, 5, 5, a^15b^20, 否], [10, 6, 4, a^14b^20, 否], [10, 7, 3, a^13b^20, 否], [10, 8, 2, a^12b^20, 否], [10, 9, 1, a^11b^20, 否], [10, 10, 0, a^10b^20, 否], [11, 1, 8, a^19b^20, 否], [11, 2, 7, a^18b^20, 否], [11, 3, 6, a^17b^20, 否], [11, 4, 5, a^16b^20, 否], [11, 5, 4, a^15b^20, 否], [11, 6, 3, a^14b^20, 否], [11, 7, 2, a^13b^20, 否], [11, 8, 1, a^12b^20, 否], [11, 9, 0, a^11b^20, 否], [12, 1, 7, a^19b^20, 否], [12, 2, 6, a^18b^20, 否], [12, 3, 5, a^17b^20, 否], [12, 4, 4, a^16b^20, 否], [12, 5, 3, a^15b^20, 否], [12, 6, 2, a^14b^20, 否], [12, 7, 1, a^13b^20, 否], [12, 8, 0, a^12b^20, 否], [13, 1, 6, a^19b^20, 否], [13, 2, 5, a^18b^20, 否], [13, 3, 4, a^17b^20, 否], [13, 4, 3, a^16b^20, 否], [13, 5, 2, a^15b^20, 否], [13, 6, 1, a^14b^20, 否], [13, 7, 0, a^13b^20, 否], [14, 1, 5, a^19b^20, 否], [14, 2, 4, a^18b^20, 否], [14, 3, 3, a^17b^20, 否], [14, 4, 2, a^16b^20, 否], [14, 5, 1, a^15b^20, 否], [14, 6, 0, a^14b^20, 否], [15, 1, 4, a^19b^20, 否], [15, 2, 3, a^18b^20, 否], [15, 3, 2, a^17b^20, 否], [15, 4, 1, a^16b^20, 否], [15, 5, 0, a^15b^20, 否], [16, 1, 3, a^19b^20, 否], [16, 2, 2, a^18b^20, 否], [16, 3, 1, a^17b^20, 否], [16, 4, 0, a^16b^20, 否], [17, 1, 2, a^19b^20, 否], [17, 2, 1, a^18b^20, 否], [17, 3, 0, a^17b^20, 否], [18, 1, 1, a^19b^20, 否], [18, 2, 0, a^18b^20, 否], [19, 1, 0, a^19b^20, 否]], 210, 210, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(20, a^20b^20, [[0, 1, 19, a^19b^20, 否], [0, 2, 18, a^18b^20, 否], [0, 3, 17, a^17b^20, 否], [0, 4, 16, a^16b^20, 否], [0, 5, 15, a^15b^20, 否], [0, 6, 14, a^14b^20, 否], [0, 7, 13, a^13b^20, 否], [0, 8, 12, a^12b^20, 否], [0, 9, 11, a^11b^20, 否], [0, 10, 10, a^10b^20, 否], [0, 11, 9, a^9b^20, 否], [0, 12, 8, a^8b^20, 否], [0, 13, 7, a^7b^20, 否], [0, 14, 6, a^6b^20, 否], [0, 15, 5, a^5b^20, 否], [0, 16, 4, a^4b^20, 否], [0, 17, 3, a^3b^20, 否], [0, 18, 2, a^2b^20, 否], [0, 19, 1, a^1b^20, 否], [0, 20, 0, a^0b^20, 否], [1, 1, 18, a^19b^20, 否], [1, 2, 17, a^18b^20, 否], [1, 3, 16, a^17b^20, 否], [1, 4, 15, a^16b^20, 否], [1, 5, 14, a^15b^20, 否], [1, 6, 13, a^14b^20, 否], [1, 7, 12, a^13b^20, 否], [1, 8, 11, a^12b^20, 否], [1, 9, 10, a^11b^20, 否], [1, 10, 9, a^10b^20, 否], [1, 11, 8, a^9b^20, 否], [1, 12, 7, a^8b^20, 否], [1, 13, 6, a^7b^20, 否], [1, 14, 5, a^6b^20, 否], [1, 15, 4, a^5b^20, 否], [1, 16, 3, a^4b^20, 否], [1, 17, 2, a^3b^20, 否], [1, 18, 1, a^2b^20, 否], [1, 19, 0, a^1b^20, 否], [2, 1, 17, a^19b^20, 否], [2, 2, 16, a^18b^20, 否], [2, 3, 15, a^17b^20, 否], [2, 4, 14, a^16b^20, 否], [2, 5, 13, a^15b^20, 否], [2, 6, 12, a^14b^20, 否], [2, 7, 11, a^13b^20, 否], [2, 8, 10, a^12b^20, 否], [2, 9, 9, a^11b^20, 否], [2, 10, 8, a^10b^20, 否], [2, 11, 7, a^9b^20, 否], [2, 12, 6, a^8b^20, 否], [2, 13, 5, a^7b^20, 否], [2, 14, 4, a^6b^20, 否], [2, 15, 3, a^5b^20, 否], [2, 16, 2, a^4b^20, 否], [2, 17, 1, a^3b^20, 否], [2, 18, 0, a^2b^20, 否], [3, 1, 16, a^19b^20, 否], [3, 2, 15, a^18b^20, 否], [3, 3, 14, a^17b^20, 否], [3, 4, 13, a^16b^20, 否], [3, 5, 12, a^15b^20, 否], [3, 6, 11, a^14b^20, 否], [3, 7, 10, a^13b^20, 否], [3, 8, 9, a^12b^20, 否], [3, 9, 8, a^11b^20, 否], [3, 10, 7, a^10b^20, 否], [3, 11, 6, a^9b^20, 否], [3, 12, 5, a^8b^20, 否], [3, 13, 4, a^7b^20, 否], [3, 14, 3, a^6b^20, 否], [3, 15, 2, a^5b^20, 否], [3, 16, 1, a^4b^20, 否], [3, 17, 0, a^3b^20, 否], [4, 1, 15, a^19b^20, 否], [4, 2, 14, a^18b^20, 否], [4, 3, 13, a^17b^20, 否], [4, 4, 12, a^16b^20, 否], [4, 5, 11, a^15b^20, 否], [4, 6, 10, a^14b^20, 否], [4, 7, 9, a^13b^20, 否], [4, 8, 8, a^12b^20, 否], [4, 9, 7, a^11b^20, 否], [4, 10, 6, a^10b^20, 否], [4, 11, 5, a^9b^20, 否], [4, 12, 4, a^8b^20, 否], [4, 13, 3, a^7b^20, 否], [4, 14, 2, a^6b^20, 否], [4, 15, 1, a^5b^20, 否], [4, 16, 0, a^4b^20, 否], [5, 1, 14, a^19b^20, 否], [5, 2, 13, a^18b^20, 否], [5, 3, 12, a^17b^20, 否], [5, 4, 11, a^16b^20, 否], [5, 5, 10, a^15b^20, 否], [5, 6, 9, a^14b^20, 否], [5, 7, 8, a^13b^20, 否], [5, 8, 7, a^12b^20, 否], [5, 9, 6, a^11b^20, 否], [5, 10, 5, a^10b^20, 否], [5, 11, 4, a^9b^20, 否], [5, 12, 3, a^8b^20, 否], [5, 13, 2, a^7b^20, 否], [5, 14, 1, a^6b^20, 否], [5, 15, 0, a^5b^20, 否], [6, 1, 13, a^19b^20, 否], [6, 2, 12, a^18b^20, 否], [6, 3, 11, a^17b^20, 否], [6, 4, 10, a^16b^20, 否], [6, 5, 9, a^15b^20, 否], [6, 6, 8, a^14b^20, 否], [6, 7, 7, a^13b^20, 否], [6, 8, 6, a^12b^20, 否], [6, 9, 5, a^11b^20, 否], [6, 10, 4, a^10b^20, 否], [6, 11, 3, a^9b^20, 否], [6, 12, 2, a^8b^20, 否], [6, 13, 1, a^7b^20, 否], [6, 14, 0, a^6b^20, 否], [7, 1, 12, a^19b^20, 否], [7, 2, 11, a^18b^20, 否], [7, 3, 10, a^17b^20, 否], [7, 4, 9, a^16b^20, 否], [7, 5, 8, a^15b^20, 否], [7, 6, 7, a^14b^20, 否], [7, 7, 6, a^13b^20, 否], [7, 8, 5, a^12b^20, 否], [7, 9, 4, a^11b^20, 否], [7, 10, 3, a^10b^20, 否], [7, 11, 2, a^9b^20, 否], [7, 12, 1, a^8b^20, 否], [7, 13, 0, a^7b^20, 否], [8, 1, 11, a^19b^20, 否], [8, 2, 10, a^18b^20, 否], [8, 3, 9, a^17b^20, 否], [8, 4, 8, a^16b^20, 否], [8, 5, 7, a^15b^20, 否], [8, 6, 6, a^14b^20, 否], [8, 7, 5, a^13b^20, 否], [8, 8, 4, a^12b^20, 否], [8, 9, 3, a^11b^20, 否], [8, 10, 2, a^10b^20, 否], [8, 11, 1, a^9b^20, 否], [8, 12, 0, a^8b^20, 否], [9, 1, 10, a^19b^20, 否], [9, 2, 9, a^18b^20, 否], [9, 3, 8, a^17b^20, 否], [9, 4, 7, a^16b^20, 否], [9, 5, 6, a^15b^20, 否], [9, 6, 5, a^14b^20, 否], [9, 7, 4, a^13b^20, 否], [9, 8, 3, a^12b^20, 否], [9, 9, 2, a^11b^20, 否], [9, 10, 1, a^10b^20, 否], [9, 11, 0, a^9b^20, 否], [10, 1, 9, a^19b^20, 否], [10, 2, 8, a^18b^20, 否], [10, 3, 7, a^17b^20, 否], [10, 4, 6, a^16b^20, 否], [10, 5, 5, a^15b^20, 否], [10, 6, 4, a^14b^20, 否], [10, 7, 3, a^13b^20, 否], [10, 8, 2, a^12b^20, 否], [10, 9, 1, a^11b^20, 否], [10, 10, 0, a^10b^20, 否], [11, 1, 8, a^19b^20, 否], [11, 2, 7, a^18b^20, 否], [11, 3, 6, a^17b^20, 否], [11, 4, 5, a^16b^20, 否], [11, 5, 4, a^15b^20, 否], [11, 6, 3, a^14b^20, 否], [11, 7, 2, a^13b^20, 否], [11, 8, 1, a^12b^20, 否], [11, 9, 0, a^11b^20, 否], [12, 1, 7, a^19b^20, 否], [12, 2, 6, a^18b^20, 否], [12, 3, 5, a^17b^20, 否], [12, 4, 4, a^16b^20, 否], [12, 5, 3, a^15b^20, 否], [12, 6, 2, a^14b^20, 否], [12, 7, 1, a^13b^20, 否], [12, 8, 0, a^12b^20, 否], [13, 1, 6, a^19b^20, 否], [13, 2, 5, a^18b^20, 否], [13, 3, 4, a^17b^20, 否], [13, 4, 3, a^16b^20, 否], [13, 5, 2, a^15b^20, 否], [13, 6, 1, a^14b^20, 否], [13, 7, 0, a^13b^20, 否], [14, 1, 5, a^19b^20, 否], [14, 2, 4, a^18b^20, 否], [14, 3, 3, a^17b^20, 否], [14, 4, 2, a^16b^20, 否], [14, 5, 1, a^15b^20, 否], [14, 6, 0, a^14b^20, 否], [15, 1, 4, a^19b^20, 否], [15, 2, 3, a^18b^20, 否], [15, 3, 2, a^17b^20, 否], [15, 4, 1, a^16b^20, 否], [15, 5, 0, a^15b^20, 否], [16, 1, 3, a^19b^20, 否], [16, 2, 2, a^18b^20, 否], [16, 3, 1, a^17b^20, 否], [16, 4, 0, a^16b^20, 否], [17, 1, 2, a^19b^20, 否], [17, 2, 1, a^18b^20, 否], [17, 3, 0, a^17b^20, 否], [18, 1, 1, a^19b^20, 否], [18, 2, 0, a^18b^20, 否], [19, 1, 0, a^19b^20, 否]], 210, 210, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

71. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":21}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

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(21, a^21b^21, [[0, 1, 20, a^20b^21, 否], [0, 2, 19, a^19b^21, 否], [0, 3, 18, a^18b^21, 否], [0, 4, 17, a^17b^21, 否], [0, 5, 16, a^16b^21, 否], [0, 6, 15, a^15b^21, 否], [0, 7, 14, a^14b^21, 否], [0, 8, 13, a^13b^21, 否], [0, 9, 12, a^12b^21, 否], [0, 10, 11, a^11b^21, 否], [0, 11, 10, a^10b^21, 否], [0, 12, 9, a^9b^21, 否], [0, 13, 8, a^8b^21, 否], [0, 14, 7, a^7b^21, 否], [0, 15, 6, a^6b^21, 否], [0, 16, 5, a^5b^21, 否], [0, 17, 4, a^4b^21, 否], [0, 18, 3, a^3b^21, 否], [0, 19, 2, a^2b^21, 否], [0, 20, 1, a^1b^21, 否], [0, 21, 0, a^0b^21, 否], [1, 1, 19, a^20b^21, 否], [1, 2, 18, a^19b^21, 否], [1, 3, 17, a^18b^21, 否], [1, 4, 16, a^17b^21, 否], [1, 5, 15, a^16b^21, 否], [1, 6, 14, a^15b^21, 否], [1, 7, 13, a^14b^21, 否], [1, 8, 12, a^13b^21, 否], [1, 9, 11, a^12b^21, 否], [1, 10, 10, a^11b^21, 否], [1, 11, 9, a^10b^21, 否], [1, 12, 8, a^9b^21, 否], [1, 13, 7, a^8b^21, 否], [1, 14, 6, a^7b^21, 否], [1, 15, 5, a^6b^21, 否], [1, 16, 4, a^5b^21, 否], [1, 17, 3, a^4b^21, 否], [1, 18, 2, a^3b^21, 否], [1, 19, 1, a^2b^21, 否], [1, 20, 0, a^1b^21, 否], [2, 1, 18, a^20b^21, 否], [2, 2, 17, a^19b^21, 否], [2, 3, 16, a^18b^21, 否], [2, 4, 15, a^17b^21, 否], [2, 5, 14, a^16b^21, 否], [2, 6, 13, a^15b^21, 否], [2, 7, 12, a^14b^21, 否], [2, 8, 11, a^13b^21, 否], [2, 9, 10, a^12b^21, 否], [2, 10, 9, a^11b^21, 否], [2, 11, 8, a^10b^21, 否], [2, 12, 7, a^9b^21, 否], [2, 13, 6, a^8b^21, 否], [2, 14, 5, a^7b^21, 否], [2, 15, 4, a^6b^21, 否], [2, 16, 3, a^5b^21, 否], [2, 17, 2, a^4b^21, 否], [2, 18, 1, a^3b^21, 否], [2, 19, 0, a^2b^21, 否], [3, 1, 17, a^20b^21, 否], [3, 2, 16, a^19b^21, 否], [3, 3, 15, a^18b^21, 否], [3, 4, 14, a^17b^21, 否], [3, 5, 13, a^16b^21, 否], [3, 6, 12, a^15b^21, 否], [3, 7, 11, a^14b^21, 否], [3, 8, 10, a^13b^21, 否], [3, 9, 9, a^12b^21, 否], [3, 10, 8, a^11b^21, 否], [3, 11, 7, a^10b^21, 否], [3, 12, 6, a^9b^21, 否], [3, 13, 5, a^8b^21, 否], [3, 14, 4, a^7b^21, 否], [3, 15, 3, a^6b^21, 否], [3, 16, 2, a^5b^21, 否], [3, 17, 1, a^4b^21, 否], [3, 18, 0, a^3b^21, 否], [4, 1, 16, a^20b^21, 否], [4, 2, 15, a^19b^21, 否], [4, 3, 14, a^18b^21, 否], [4, 4, 13, a^17b^21, 否], [4, 5, 12, a^16b^21, 否], [4, 6, 11, a^15b^21, 否], [4, 7, 10, a^14b^21, 否], [4, 8, 9, a^13b^21, 否], [4, 9, 8, a^12b^21, 否], [4, 10, 7, a^11b^21, 否], [4, 11, 6, a^10b^21, 否], [4, 12, 5, a^9b^21, 否], [4, 13, 4, a^8b^21, 否], [4, 14, 3, a^7b^21, 否], [4, 15, 2, a^6b^21, 否], [4, 16, 1, a^5b^21, 否], [4, 17, 0, a^4b^21, 否], [5, 1, 15, a^20b^21, 否], [5, 2, 14, a^19b^21, 否], [5, 3, 13, a^18b^21, 否], [5, 4, 12, a^17b^21, 否], [5, 5, 11, a^16b^21, 否], [5, 6, 10, a^15b^21, 否], [5, 7, 9, a^14b^21, 否], [5, 8, 8, a^13b^21, 否], [5, 9, 7, a^12b^21, 否], [5, 10, 6, a^11b^21, 否], [5, 11, 5, a^10b^21, 否], [5, 12, 4, a^9b^21, 否], [5, 13, 3, a^8b^21, 否], [5, 14, 2, a^7b^21, 否], [5, 15, 1, a^6b^21, 否], [5, 16, 0, a^5b^21, 否], [6, 1, 14, a^20b^21, 否], [6, 2, 13, a^19b^21, 否], [6, 3, 12, a^18b^21, 否], [6, 4, 11, a^17b^21, 否], [6, 5, 10, a^16b^21, 否], [6, 6, 9, a^15b^21, 否], [6, 7, 8, a^14b^21, 否], [6, 8, 7, a^13b^21, 否], [6, 9, 6, a^12b^21, 否], [6, 10, 5, a^11b^21, 否], [6, 11, 4, a^10b^21, 否], [6, 12, 3, a^9b^21, 否], [6, 13, 2, a^8b^21, 否], [6, 14, 1, a^7b^21, 否], [6, 15, 0, a^6b^21, 否], [7, 1, 13, a^20b^21, 否], [7, 2, 12, a^19b^21, 否], [7, 3, 11, a^18b^21, 否], [7, 4, 10, a^17b^21, 否], [7, 5, 9, a^16b^21, 否], [7, 6, 8, a^15b^21, 否], [7, 7, 7, a^14b^21, 否], [7, 8, 6, a^13b^21, 否], [7, 9, 5, a^12b^21, 否], [7, 10, 4, a^11b^21, 否], [7, 11, 3, a^10b^21, 否], [7, 12, 2, a^9b^21, 否], [7, 13, 1, a^8b^21, 否], [7, 14, 0, a^7b^21, 否], [8, 1, 12, a^20b^21, 否], [8, 2, 11, a^19b^21, 否], [8, 3, 10, a^18b^21, 否], [8, 4, 9, a^17b^21, 否], [8, 5, 8, a^16b^21, 否], [8, 6, 7, a^15b^21, 否], [8, 7, 6, a^14b^21, 否], [8, 8, 5, a^13b^21, 否], [8, 9, 4, a^12b^21, 否], [8, 10, 3, a^11b^21, 否], [8, 11, 2, a^10b^21, 否], [8, 12, 1, a^9b^21, 否], [8, 13, 0, a^8b^21, 否], [9, 1, 11, a^20b^21, 否], [9, 2, 10, a^19b^21, 否], [9, 3, 9, a^18b^21, 否], [9, 4, 8, a^17b^21, 否], [9, 5, 7, a^16b^21, 否], [9, 6, 6, a^15b^21, 否], [9, 7, 5, a^14b^21, 否], [9, 8, 4, a^13b^21, 否], [9, 9, 3, a^12b^21, 否], [9, 10, 2, a^11b^21, 否], [9, 11, 1, a^10b^21, 否], [9, 12, 0, a^9b^21, 否], [10, 1, 10, a^20b^21, 否], [10, 2, 9, a^19b^21, 否], [10, 3, 8, a^18b^21, 否], [10, 4, 7, a^17b^21, 否], [10, 5, 6, a^16b^21, 否], [10, 6, 5, a^15b^21, 否], [10, 7, 4, a^14b^21, 否], [10, 8, 3, a^13b^21, 否], [10, 9, 2, a^12b^21, 否], [10, 10, 1, a^11b^21, 否], [10, 11, 0, a^10b^21, 否], [11, 1, 9, a^20b^21, 否], [11, 2, 8, a^19b^21, 否], [11, 3, 7, a^18b^21, 否], [11, 4, 6, a^17b^21, 否], [11, 5, 5, a^16b^21, 否], [11, 6, 4, a^15b^21, 否], [11, 7, 3, a^14b^21, 否], [11, 8, 2, a^13b^21, 否], [11, 9, 1, a^12b^21, 否], [11, 10, 0, a^11b^21, 否], [12, 1, 8, a^20b^21, 否], [12, 2, 7, a^19b^21, 否], [12, 3, 6, a^18b^21, 否], [12, 4, 5, a^17b^21, 否], [12, 5, 4, a^16b^21, 否], [12, 6, 3, a^15b^21, 否], [12, 7, 2, a^14b^21, 否], [12, 8, 1, a^13b^21, 否], [12, 9, 0, a^12b^21, 否], [13, 1, 7, a^20b^21, 否], [13, 2, 6, a^19b^21, 否], [13, 3, 5, a^18b^21, 否], [13, 4, 4, a^17b^21, 否], [13, 5, 3, a^16b^21, 否], [13, 6, 2, a^15b^21, 否], [13, 7, 1, a^14b^21, 否], [13, 8, 0, a^13b^21, 否], [14, 1, 6, a^20b^21, 否], [14, 2, 5, a^19b^21, 否], [14, 3, 4, a^18b^21, 否], [14, 4, 3, a^17b^21, 否], [14, 5, 2, a^16b^21, 否], [14, 6, 1, a^15b^21, 否], [14, 7, 0, a^14b^21, 否], [15, 1, 5, a^20b^21, 否], [15, 2, 4, a^19b^21, 否], [15, 3, 3, a^18b^21, 否], [15, 4, 2, a^17b^21, 否], [15, 5, 1, a^16b^21, 否], [15, 6, 0, a^15b^21, 否], [16, 1, 4, a^20b^21, 否], [16, 2, 3, a^19b^21, 否], [16, 3, 2, a^18b^21, 否], [16, 4, 1, a^17b^21, 否], [16, 5, 0, a^16b^21, 否], [17, 1, 3, a^20b^21, 否], [17, 2, 2, a^19b^21, 否], [17, 3, 1, a^18b^21, 否], [17, 4, 0, a^17b^21, 否], [18, 1, 2, a^20b^21, 否], [18, 2, 1, a^19b^21, 否], [18, 3, 0, a^18b^21, 否], [19, 1, 1, a^20b^21, 否], [19, 2, 0, a^19b^21, 否], [20, 1, 0, a^20b^21, 否]], 231, 231, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(21, a^21b^21, [[0, 1, 20, a^20b^21, 否], [0, 2, 19, a^19b^21, 否], [0, 3, 18, a^18b^21, 否], [0, 4, 17, a^17b^21, 否], [0, 5, 16, a^16b^21, 否], [0, 6, 15, a^15b^21, 否], [0, 7, 14, a^14b^21, 否], [0, 8, 13, a^13b^21, 否], [0, 9, 12, a^12b^21, 否], [0, 10, 11, a^11b^21, 否], [0, 11, 10, a^10b^21, 否], [0, 12, 9, a^9b^21, 否], [0, 13, 8, a^8b^21, 否], [0, 14, 7, a^7b^21, 否], [0, 15, 6, a^6b^21, 否], [0, 16, 5, a^5b^21, 否], [0, 17, 4, a^4b^21, 否], [0, 18, 3, a^3b^21, 否], [0, 19, 2, a^2b^21, 否], [0, 20, 1, a^1b^21, 否], [0, 21, 0, a^0b^21, 否], [1, 1, 19, a^20b^21, 否], [1, 2, 18, a^19b^21, 否], [1, 3, 17, a^18b^21, 否], [1, 4, 16, a^17b^21, 否], [1, 5, 15, a^16b^21, 否], [1, 6, 14, a^15b^21, 否], [1, 7, 13, a^14b^21, 否], [1, 8, 12, a^13b^21, 否], [1, 9, 11, a^12b^21, 否], [1, 10, 10, a^11b^21, 否], [1, 11, 9, a^10b^21, 否], [1, 12, 8, a^9b^21, 否], [1, 13, 7, a^8b^21, 否], [1, 14, 6, a^7b^21, 否], [1, 15, 5, a^6b^21, 否], [1, 16, 4, a^5b^21, 否], [1, 17, 3, a^4b^21, 否], [1, 18, 2, a^3b^21, 否], [1, 19, 1, a^2b^21, 否], [1, 20, 0, a^1b^21, 否], [2, 1, 18, a^20b^21, 否], [2, 2, 17, a^19b^21, 否], [2, 3, 16, a^18b^21, 否], [2, 4, 15, a^17b^21, 否], [2, 5, 14, a^16b^21, 否], [2, 6, 13, a^15b^21, 否], [2, 7, 12, a^14b^21, 否], [2, 8, 11, a^13b^21, 否], [2, 9, 10, a^12b^21, 否], [2, 10, 9, a^11b^21, 否], [2, 11, 8, a^10b^21, 否], [2, 12, 7, a^9b^21, 否], [2, 13, 6, a^8b^21, 否], [2, 14, 5, a^7b^21, 否], [2, 15, 4, a^6b^21, 否], [2, 16, 3, a^5b^21, 否], [2, 17, 2, a^4b^21, 否], [2, 18, 1, a^3b^21, 否], [2, 19, 0, a^2b^21, 否], [3, 1, 17, a^20b^21, 否], [3, 2, 16, a^19b^21, 否], [3, 3, 15, a^18b^21, 否], [3, 4, 14, a^17b^21, 否], [3, 5, 13, a^16b^21, 否], [3, 6, 12, a^15b^21, 否], [3, 7, 11, a^14b^21, 否], [3, 8, 10, a^13b^21, 否], [3, 9, 9, a^12b^21, 否], [3, 10, 8, a^11b^21, 否], [3, 11, 7, a^10b^21, 否], [3, 12, 6, a^9b^21, 否], [3, 13, 5, a^8b^21, 否], [3, 14, 4, a^7b^21, 否], [3, 15, 3, a^6b^21, 否], [3, 16, 2, a^5b^21, 否], [3, 17, 1, a^4b^21, 否], [3, 18, 0, a^3b^21, 否], [4, 1, 16, a^20b^21, 否], [4, 2, 15, a^19b^21, 否], [4, 3, 14, a^18b^21, 否], [4, 4, 13, a^17b^21, 否], [4, 5, 12, a^16b^21, 否], [4, 6, 11, a^15b^21, 否], [4, 7, 10, a^14b^21, 否], [4, 8, 9, a^13b^21, 否], [4, 9, 8, a^12b^21, 否], [4, 10, 7, a^11b^21, 否], [4, 11, 6, a^10b^21, 否], [4, 12, 5, a^9b^21, 否], [4, 13, 4, a^8b^21, 否], [4, 14, 3, a^7b^21, 否], [4, 15, 2, a^6b^21, 否], [4, 16, 1, a^5b^21, 否], [4, 17, 0, a^4b^21, 否], [5, 1, 15, a^20b^21, 否], [5, 2, 14, a^19b^21, 否], [5, 3, 13, a^18b^21, 否], [5, 4, 12, a^17b^21, 否], [5, 5, 11, a^16b^21, 否], [5, 6, 10, a^15b^21, 否], [5, 7, 9, a^14b^21, 否], [5, 8, 8, a^13b^21, 否], [5, 9, 7, a^12b^21, 否], [5, 10, 6, a^11b^21, 否], [5, 11, 5, a^10b^21, 否], [5, 12, 4, a^9b^21, 否], [5, 13, 3, a^8b^21, 否], [5, 14, 2, a^7b^21, 否], [5, 15, 1, a^6b^21, 否], [5, 16, 0, a^5b^21, 否], [6, 1, 14, a^20b^21, 否], [6, 2, 13, a^19b^21, 否], [6, 3, 12, a^18b^21, 否], [6, 4, 11, a^17b^21, 否], [6, 5, 10, a^16b^21, 否], [6, 6, 9, a^15b^21, 否], [6, 7, 8, a^14b^21, 否], [6, 8, 7, a^13b^21, 否], [6, 9, 6, a^12b^21, 否], [6, 10, 5, a^11b^21, 否], [6, 11, 4, a^10b^21, 否], [6, 12, 3, a^9b^21, 否], [6, 13, 2, a^8b^21, 否], [6, 14, 1, a^7b^21, 否], [6, 15, 0, a^6b^21, 否], [7, 1, 13, a^20b^21, 否], [7, 2, 12, a^19b^21, 否], [7, 3, 11, a^18b^21, 否], [7, 4, 10, a^17b^21, 否], [7, 5, 9, a^16b^21, 否], [7, 6, 8, a^15b^21, 否], [7, 7, 7, a^14b^21, 否], [7, 8, 6, a^13b^21, 否], [7, 9, 5, a^12b^21, 否], [7, 10, 4, a^11b^21, 否], [7, 11, 3, a^10b^21, 否], [7, 12, 2, a^9b^21, 否], [7, 13, 1, a^8b^21, 否], [7, 14, 0, a^7b^21, 否], [8, 1, 12, a^20b^21, 否], [8, 2, 11, a^19b^21, 否], [8, 3, 10, a^18b^21, 否], [8, 4, 9, a^17b^21, 否], [8, 5, 8, a^16b^21, 否], [8, 6, 7, a^15b^21, 否], [8, 7, 6, a^14b^21, 否], [8, 8, 5, a^13b^21, 否], [8, 9, 4, a^12b^21, 否], [8, 10, 3, a^11b^21, 否], [8, 11, 2, a^10b^21, 否], [8, 12, 1, a^9b^21, 否], [8, 13, 0, a^8b^21, 否], [9, 1, 11, a^20b^21, 否], [9, 2, 10, a^19b^21, 否], [9, 3, 9, a^18b^21, 否], [9, 4, 8, a^17b^21, 否], [9, 5, 7, a^16b^21, 否], [9, 6, 6, a^15b^21, 否], [9, 7, 5, a^14b^21, 否], [9, 8, 4, a^13b^21, 否], [9, 9, 3, a^12b^21, 否], [9, 10, 2, a^11b^21, 否], [9, 11, 1, a^10b^21, 否], [9, 12, 0, a^9b^21, 否], [10, 1, 10, a^20b^21, 否], [10, 2, 9, a^19b^21, 否], [10, 3, 8, a^18b^21, 否], [10, 4, 7, a^17b^21, 否], [10, 5, 6, a^16b^21, 否], [10, 6, 5, a^15b^21, 否], [10, 7, 4, a^14b^21, 否], [10, 8, 3, a^13b^21, 否], [10, 9, 2, a^12b^21, 否], [10, 10, 1, a^11b^21, 否], [10, 11, 0, a^10b^21, 否], [11, 1, 9, a^20b^21, 否], [11, 2, 8, a^19b^21, 否], [11, 3, 7, a^18b^21, 否], [11, 4, 6, a^17b^21, 否], [11, 5, 5, a^16b^21, 否], [11, 6, 4, a^15b^21, 否], [11, 7, 3, a^14b^21, 否], [11, 8, 2, a^13b^21, 否], [11, 9, 1, a^12b^21, 否], [11, 10, 0, a^11b^21, 否], [12, 1, 8, a^20b^21, 否], [12, 2, 7, a^19b^21, 否], [12, 3, 6, a^18b^21, 否], [12, 4, 5, a^17b^21, 否], [12, 5, 4, a^16b^21, 否], [12, 6, 3, a^15b^21, 否], [12, 7, 2, a^14b^21, 否], [12, 8, 1, a^13b^21, 否], [12, 9, 0, a^12b^21, 否], [13, 1, 7, a^20b^21, 否], [13, 2, 6, a^19b^21, 否], [13, 3, 5, a^18b^21, 否], [13, 4, 4, a^17b^21, 否], [13, 5, 3, a^16b^21, 否], [13, 6, 2, a^15b^21, 否], [13, 7, 1, a^14b^21, 否], [13, 8, 0, a^13b^21, 否], [14, 1, 6, a^20b^21, 否], [14, 2, 5, a^19b^21, 否], [14, 3, 4, a^18b^21, 否], [14, 4, 3, a^17b^21, 否], [14, 5, 2, a^16b^21, 否], [14, 6, 1, a^15b^21, 否], [14, 7, 0, a^14b^21, 否], [15, 1, 5, a^20b^21, 否], [15, 2, 4, a^19b^21, 否], [15, 3, 3, a^18b^21, 否], [15, 4, 2, a^17b^21, 否], [15, 5, 1, a^16b^21, 否], [15, 6, 0, a^15b^21, 否], [16, 1, 4, a^20b^21, 否], [16, 2, 3, a^19b^21, 否], [16, 3, 2, a^18b^21, 否], [16, 4, 1, a^17b^21, 否], [16, 5, 0, a^16b^21, 否], [17, 1, 3, a^20b^21, 否], [17, 2, 2, a^19b^21, 否], [17, 3, 1, a^18b^21, 否], [17, 4, 0, a^17b^21, 否], [18, 1, 2, a^20b^21, 否], [18, 2, 1, a^19b^21, 否], [18, 3, 0, a^18b^21, 否], [19, 1, 1, a^20b^21, 否], [19, 2, 0, a^19b^21, 否], [20, 1, 0, a^20b^21, 否]], 231, 231, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

72. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":22}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(22, a^22b^22, [[0, 1, 21, a^21b^22, 否], [0, 2, 20, a^20b^22, 否], [0, 3, 19, a^19b^22, 否], [0, 4, 18, a^18b^22, 否], [0, 5, 17, a^17b^22, 否], [0, 6, 16, a^16b^22, 否], [0, 7, 15, a^15b^22, 否], [0, 8, 14, a^14b^22, 否], [0, 9, 13, a^13b^22, 否], [0, 10, 12, a^12b^22, 否], [0, 11, 11, a^11b^22, 否], [0, 12, 10, a^10b^22, 否], [0, 13, 9, a^9b^22, 否], [0, 14, 8, a^8b^22, 否], [0, 15, 7, a^7b^22, 否], [0, 16, 6, a^6b^22, 否], [0, 17, 5, a^5b^22, 否], [0, 18, 4, a^4b^22, 否], [0, 19, 3, a^3b^22, 否], [0, 20, 2, a^2b^22, 否], [0, 21, 1, a^1b^22, 否], [0, 22, 0, a^0b^22, 否], [1, 1, 20, a^21b^22, 否], [1, 2, 19, a^20b^22, 否], [1, 3, 18, a^19b^22, 否], [1, 4, 17, a^18b^22, 否], [1, 5, 16, a^17b^22, 否], [1, 6, 15, a^16b^22, 否], [1, 7, 14, a^15b^22, 否], [1, 8, 13, a^14b^22, 否], [1, 9, 12, a^13b^22, 否], [1, 10, 11, a^12b^22, 否], [1, 11, 10, a^11b^22, 否], [1, 12, 9, a^10b^22, 否], [1, 13, 8, a^9b^22, 否], [1, 14, 7, a^8b^22, 否], [1, 15, 6, a^7b^22, 否], [1, 16, 5, a^6b^22, 否], [1, 17, 4, a^5b^22, 否], [1, 18, 3, a^4b^22, 否], [1, 19, 2, a^3b^22, 否], [1, 20, 1, a^2b^22, 否], [1, 21, 0, a^1b^22, 否], [2, 1, 19, a^21b^22, 否], [2, 2, 18, a^20b^22, 否], [2, 3, 17, a^19b^22, 否], [2, 4, 16, a^18b^22, 否], [2, 5, 15, a^17b^22, 否], [2, 6, 14, a^16b^22, 否], [2, 7, 13, a^15b^22, 否], [2, 8, 12, a^14b^22, 否], [2, 9, 11, a^13b^22, 否], [2, 10, 10, a^12b^22, 否], [2, 11, 9, a^11b^22, 否], [2, 12, 8, a^10b^22, 否], [2, 13, 7, a^9b^22, 否], [2, 14, 6, a^8b^22, 否], [2, 15, 5, a^7b^22, 否], [2, 16, 4, a^6b^22, 否], [2, 17, 3, a^5b^22, 否], [2, 18, 2, a^4b^22, 否], [2, 19, 1, a^3b^22, 否], [2, 20, 0, a^2b^22, 否], [3, 1, 18, a^21b^22, 否], [3, 2, 17, a^20b^22, 否], [3, 3, 16, a^19b^22, 否], [3, 4, 15, a^18b^22, 否], [3, 5, 14, a^17b^22, 否], [3, 6, 13, a^16b^22, 否], [3, 7, 12, a^15b^22, 否], [3, 8, 11, a^14b^22, 否], [3, 9, 10, a^13b^22, 否], [3, 10, 9, a^12b^22, 否], [3, 11, 8, a^11b^22, 否], [3, 12, 7, a^10b^22, 否], [3, 13, 6, a^9b^22, 否], [3, 14, 5, a^8b^22, 否], [3, 15, 4, a^7b^22, 否], [3, 16, 3, a^6b^22, 否], [3, 17, 2, a^5b^22, 否], [3, 18, 1, a^4b^22, 否], [3, 19, 0, a^3b^22, 否], [4, 1, 17, a^21b^22, 否], [4, 2, 16, a^20b^22, 否], [4, 3, 15, a^19b^22, 否], [4, 4, 14, a^18b^22, 否], [4, 5, 13, a^17b^22, 否], [4, 6, 12, a^16b^22, 否], [4, 7, 11, a^15b^22, 否], [4, 8, 10, a^14b^22, 否], [4, 9, 9, a^13b^22, 否], [4, 10, 8, a^12b^22, 否], [4, 11, 7, a^11b^22, 否], [4, 12, 6, a^10b^22, 否], [4, 13, 5, a^9b^22, 否], [4, 14, 4, a^8b^22, 否], [4, 15, 3, a^7b^22, 否], [4, 16, 2, a^6b^22, 否], [4, 17, 1, a^5b^22, 否], [4, 18, 0, a^4b^22, 否], [5, 1, 16, a^21b^22, 否], [5, 2, 15, a^20b^22, 否], [5, 3, 14, a^19b^22, 否], [5, 4, 13, a^18b^22, 否], [5, 5, 12, a^17b^22, 否], [5, 6, 11, a^16b^22, 否], [5, 7, 10, a^15b^22, 否], [5, 8, 9, a^14b^22, 否], [5, 9, 8, a^13b^22, 否], [5, 10, 7, a^12b^22, 否], [5, 11, 6, a^11b^22, 否], [5, 12, 5, a^10b^22, 否], [5, 13, 4, a^9b^22, 否], [5, 14, 3, a^8b^22, 否], [5, 15, 2, a^7b^22, 否], [5, 16, 1, a^6b^22, 否], [5, 17, 0, a^5b^22, 否], [6, 1, 15, a^21b^22, 否], [6, 2, 14, a^20b^22, 否], [6, 3, 13, a^19b^22, 否], [6, 4, 12, a^18b^22, 否], [6, 5, 11, a^17b^22, 否], [6, 6, 10, a^16b^22, 否], [6, 7, 9, a^15b^22, 否], [6, 8, 8, a^14b^22, 否], [6, 9, 7, a^13b^22, 否], [6, 10, 6, a^12b^22, 否], [6, 11, 5, a^11b^22, 否], [6, 12, 4, a^10b^22, 否], [6, 13, 3, a^9b^22, 否], [6, 14, 2, a^8b^22, 否], [6, 15, 1, a^7b^22, 否], [6, 16, 0, a^6b^22, 否], [7, 1, 14, a^21b^22, 否], [7, 2, 13, a^20b^22, 否], [7, 3, 12, a^19b^22, 否], [7, 4, 11, a^18b^22, 否], [7, 5, 10, a^17b^22, 否], [7, 6, 9, a^16b^22, 否], [7, 7, 8, a^15b^22, 否], [7, 8, 7, a^14b^22, 否], [7, 9, 6, a^13b^22, 否], [7, 10, 5, a^12b^22, 否], [7, 11, 4, a^11b^22, 否], [7, 12, 3, a^10b^22, 否], [7, 13, 2, a^9b^22, 否], [7, 14, 1, a^8b^22, 否], [7, 15, 0, a^7b^22, 否], [8, 1, 13, a^21b^22, 否], [8, 2, 12, a^20b^22, 否], [8, 3, 11, a^19b^22, 否], [8, 4, 10, a^18b^22, 否], [8, 5, 9, a^17b^22, 否], [8, 6, 8, a^16b^22, 否], [8, 7, 7, a^15b^22, 否], [8, 8, 6, a^14b^22, 否], [8, 9, 5, a^13b^22, 否], [8, 10, 4, a^12b^22, 否], [8, 11, 3, a^11b^22, 否], [8, 12, 2, a^10b^22, 否], [8, 13, 1, a^9b^22, 否], [8, 14, 0, a^8b^22, 否], [9, 1, 12, a^21b^22, 否], [9, 2, 11, a^20b^22, 否], [9, 3, 10, a^19b^22, 否], [9, 4, 9, a^18b^22, 否], [9, 5, 8, a^17b^22, 否], [9, 6, 7, a^16b^22, 否], [9, 7, 6, a^15b^22, 否], [9, 8, 5, a^14b^22, 否], [9, 9, 4, a^13b^22, 否], [9, 10, 3, a^12b^22, 否], [9, 11, 2, a^11b^22, 否], [9, 12, 1, a^10b^22, 否], [9, 13, 0, a^9b^22, 否], [10, 1, 11, a^21b^22, 否], [10, 2, 10, a^20b^22, 否], [10, 3, 9, a^19b^22, 否], [10, 4, 8, a^18b^22, 否], [10, 5, 7, a^17b^22, 否], [10, 6, 6, a^16b^22, 否], [10, 7, 5, a^15b^22, 否], [10, 8, 4, a^14b^22, 否], [10, 9, 3, a^13b^22, 否], [10, 10, 2, a^12b^22, 否], [10, 11, 1, a^11b^22, 否], [10, 12, 0, a^10b^22, 否], [11, 1, 10, a^21b^22, 否], [11, 2, 9, a^20b^22, 否], [11, 3, 8, a^19b^22, 否], [11, 4, 7, a^18b^22, 否], [11, 5, 6, a^17b^22, 否], [11, 6, 5, a^16b^22, 否], [11, 7, 4, a^15b^22, 否], [11, 8, 3, a^14b^22, 否], [11, 9, 2, a^13b^22, 否], [11, 10, 1, a^12b^22, 否], [11, 11, 0, a^11b^22, 否], [12, 1, 9, a^21b^22, 否], [12, 2, 8, a^20b^22, 否], [12, 3, 7, a^19b^22, 否], [12, 4, 6, a^18b^22, 否], [12, 5, 5, a^17b^22, 否], [12, 6, 4, a^16b^22, 否], [12, 7, 3, a^15b^22, 否], [12, 8, 2, a^14b^22, 否], [12, 9, 1, a^13b^22, 否], [12, 10, 0, a^12b^22, 否], [13, 1, 8, a^21b^22, 否], [13, 2, 7, a^20b^22, 否], [13, 3, 6, a^19b^22, 否], [13, 4, 5, a^18b^22, 否], [13, 5, 4, a^17b^22, 否], [13, 6, 3, a^16b^22, 否], [13, 7, 2, a^15b^22, 否], [13, 8, 1, a^14b^22, 否], [13, 9, 0, a^13b^22, 否], [14, 1, 7, a^21b^22, 否], [14, 2, 6, a^20b^22, 否], [14, 3, 5, a^19b^22, 否], [14, 4, 4, a^18b^22, 否], [14, 5, 3, a^17b^22, 否], [14, 6, 2, a^16b^22, 否], [14, 7, 1, a^15b^22, 否], [14, 8, 0, a^14b^22, 否], [15, 1, 6, a^21b^22, 否], [15, 2, 5, a^20b^22, 否], [15, 3, 4, a^19b^22, 否], [15, 4, 3, a^18b^22, 否], [15, 5, 2, a^17b^22, 否], [15, 6, 1, a^16b^22, 否], [15, 7, 0, a^15b^22, 否], [16, 1, 5, a^21b^22, 否], [16, 2, 4, a^20b^22, 否], [16, 3, 3, a^19b^22, 否], [16, 4, 2, a^18b^22, 否], [16, 5, 1, a^17b^22, 否], [16, 6, 0, a^16b^22, 否], [17, 1, 4, a^21b^22, 否], [17, 2, 3, a^20b^22, 否], [17, 3, 2, a^19b^22, 否], [17, 4, 1, a^18b^22, 否], [17, 5, 0, a^17b^22, 否], [18, 1, 3, a^21b^22, 否], [18, 2, 2, a^20b^22, 否], [18, 3, 1, a^19b^22, 否], [18, 4, 0, a^18b^22, 否], [19, 1, 2, a^21b^22, 否], [19, 2, 1, a^20b^22, 否], [19, 3, 0, a^19b^22, 否], [20, 1, 1, a^21b^22, 否], [20, 2, 0, a^20b^22, 否], [21, 1, 0, a^21b^22, 否]], 253, 253, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(22, a^22b^22, [[0, 1, 21, a^21b^22, 否], [0, 2, 20, a^20b^22, 否], [0, 3, 19, a^19b^22, 否], [0, 4, 18, a^18b^22, 否], [0, 5, 17, a^17b^22, 否], [0, 6, 16, a^16b^22, 否], [0, 7, 15, a^15b^22, 否], [0, 8, 14, a^14b^22, 否], [0, 9, 13, a^13b^22, 否], [0, 10, 12, a^12b^22, 否], [0, 11, 11, a^11b^22, 否], [0, 12, 10, a^10b^22, 否], [0, 13, 9, a^9b^22, 否], [0, 14, 8, a^8b^22, 否], [0, 15, 7, a^7b^22, 否], [0, 16, 6, a^6b^22, 否], [0, 17, 5, a^5b^22, 否], [0, 18, 4, a^4b^22, 否], [0, 19, 3, a^3b^22, 否], [0, 20, 2, a^2b^22, 否], [0, 21, 1, a^1b^22, 否], [0, 22, 0, a^0b^22, 否], [1, 1, 20, a^21b^22, 否], [1, 2, 19, a^20b^22, 否], [1, 3, 18, a^19b^22, 否], [1, 4, 17, a^18b^22, 否], [1, 5, 16, a^17b^22, 否], [1, 6, 15, a^16b^22, 否], [1, 7, 14, a^15b^22, 否], [1, 8, 13, a^14b^22, 否], [1, 9, 12, a^13b^22, 否], [1, 10, 11, a^12b^22, 否], [1, 11, 10, a^11b^22, 否], [1, 12, 9, a^10b^22, 否], [1, 13, 8, a^9b^22, 否], [1, 14, 7, a^8b^22, 否], [1, 15, 6, a^7b^22, 否], [1, 16, 5, a^6b^22, 否], [1, 17, 4, a^5b^22, 否], [1, 18, 3, a^4b^22, 否], [1, 19, 2, a^3b^22, 否], [1, 20, 1, a^2b^22, 否], [1, 21, 0, a^1b^22, 否], [2, 1, 19, a^21b^22, 否], [2, 2, 18, a^20b^22, 否], [2, 3, 17, a^19b^22, 否], [2, 4, 16, a^18b^22, 否], [2, 5, 15, a^17b^22, 否], [2, 6, 14, a^16b^22, 否], [2, 7, 13, a^15b^22, 否], [2, 8, 12, a^14b^22, 否], [2, 9, 11, a^13b^22, 否], [2, 10, 10, a^12b^22, 否], [2, 11, 9, a^11b^22, 否], [2, 12, 8, a^10b^22, 否], [2, 13, 7, a^9b^22, 否], [2, 14, 6, a^8b^22, 否], [2, 15, 5, a^7b^22, 否], [2, 16, 4, a^6b^22, 否], [2, 17, 3, a^5b^22, 否], [2, 18, 2, a^4b^22, 否], [2, 19, 1, a^3b^22, 否], [2, 20, 0, a^2b^22, 否], [3, 1, 18, a^21b^22, 否], [3, 2, 17, a^20b^22, 否], [3, 3, 16, a^19b^22, 否], [3, 4, 15, a^18b^22, 否], [3, 5, 14, a^17b^22, 否], [3, 6, 13, a^16b^22, 否], [3, 7, 12, a^15b^22, 否], [3, 8, 11, a^14b^22, 否], [3, 9, 10, a^13b^22, 否], [3, 10, 9, a^12b^22, 否], [3, 11, 8, a^11b^22, 否], [3, 12, 7, a^10b^22, 否], [3, 13, 6, a^9b^22, 否], [3, 14, 5, a^8b^22, 否], [3, 15, 4, a^7b^22, 否], [3, 16, 3, a^6b^22, 否], [3, 17, 2, a^5b^22, 否], [3, 18, 1, a^4b^22, 否], [3, 19, 0, a^3b^22, 否], [4, 1, 17, a^21b^22, 否], [4, 2, 16, a^20b^22, 否], [4, 3, 15, a^19b^22, 否], [4, 4, 14, a^18b^22, 否], [4, 5, 13, a^17b^22, 否], [4, 6, 12, a^16b^22, 否], [4, 7, 11, a^15b^22, 否], [4, 8, 10, a^14b^22, 否], [4, 9, 9, a^13b^22, 否], [4, 10, 8, a^12b^22, 否], [4, 11, 7, a^11b^22, 否], [4, 12, 6, a^10b^22, 否], [4, 13, 5, a^9b^22, 否], [4, 14, 4, a^8b^22, 否], [4, 15, 3, a^7b^22, 否], [4, 16, 2, a^6b^22, 否], [4, 17, 1, a^5b^22, 否], [4, 18, 0, a^4b^22, 否], [5, 1, 16, a^21b^22, 否], [5, 2, 15, a^20b^22, 否], [5, 3, 14, a^19b^22, 否], [5, 4, 13, a^18b^22, 否], [5, 5, 12, a^17b^22, 否], [5, 6, 11, a^16b^22, 否], [5, 7, 10, a^15b^22, 否], [5, 8, 9, a^14b^22, 否], [5, 9, 8, a^13b^22, 否], [5, 10, 7, a^12b^22, 否], [5, 11, 6, a^11b^22, 否], [5, 12, 5, a^10b^22, 否], [5, 13, 4, a^9b^22, 否], [5, 14, 3, a^8b^22, 否], [5, 15, 2, a^7b^22, 否], [5, 16, 1, a^6b^22, 否], [5, 17, 0, a^5b^22, 否], [6, 1, 15, a^21b^22, 否], [6, 2, 14, a^20b^22, 否], [6, 3, 13, a^19b^22, 否], [6, 4, 12, a^18b^22, 否], [6, 5, 11, a^17b^22, 否], [6, 6, 10, a^16b^22, 否], [6, 7, 9, a^15b^22, 否], [6, 8, 8, a^14b^22, 否], [6, 9, 7, a^13b^22, 否], [6, 10, 6, a^12b^22, 否], [6, 11, 5, a^11b^22, 否], [6, 12, 4, a^10b^22, 否], [6, 13, 3, a^9b^22, 否], [6, 14, 2, a^8b^22, 否], [6, 15, 1, a^7b^22, 否], [6, 16, 0, a^6b^22, 否], [7, 1, 14, a^21b^22, 否], [7, 2, 13, a^20b^22, 否], [7, 3, 12, a^19b^22, 否], [7, 4, 11, a^18b^22, 否], [7, 5, 10, a^17b^22, 否], [7, 6, 9, a^16b^22, 否], [7, 7, 8, a^15b^22, 否], [7, 8, 7, a^14b^22, 否], [7, 9, 6, a^13b^22, 否], [7, 10, 5, a^12b^22, 否], [7, 11, 4, a^11b^22, 否], [7, 12, 3, a^10b^22, 否], [7, 13, 2, a^9b^22, 否], [7, 14, 1, a^8b^22, 否], [7, 15, 0, a^7b^22, 否], [8, 1, 13, a^21b^22, 否], [8, 2, 12, a^20b^22, 否], [8, 3, 11, a^19b^22, 否], [8, 4, 10, a^18b^22, 否], [8, 5, 9, a^17b^22, 否], [8, 6, 8, a^16b^22, 否], [8, 7, 7, a^15b^22, 否], [8, 8, 6, a^14b^22, 否], [8, 9, 5, a^13b^22, 否], [8, 10, 4, a^12b^22, 否], [8, 11, 3, a^11b^22, 否], [8, 12, 2, a^10b^22, 否], [8, 13, 1, a^9b^22, 否], [8, 14, 0, a^8b^22, 否], [9, 1, 12, a^21b^22, 否], [9, 2, 11, a^20b^22, 否], [9, 3, 10, a^19b^22, 否], [9, 4, 9, a^18b^22, 否], [9, 5, 8, a^17b^22, 否], [9, 6, 7, a^16b^22, 否], [9, 7, 6, a^15b^22, 否], [9, 8, 5, a^14b^22, 否], [9, 9, 4, a^13b^22, 否], [9, 10, 3, a^12b^22, 否], [9, 11, 2, a^11b^22, 否], [9, 12, 1, a^10b^22, 否], [9, 13, 0, a^9b^22, 否], [10, 1, 11, a^21b^22, 否], [10, 2, 10, a^20b^22, 否], [10, 3, 9, a^19b^22, 否], [10, 4, 8, a^18b^22, 否], [10, 5, 7, a^17b^22, 否], [10, 6, 6, a^16b^22, 否], [10, 7, 5, a^15b^22, 否], [10, 8, 4, a^14b^22, 否], [10, 9, 3, a^13b^22, 否], [10, 10, 2, a^12b^22, 否], [10, 11, 1, a^11b^22, 否], [10, 12, 0, a^10b^22, 否], [11, 1, 10, a^21b^22, 否], [11, 2, 9, a^20b^22, 否], [11, 3, 8, a^19b^22, 否], [11, 4, 7, a^18b^22, 否], [11, 5, 6, a^17b^22, 否], [11, 6, 5, a^16b^22, 否], [11, 7, 4, a^15b^22, 否], [11, 8, 3, a^14b^22, 否], [11, 9, 2, a^13b^22, 否], [11, 10, 1, a^12b^22, 否], [11, 11, 0, a^11b^22, 否], [12, 1, 9, a^21b^22, 否], [12, 2, 8, a^20b^22, 否], [12, 3, 7, a^19b^22, 否], [12, 4, 6, a^18b^22, 否], [12, 5, 5, a^17b^22, 否], [12, 6, 4, a^16b^22, 否], [12, 7, 3, a^15b^22, 否], [12, 8, 2, a^14b^22, 否], [12, 9, 1, a^13b^22, 否], [12, 10, 0, a^12b^22, 否], [13, 1, 8, a^21b^22, 否], [13, 2, 7, a^20b^22, 否], [13, 3, 6, a^19b^22, 否], [13, 4, 5, a^18b^22, 否], [13, 5, 4, a^17b^22, 否], [13, 6, 3, a^16b^22, 否], [13, 7, 2, a^15b^22, 否], [13, 8, 1, a^14b^22, 否], [13, 9, 0, a^13b^22, 否], [14, 1, 7, a^21b^22, 否], [14, 2, 6, a^20b^22, 否], [14, 3, 5, a^19b^22, 否], [14, 4, 4, a^18b^22, 否], [14, 5, 3, a^17b^22, 否], [14, 6, 2, a^16b^22, 否], [14, 7, 1, a^15b^22, 否], [14, 8, 0, a^14b^22, 否], [15, 1, 6, a^21b^22, 否], [15, 2, 5, a^20b^22, 否], [15, 3, 4, a^19b^22, 否], [15, 4, 3, a^18b^22, 否], [15, 5, 2, a^17b^22, 否], [15, 6, 1, a^16b^22, 否], [15, 7, 0, a^15b^22, 否], [16, 1, 5, a^21b^22, 否], [16, 2, 4, a^20b^22, 否], [16, 3, 3, a^19b^22, 否], [16, 4, 2, a^18b^22, 否], [16, 5, 1, a^17b^22, 否], [16, 6, 0, a^16b^22, 否], [17, 1, 4, a^21b^22, 否], [17, 2, 3, a^20b^22, 否], [17, 3, 2, a^19b^22, 否], [17, 4, 1, a^18b^22, 否], [17, 5, 0, a^17b^22, 否], [18, 1, 3, a^21b^22, 否], [18, 2, 2, a^20b^22, 否], [18, 3, 1, a^19b^22, 否], [18, 4, 0, a^18b^22, 否], [19, 1, 2, a^21b^22, 否], [19, 2, 1, a^20b^22, 否], [19, 3, 0, a^19b^22, 否], [20, 1, 1, a^21b^22, 否], [20, 2, 0, a^20b^22, 否], [21, 1, 0, a^21b^22, 否]], 253, 253, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

73. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":23}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(23, a^23b^23, [[0, 1, 22, a^22b^23, 否], [0, 2, 21, a^21b^23, 否], [0, 3, 20, a^20b^23, 否], [0, 4, 19, a^19b^23, 否], [0, 5, 18, a^18b^23, 否], [0, 6, 17, a^17b^23, 否], [0, 7, 16, a^16b^23, 否], [0, 8, 15, a^15b^23, 否], [0, 9, 14, a^14b^23, 否], [0, 10, 13, a^13b^23, 否], [0, 11, 12, a^12b^23, 否], [0, 12, 11, a^11b^23, 否], [0, 13, 10, a^10b^23, 否], [0, 14, 9, a^9b^23, 否], [0, 15, 8, a^8b^23, 否], [0, 16, 7, a^7b^23, 否], [0, 17, 6, a^6b^23, 否], [0, 18, 5, a^5b^23, 否], [0, 19, 4, a^4b^23, 否], [0, 20, 3, a^3b^23, 否], [0, 21, 2, a^2b^23, 否], [0, 22, 1, a^1b^23, 否], [0, 23, 0, a^0b^23, 否], [1, 1, 21, a^22b^23, 否], [1, 2, 20, a^21b^23, 否], [1, 3, 19, a^20b^23, 否], [1, 4, 18, a^19b^23, 否], [1, 5, 17, a^18b^23, 否], [1, 6, 16, a^17b^23, 否], [1, 7, 15, a^16b^23, 否], [1, 8, 14, a^15b^23, 否], [1, 9, 13, a^14b^23, 否], [1, 10, 12, a^13b^23, 否], [1, 11, 11, a^12b^23, 否], [1, 12, 10, a^11b^23, 否], [1, 13, 9, a^10b^23, 否], [1, 14, 8, a^9b^23, 否], [1, 15, 7, a^8b^23, 否], [1, 16, 6, a^7b^23, 否], [1, 17, 5, a^6b^23, 否], [1, 18, 4, a^5b^23, 否], [1, 19, 3, a^4b^23, 否], [1, 20, 2, a^3b^23, 否], [1, 21, 1, a^2b^23, 否], [1, 22, 0, a^1b^23, 否], [2, 1, 20, a^22b^23, 否], [2, 2, 19, a^21b^23, 否], [2, 3, 18, a^20b^23, 否], [2, 4, 17, a^19b^23, 否], [2, 5, 16, a^18b^23, 否], [2, 6, 15, a^17b^23, 否], [2, 7, 14, a^16b^23, 否], [2, 8, 13, a^15b^23, 否], [2, 9, 12, a^14b^23, 否], [2, 10, 11, a^13b^23, 否], [2, 11, 10, a^12b^23, 否], [2, 12, 9, a^11b^23, 否], [2, 13, 8, a^10b^23, 否], [2, 14, 7, a^9b^23, 否], [2, 15, 6, a^8b^23, 否], [2, 16, 5, a^7b^23, 否], [2, 17, 4, a^6b^23, 否], [2, 18, 3, a^5b^23, 否], [2, 19, 2, a^4b^23, 否], [2, 20, 1, a^3b^23, 否], [2, 21, 0, a^2b^23, 否], [3, 1, 19, a^22b^23, 否], [3, 2, 18, a^21b^23, 否], [3, 3, 17, a^20b^23, 否], [3, 4, 16, a^19b^23, 否], [3, 5, 15, a^18b^23, 否], [3, 6, 14, a^17b^23, 否], [3, 7, 13, a^16b^23, 否], [3, 8, 12, a^15b^23, 否], [3, 9, 11, a^14b^23, 否], [3, 10, 10, a^13b^23, 否], [3, 11, 9, a^12b^23, 否], [3, 12, 8, a^11b^23, 否], [3, 13, 7, a^10b^23, 否], [3, 14, 6, a^9b^23, 否], [3, 15, 5, a^8b^23, 否], [3, 16, 4, a^7b^23, 否], [3, 17, 3, a^6b^23, 否], [3, 18, 2, a^5b^23, 否], [3, 19, 1, a^4b^23, 否], [3, 20, 0, a^3b^23, 否], [4, 1, 18, a^22b^23, 否], [4, 2, 17, a^21b^23, 否], [4, 3, 16, a^20b^23, 否], [4, 4, 15, a^19b^23, 否], [4, 5, 14, a^18b^23, 否], [4, 6, 13, a^17b^23, 否], [4, 7, 12, a^16b^23, 否], [4, 8, 11, a^15b^23, 否], [4, 9, 10, a^14b^23, 否], [4, 10, 9, a^13b^23, 否], [4, 11, 8, a^12b^23, 否], [4, 12, 7, a^11b^23, 否], [4, 13, 6, a^10b^23, 否], [4, 14, 5, a^9b^23, 否], [4, 15, 4, a^8b^23, 否], [4, 16, 3, a^7b^23, 否], [4, 17, 2, a^6b^23, 否], [4, 18, 1, a^5b^23, 否], [4, 19, 0, a^4b^23, 否], [5, 1, 17, a^22b^23, 否], [5, 2, 16, a^21b^23, 否], [5, 3, 15, a^20b^23, 否], [5, 4, 14, a^19b^23, 否], [5, 5, 13, a^18b^23, 否], [5, 6, 12, a^17b^23, 否], [5, 7, 11, a^16b^23, 否], [5, 8, 10, a^15b^23, 否], [5, 9, 9, a^14b^23, 否], [5, 10, 8, a^13b^23, 否], [5, 11, 7, a^12b^23, 否], [5, 12, 6, a^11b^23, 否], [5, 13, 5, a^10b^23, 否], [5, 14, 4, a^9b^23, 否], [5, 15, 3, a^8b^23, 否], [5, 16, 2, a^7b^23, 否], [5, 17, 1, a^6b^23, 否], [5, 18, 0, a^5b^23, 否], [6, 1, 16, a^22b^23, 否], [6, 2, 15, a^21b^23, 否], [6, 3, 14, a^20b^23, 否], [6, 4, 13, a^19b^23, 否], [6, 5, 12, a^18b^23, 否], [6, 6, 11, a^17b^23, 否], [6, 7, 10, a^16b^23, 否], [6, 8, 9, a^15b^23, 否], [6, 9, 8, a^14b^23, 否], [6, 10, 7, a^13b^23, 否], [6, 11, 6, a^12b^23, 否], [6, 12, 5, a^11b^23, 否], [6, 13, 4, a^10b^23, 否], [6, 14, 3, a^9b^23, 否], [6, 15, 2, a^8b^23, 否], [6, 16, 1, a^7b^23, 否], [6, 17, 0, a^6b^23, 否], [7, 1, 15, a^22b^23, 否], [7, 2, 14, a^21b^23, 否], [7, 3, 13, a^20b^23, 否], [7, 4, 12, a^19b^23, 否], [7, 5, 11, a^18b^23, 否], [7, 6, 10, a^17b^23, 否], [7, 7, 9, a^16b^23, 否], [7, 8, 8, a^15b^23, 否], [7, 9, 7, a^14b^23, 否], [7, 10, 6, a^13b^23, 否], [7, 11, 5, a^12b^23, 否], [7, 12, 4, a^11b^23, 否], [7, 13, 3, a^10b^23, 否], [7, 14, 2, a^9b^23, 否], [7, 15, 1, a^8b^23, 否], [7, 16, 0, a^7b^23, 否], [8, 1, 14, a^22b^23, 否], [8, 2, 13, a^21b^23, 否], [8, 3, 12, a^20b^23, 否], [8, 4, 11, a^19b^23, 否], [8, 5, 10, a^18b^23, 否], [8, 6, 9, a^17b^23, 否], [8, 7, 8, a^16b^23, 否], [8, 8, 7, a^15b^23, 否], [8, 9, 6, a^14b^23, 否], [8, 10, 5, a^13b^23, 否], [8, 11, 4, a^12b^23, 否], [8, 12, 3, a^11b^23, 否], [8, 13, 2, a^10b^23, 否], [8, 14, 1, a^9b^23, 否], [8, 15, 0, a^8b^23, 否], [9, 1, 13, a^22b^23, 否], [9, 2, 12, a^21b^23, 否], [9, 3, 11, a^20b^23, 否], [9, 4, 10, a^19b^23, 否], [9, 5, 9, a^18b^23, 否], [9, 6, 8, a^17b^23, 否], [9, 7, 7, a^16b^23, 否], [9, 8, 6, a^15b^23, 否], [9, 9, 5, a^14b^23, 否], [9, 10, 4, a^13b^23, 否], [9, 11, 3, a^12b^23, 否], [9, 12, 2, a^11b^23, 否], [9, 13, 1, a^10b^23, 否], [9, 14, 0, a^9b^23, 否], [10, 1, 12, a^22b^23, 否], [10, 2, 11, a^21b^23, 否], [10, 3, 10, a^20b^23, 否], [10, 4, 9, a^19b^23, 否], [10, 5, 8, a^18b^23, 否], [10, 6, 7, a^17b^23, 否], [10, 7, 6, a^16b^23, 否], [10, 8, 5, a^15b^23, 否], [10, 9, 4, a^14b^23, 否], [10, 10, 3, a^13b^23, 否], [10, 11, 2, a^12b^23, 否], [10, 12, 1, a^11b^23, 否], [10, 13, 0, a^10b^23, 否], [11, 1, 11, a^22b^23, 否], [11, 2, 10, a^21b^23, 否], [11, 3, 9, a^20b^23, 否], [11, 4, 8, a^19b^23, 否], [11, 5, 7, a^18b^23, 否], [11, 6, 6, a^17b^23, 否], [11, 7, 5, a^16b^23, 否], [11, 8, 4, a^15b^23, 否], [11, 9, 3, a^14b^23, 否], [11, 10, 2, a^13b^23, 否], [11, 11, 1, a^12b^23, 否], [11, 12, 0, a^11b^23, 否], [12, 1, 10, a^22b^23, 否], [12, 2, 9, a^21b^23, 否], [12, 3, 8, a^20b^23, 否], [12, 4, 7, a^19b^23, 否], [12, 5, 6, a^18b^23, 否], [12, 6, 5, a^17b^23, 否], [12, 7, 4, a^16b^23, 否], [12, 8, 3, a^15b^23, 否], [12, 9, 2, a^14b^23, 否], [12, 10, 1, a^13b^23, 否], [12, 11, 0, a^12b^23, 否], [13, 1, 9, a^22b^23, 否], [13, 2, 8, a^21b^23, 否], [13, 3, 7, a^20b^23, 否], [13, 4, 6, a^19b^23, 否], [13, 5, 5, a^18b^23, 否], [13, 6, 4, a^17b^23, 否], [13, 7, 3, a^16b^23, 否], [13, 8, 2, a^15b^23, 否], [13, 9, 1, a^14b^23, 否], [13, 10, 0, a^13b^23, 否], [14, 1, 8, a^22b^23, 否], [14, 2, 7, a^21b^23, 否], [14, 3, 6, a^20b^23, 否], [14, 4, 5, a^19b^23, 否], [14, 5, 4, a^18b^23, 否], [14, 6, 3, a^17b^23, 否], [14, 7, 2, a^16b^23, 否], [14, 8, 1, a^15b^23, 否], [14, 9, 0, a^14b^23, 否], [15, 1, 7, a^22b^23, 否], [15, 2, 6, a^21b^23, 否], [15, 3, 5, a^20b^23, 否], [15, 4, 4, a^19b^23, 否], [15, 5, 3, a^18b^23, 否], [15, 6, 2, a^17b^23, 否], [15, 7, 1, a^16b^23, 否], [15, 8, 0, a^15b^23, 否], [16, 1, 6, a^22b^23, 否], [16, 2, 5, a^21b^23, 否], [16, 3, 4, a^20b^23, 否], [16, 4, 3, a^19b^23, 否], [16, 5, 2, a^18b^23, 否], [16, 6, 1, a^17b^23, 否], [16, 7, 0, a^16b^23, 否], [17, 1, 5, a^22b^23, 否], [17, 2, 4, a^21b^23, 否], [17, 3, 3, a^20b^23, 否], [17, 4, 2, a^19b^23, 否], [17, 5, 1, a^18b^23, 否], [17, 6, 0, a^17b^23, 否], [18, 1, 4, a^22b^23, 否], [18, 2, 3, a^21b^23, 否], [18, 3, 2, a^20b^23, 否], [18, 4, 1, a^19b^23, 否], [18, 5, 0, a^18b^23, 否], [19, 1, 3, a^22b^23, 否], [19, 2, 2, a^21b^23, 否], [19, 3, 1, a^20b^23, 否], [19, 4, 0, a^19b^23, 否], [20, 1, 2, a^22b^23, 否], [20, 2, 1, a^21b^23, 否], [20, 3, 0, a^20b^23, 否], [21, 1, 1, a^22b^23, 否], [21, 2, 0, a^21b^23, 否], [22, 1, 0, a^22b^23, 否]], 276, 276, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(23, a^23b^23, [[0, 1, 22, a^22b^23, 否], [0, 2, 21, a^21b^23, 否], [0, 3, 20, a^20b^23, 否], [0, 4, 19, a^19b^23, 否], [0, 5, 18, a^18b^23, 否], [0, 6, 17, a^17b^23, 否], [0, 7, 16, a^16b^23, 否], [0, 8, 15, a^15b^23, 否], [0, 9, 14, a^14b^23, 否], [0, 10, 13, a^13b^23, 否], [0, 11, 12, a^12b^23, 否], [0, 12, 11, a^11b^23, 否], [0, 13, 10, a^10b^23, 否], [0, 14, 9, a^9b^23, 否], [0, 15, 8, a^8b^23, 否], [0, 16, 7, a^7b^23, 否], [0, 17, 6, a^6b^23, 否], [0, 18, 5, a^5b^23, 否], [0, 19, 4, a^4b^23, 否], [0, 20, 3, a^3b^23, 否], [0, 21, 2, a^2b^23, 否], [0, 22, 1, a^1b^23, 否], [0, 23, 0, a^0b^23, 否], [1, 1, 21, a^22b^23, 否], [1, 2, 20, a^21b^23, 否], [1, 3, 19, a^20b^23, 否], [1, 4, 18, a^19b^23, 否], [1, 5, 17, a^18b^23, 否], [1, 6, 16, a^17b^23, 否], [1, 7, 15, a^16b^23, 否], [1, 8, 14, a^15b^23, 否], [1, 9, 13, a^14b^23, 否], [1, 10, 12, a^13b^23, 否], [1, 11, 11, a^12b^23, 否], [1, 12, 10, a^11b^23, 否], [1, 13, 9, a^10b^23, 否], [1, 14, 8, a^9b^23, 否], [1, 15, 7, a^8b^23, 否], [1, 16, 6, a^7b^23, 否], [1, 17, 5, a^6b^23, 否], [1, 18, 4, a^5b^23, 否], [1, 19, 3, a^4b^23, 否], [1, 20, 2, a^3b^23, 否], [1, 21, 1, a^2b^23, 否], [1, 22, 0, a^1b^23, 否], [2, 1, 20, a^22b^23, 否], [2, 2, 19, a^21b^23, 否], [2, 3, 18, a^20b^23, 否], [2, 4, 17, a^19b^23, 否], [2, 5, 16, a^18b^23, 否], [2, 6, 15, a^17b^23, 否], [2, 7, 14, a^16b^23, 否], [2, 8, 13, a^15b^23, 否], [2, 9, 12, a^14b^23, 否], [2, 10, 11, a^13b^23, 否], [2, 11, 10, a^12b^23, 否], [2, 12, 9, a^11b^23, 否], [2, 13, 8, a^10b^23, 否], [2, 14, 7, a^9b^23, 否], [2, 15, 6, a^8b^23, 否], [2, 16, 5, a^7b^23, 否], [2, 17, 4, a^6b^23, 否], [2, 18, 3, a^5b^23, 否], [2, 19, 2, a^4b^23, 否], [2, 20, 1, a^3b^23, 否], [2, 21, 0, a^2b^23, 否], [3, 1, 19, a^22b^23, 否], [3, 2, 18, a^21b^23, 否], [3, 3, 17, a^20b^23, 否], [3, 4, 16, a^19b^23, 否], [3, 5, 15, a^18b^23, 否], [3, 6, 14, a^17b^23, 否], [3, 7, 13, a^16b^23, 否], [3, 8, 12, a^15b^23, 否], [3, 9, 11, a^14b^23, 否], [3, 10, 10, a^13b^23, 否], [3, 11, 9, a^12b^23, 否], [3, 12, 8, a^11b^23, 否], [3, 13, 7, a^10b^23, 否], [3, 14, 6, a^9b^23, 否], [3, 15, 5, a^8b^23, 否], [3, 16, 4, a^7b^23, 否], [3, 17, 3, a^6b^23, 否], [3, 18, 2, a^5b^23, 否], [3, 19, 1, a^4b^23, 否], [3, 20, 0, a^3b^23, 否], [4, 1, 18, a^22b^23, 否], [4, 2, 17, a^21b^23, 否], [4, 3, 16, a^20b^23, 否], [4, 4, 15, a^19b^23, 否], [4, 5, 14, a^18b^23, 否], [4, 6, 13, a^17b^23, 否], [4, 7, 12, a^16b^23, 否], [4, 8, 11, a^15b^23, 否], [4, 9, 10, a^14b^23, 否], [4, 10, 9, a^13b^23, 否], [4, 11, 8, a^12b^23, 否], [4, 12, 7, a^11b^23, 否], [4, 13, 6, a^10b^23, 否], [4, 14, 5, a^9b^23, 否], [4, 15, 4, a^8b^23, 否], [4, 16, 3, a^7b^23, 否], [4, 17, 2, a^6b^23, 否], [4, 18, 1, a^5b^23, 否], [4, 19, 0, a^4b^23, 否], [5, 1, 17, a^22b^23, 否], [5, 2, 16, a^21b^23, 否], [5, 3, 15, a^20b^23, 否], [5, 4, 14, a^19b^23, 否], [5, 5, 13, a^18b^23, 否], [5, 6, 12, a^17b^23, 否], [5, 7, 11, a^16b^23, 否], [5, 8, 10, a^15b^23, 否], [5, 9, 9, a^14b^23, 否], [5, 10, 8, a^13b^23, 否], [5, 11, 7, a^12b^23, 否], [5, 12, 6, a^11b^23, 否], [5, 13, 5, a^10b^23, 否], [5, 14, 4, a^9b^23, 否], [5, 15, 3, a^8b^23, 否], [5, 16, 2, a^7b^23, 否], [5, 17, 1, a^6b^23, 否], [5, 18, 0, a^5b^23, 否], [6, 1, 16, a^22b^23, 否], [6, 2, 15, a^21b^23, 否], [6, 3, 14, a^20b^23, 否], [6, 4, 13, a^19b^23, 否], [6, 5, 12, a^18b^23, 否], [6, 6, 11, a^17b^23, 否], [6, 7, 10, a^16b^23, 否], [6, 8, 9, a^15b^23, 否], [6, 9, 8, a^14b^23, 否], [6, 10, 7, a^13b^23, 否], [6, 11, 6, a^12b^23, 否], [6, 12, 5, a^11b^23, 否], [6, 13, 4, a^10b^23, 否], [6, 14, 3, a^9b^23, 否], [6, 15, 2, a^8b^23, 否], [6, 16, 1, a^7b^23, 否], [6, 17, 0, a^6b^23, 否], [7, 1, 15, a^22b^23, 否], [7, 2, 14, a^21b^23, 否], [7, 3, 13, a^20b^23, 否], [7, 4, 12, a^19b^23, 否], [7, 5, 11, a^18b^23, 否], [7, 6, 10, a^17b^23, 否], [7, 7, 9, a^16b^23, 否], [7, 8, 8, a^15b^23, 否], [7, 9, 7, a^14b^23, 否], [7, 10, 6, a^13b^23, 否], [7, 11, 5, a^12b^23, 否], [7, 12, 4, a^11b^23, 否], [7, 13, 3, a^10b^23, 否], [7, 14, 2, a^9b^23, 否], [7, 15, 1, a^8b^23, 否], [7, 16, 0, a^7b^23, 否], [8, 1, 14, a^22b^23, 否], [8, 2, 13, a^21b^23, 否], [8, 3, 12, a^20b^23, 否], [8, 4, 11, a^19b^23, 否], [8, 5, 10, a^18b^23, 否], [8, 6, 9, a^17b^23, 否], [8, 7, 8, a^16b^23, 否], [8, 8, 7, a^15b^23, 否], [8, 9, 6, a^14b^23, 否], [8, 10, 5, a^13b^23, 否], [8, 11, 4, a^12b^23, 否], [8, 12, 3, a^11b^23, 否], [8, 13, 2, a^10b^23, 否], [8, 14, 1, a^9b^23, 否], [8, 15, 0, a^8b^23, 否], [9, 1, 13, a^22b^23, 否], [9, 2, 12, a^21b^23, 否], [9, 3, 11, a^20b^23, 否], [9, 4, 10, a^19b^23, 否], [9, 5, 9, a^18b^23, 否], [9, 6, 8, a^17b^23, 否], [9, 7, 7, a^16b^23, 否], [9, 8, 6, a^15b^23, 否], [9, 9, 5, a^14b^23, 否], [9, 10, 4, a^13b^23, 否], [9, 11, 3, a^12b^23, 否], [9, 12, 2, a^11b^23, 否], [9, 13, 1, a^10b^23, 否], [9, 14, 0, a^9b^23, 否], [10, 1, 12, a^22b^23, 否], [10, 2, 11, a^21b^23, 否], [10, 3, 10, a^20b^23, 否], [10, 4, 9, a^19b^23, 否], [10, 5, 8, a^18b^23, 否], [10, 6, 7, a^17b^23, 否], [10, 7, 6, a^16b^23, 否], [10, 8, 5, a^15b^23, 否], [10, 9, 4, a^14b^23, 否], [10, 10, 3, a^13b^23, 否], [10, 11, 2, a^12b^23, 否], [10, 12, 1, a^11b^23, 否], [10, 13, 0, a^10b^23, 否], [11, 1, 11, a^22b^23, 否], [11, 2, 10, a^21b^23, 否], [11, 3, 9, a^20b^23, 否], [11, 4, 8, a^19b^23, 否], [11, 5, 7, a^18b^23, 否], [11, 6, 6, a^17b^23, 否], [11, 7, 5, a^16b^23, 否], [11, 8, 4, a^15b^23, 否], [11, 9, 3, a^14b^23, 否], [11, 10, 2, a^13b^23, 否], [11, 11, 1, a^12b^23, 否], [11, 12, 0, a^11b^23, 否], [12, 1, 10, a^22b^23, 否], [12, 2, 9, a^21b^23, 否], [12, 3, 8, a^20b^23, 否], [12, 4, 7, a^19b^23, 否], [12, 5, 6, a^18b^23, 否], [12, 6, 5, a^17b^23, 否], [12, 7, 4, a^16b^23, 否], [12, 8, 3, a^15b^23, 否], [12, 9, 2, a^14b^23, 否], [12, 10, 1, a^13b^23, 否], [12, 11, 0, a^12b^23, 否], [13, 1, 9, a^22b^23, 否], [13, 2, 8, a^21b^23, 否], [13, 3, 7, a^20b^23, 否], [13, 4, 6, a^19b^23, 否], [13, 5, 5, a^18b^23, 否], [13, 6, 4, a^17b^23, 否], [13, 7, 3, a^16b^23, 否], [13, 8, 2, a^15b^23, 否], [13, 9, 1, a^14b^23, 否], [13, 10, 0, a^13b^23, 否], [14, 1, 8, a^22b^23, 否], [14, 2, 7, a^21b^23, 否], [14, 3, 6, a^20b^23, 否], [14, 4, 5, a^19b^23, 否], [14, 5, 4, a^18b^23, 否], [14, 6, 3, a^17b^23, 否], [14, 7, 2, a^16b^23, 否], [14, 8, 1, a^15b^23, 否], [14, 9, 0, a^14b^23, 否], [15, 1, 7, a^22b^23, 否], [15, 2, 6, a^21b^23, 否], [15, 3, 5, a^20b^23, 否], [15, 4, 4, a^19b^23, 否], [15, 5, 3, a^18b^23, 否], [15, 6, 2, a^17b^23, 否], [15, 7, 1, a^16b^23, 否], [15, 8, 0, a^15b^23, 否], [16, 1, 6, a^22b^23, 否], [16, 2, 5, a^21b^23, 否], [16, 3, 4, a^20b^23, 否], [16, 4, 3, a^19b^23, 否], [16, 5, 2, a^18b^23, 否], [16, 6, 1, a^17b^23, 否], [16, 7, 0, a^16b^23, 否], [17, 1, 5, a^22b^23, 否], [17, 2, 4, a^21b^23, 否], [17, 3, 3, a^20b^23, 否], [17, 4, 2, a^19b^23, 否], [17, 5, 1, a^18b^23, 否], [17, 6, 0, a^17b^23, 否], [18, 1, 4, a^22b^23, 否], [18, 2, 3, a^21b^23, 否], [18, 3, 2, a^20b^23, 否], [18, 4, 1, a^19b^23, 否], [18, 5, 0, a^18b^23, 否], [19, 1, 3, a^22b^23, 否], [19, 2, 2, a^21b^23, 否], [19, 3, 1, a^20b^23, 否], [19, 4, 0, a^19b^23, 否], [20, 1, 2, a^22b^23, 否], [20, 2, 1, a^21b^23, 否], [20, 3, 0, a^20b^23, 否], [21, 1, 1, a^22b^23, 否], [21, 2, 0, a^21b^23, 否], [22, 1, 0, a^22b^23, 否]], 276, 276, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

74. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":24}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(24, a^24b^24, [[0, 1, 23, a^23b^24, 否], [0, 2, 22, a^22b^24, 否], [0, 3, 21, a^21b^24, 否], [0, 4, 20, a^20b^24, 否], [0, 5, 19, a^19b^24, 否], [0, 6, 18, a^18b^24, 否], [0, 7, 17, a^17b^24, 否], [0, 8, 16, a^16b^24, 否], [0, 9, 15, a^15b^24, 否], [0, 10, 14, a^14b^24, 否], [0, 11, 13, a^13b^24, 否], [0, 12, 12, a^12b^24, 否], [0, 13, 11, a^11b^24, 否], [0, 14, 10, a^10b^24, 否], [0, 15, 9, a^9b^24, 否], [0, 16, 8, a^8b^24, 否], [0, 17, 7, a^7b^24, 否], [0, 18, 6, a^6b^24, 否], [0, 19, 5, a^5b^24, 否], [0, 20, 4, a^4b^24, 否], [0, 21, 3, a^3b^24, 否], [0, 22, 2, a^2b^24, 否], [0, 23, 1, a^1b^24, 否], [0, 24, 0, a^0b^24, 否], [1, 1, 22, a^23b^24, 否], [1, 2, 21, a^22b^24, 否], [1, 3, 20, a^21b^24, 否], [1, 4, 19, a^20b^24, 否], [1, 5, 18, a^19b^24, 否], [1, 6, 17, a^18b^24, 否], [1, 7, 16, a^17b^24, 否], [1, 8, 15, a^16b^24, 否], [1, 9, 14, a^15b^24, 否], [1, 10, 13, a^14b^24, 否], [1, 11, 12, a^13b^24, 否], [1, 12, 11, a^12b^24, 否], [1, 13, 10, a^11b^24, 否], [1, 14, 9, a^10b^24, 否], [1, 15, 8, a^9b^24, 否], [1, 16, 7, a^8b^24, 否], [1, 17, 6, a^7b^24, 否], [1, 18, 5, a^6b^24, 否], [1, 19, 4, a^5b^24, 否], [1, 20, 3, a^4b^24, 否], [1, 21, 2, a^3b^24, 否], [1, 22, 1, a^2b^24, 否], [1, 23, 0, a^1b^24, 否], [2, 1, 21, a^23b^24, 否], [2, 2, 20, a^22b^24, 否], [2, 3, 19, a^21b^24, 否], [2, 4, 18, a^20b^24, 否], [2, 5, 17, a^19b^24, 否], [2, 6, 16, a^18b^24, 否], [2, 7, 15, a^17b^24, 否], [2, 8, 14, a^16b^24, 否], [2, 9, 13, a^15b^24, 否], [2, 10, 12, a^14b^24, 否], [2, 11, 11, a^13b^24, 否], [2, 12, 10, a^12b^24, 否], [2, 13, 9, a^11b^24, 否], [2, 14, 8, a^10b^24, 否], [2, 15, 7, a^9b^24, 否], [2, 16, 6, a^8b^24, 否], [2, 17, 5, a^7b^24, 否], [2, 18, 4, a^6b^24, 否], [2, 19, 3, a^5b^24, 否], [2, 20, 2, a^4b^24, 否], [2, 21, 1, a^3b^24, 否], [2, 22, 0, a^2b^24, 否], [3, 1, 20, a^23b^24, 否], [3, 2, 19, a^22b^24, 否], [3, 3, 18, a^21b^24, 否], [3, 4, 17, a^20b^24, 否], [3, 5, 16, a^19b^24, 否], [3, 6, 15, a^18b^24, 否], [3, 7, 14, a^17b^24, 否], [3, 8, 13, a^16b^24, 否], [3, 9, 12, a^15b^24, 否], [3, 10, 11, a^14b^24, 否], [3, 11, 10, a^13b^24, 否], [3, 12, 9, a^12b^24, 否], [3, 13, 8, a^11b^24, 否], [3, 14, 7, a^10b^24, 否], [3, 15, 6, a^9b^24, 否], [3, 16, 5, a^8b^24, 否], [3, 17, 4, a^7b^24, 否], [3, 18, 3, a^6b^24, 否], [3, 19, 2, a^5b^24, 否], [3, 20, 1, a^4b^24, 否], [3, 21, 0, a^3b^24, 否], [4, 1, 19, a^23b^24, 否], [4, 2, 18, a^22b^24, 否], [4, 3, 17, a^21b^24, 否], [4, 4, 16, a^20b^24, 否], [4, 5, 15, a^19b^24, 否], [4, 6, 14, a^18b^24, 否], [4, 7, 13, a^17b^24, 否], [4, 8, 12, a^16b^24, 否], [4, 9, 11, a^15b^24, 否], [4, 10, 10, a^14b^24, 否], [4, 11, 9, a^13b^24, 否], [4, 12, 8, a^12b^24, 否], [4, 13, 7, a^11b^24, 否], [4, 14, 6, a^10b^24, 否], [4, 15, 5, a^9b^24, 否], [4, 16, 4, a^8b^24, 否], [4, 17, 3, a^7b^24, 否], [4, 18, 2, a^6b^24, 否], [4, 19, 1, a^5b^24, 否], [4, 20, 0, a^4b^24, 否], [5, 1, 18, a^23b^24, 否], [5, 2, 17, a^22b^24, 否], [5, 3, 16, a^21b^24, 否], [5, 4, 15, a^20b^24, 否], [5, 5, 14, a^19b^24, 否], [5, 6, 13, a^18b^24, 否], [5, 7, 12, a^17b^24, 否], [5, 8, 11, a^16b^24, 否], [5, 9, 10, a^15b^24, 否], [5, 10, 9, a^14b^24, 否], [5, 11, 8, a^13b^24, 否], [5, 12, 7, a^12b^24, 否], [5, 13, 6, a^11b^24, 否], [5, 14, 5, a^10b^24, 否], [5, 15, 4, a^9b^24, 否], [5, 16, 3, a^8b^24, 否], [5, 17, 2, a^7b^24, 否], [5, 18, 1, a^6b^24, 否], [5, 19, 0, a^5b^24, 否], [6, 1, 17, a^23b^24, 否], [6, 2, 16, a^22b^24, 否], [6, 3, 15, a^21b^24, 否], [6, 4, 14, a^20b^24, 否], [6, 5, 13, a^19b^24, 否], [6, 6, 12, a^18b^24, 否], [6, 7, 11, a^17b^24, 否], [6, 8, 10, a^16b^24, 否], [6, 9, 9, a^15b^24, 否], [6, 10, 8, a^14b^24, 否], [6, 11, 7, a^13b^24, 否], [6, 12, 6, a^12b^24, 否], [6, 13, 5, a^11b^24, 否], [6, 14, 4, a^10b^24, 否], [6, 15, 3, a^9b^24, 否], [6, 16, 2, a^8b^24, 否], [6, 17, 1, a^7b^24, 否], [6, 18, 0, a^6b^24, 否], [7, 1, 16, a^23b^24, 否], [7, 2, 15, a^22b^24, 否], [7, 3, 14, a^21b^24, 否], [7, 4, 13, a^20b^24, 否], [7, 5, 12, a^19b^24, 否], [7, 6, 11, a^18b^24, 否], [7, 7, 10, a^17b^24, 否], [7, 8, 9, a^16b^24, 否], [7, 9, 8, a^15b^24, 否], [7, 10, 7, a^14b^24, 否], [7, 11, 6, a^13b^24, 否], [7, 12, 5, a^12b^24, 否], [7, 13, 4, a^11b^24, 否], [7, 14, 3, a^10b^24, 否], [7, 15, 2, a^9b^24, 否], [7, 16, 1, a^8b^24, 否], [7, 17, 0, a^7b^24, 否], [8, 1, 15, a^23b^24, 否], [8, 2, 14, a^22b^24, 否], [8, 3, 13, a^21b^24, 否], [8, 4, 12, a^20b^24, 否], [8, 5, 11, a^19b^24, 否], [8, 6, 10, a^18b^24, 否], [8, 7, 9, a^17b^24, 否], [8, 8, 8, a^16b^24, 否], [8, 9, 7, a^15b^24, 否], [8, 10, 6, a^14b^24, 否], [8, 11, 5, a^13b^24, 否], [8, 12, 4, a^12b^24, 否], [8, 13, 3, a^11b^24, 否], [8, 14, 2, a^10b^24, 否], [8, 15, 1, a^9b^24, 否], [8, 16, 0, a^8b^24, 否], [9, 1, 14, a^23b^24, 否], [9, 2, 13, a^22b^24, 否], [9, 3, 12, a^21b^24, 否], [9, 4, 11, a^20b^24, 否], [9, 5, 10, a^19b^24, 否], [9, 6, 9, a^18b^24, 否], [9, 7, 8, a^17b^24, 否], [9, 8, 7, a^16b^24, 否], [9, 9, 6, a^15b^24, 否], [9, 10, 5, a^14b^24, 否], [9, 11, 4, a^13b^24, 否], [9, 12, 3, a^12b^24, 否], [9, 13, 2, a^11b^24, 否], [9, 14, 1, a^10b^24, 否], [9, 15, 0, a^9b^24, 否], [10, 1, 13, a^23b^24, 否], [10, 2, 12, a^22b^24, 否], [10, 3, 11, a^21b^24, 否], [10, 4, 10, a^20b^24, 否], [10, 5, 9, a^19b^24, 否], [10, 6, 8, a^18b^24, 否], [10, 7, 7, a^17b^24, 否], [10, 8, 6, a^16b^24, 否], [10, 9, 5, a^15b^24, 否], [10, 10, 4, a^14b^24, 否], [10, 11, 3, a^13b^24, 否], [10, 12, 2, a^12b^24, 否], [10, 13, 1, a^11b^24, 否], [10, 14, 0, a^10b^24, 否], [11, 1, 12, a^23b^24, 否], [11, 2, 11, a^22b^24, 否], [11, 3, 10, a^21b^24, 否], [11, 4, 9, a^20b^24, 否], [11, 5, 8, a^19b^24, 否], [11, 6, 7, a^18b^24, 否], [11, 7, 6, a^17b^24, 否], [11, 8, 5, a^16b^24, 否], [11, 9, 4, a^15b^24, 否], [11, 10, 3, a^14b^24, 否], [11, 11, 2, a^13b^24, 否], [11, 12, 1, a^12b^24, 否], [11, 13, 0, a^11b^24, 否], [12, 1, 11, a^23b^24, 否], [12, 2, 10, a^22b^24, 否], [12, 3, 9, a^21b^24, 否], [12, 4, 8, a^20b^24, 否], [12, 5, 7, a^19b^24, 否], [12, 6, 6, a^18b^24, 否], [12, 7, 5, a^17b^24, 否], [12, 8, 4, a^16b^24, 否], [12, 9, 3, a^15b^24, 否], [12, 10, 2, a^14b^24, 否], [12, 11, 1, a^13b^24, 否], [12, 12, 0, a^12b^24, 否], [13, 1, 10, a^23b^24, 否], [13, 2, 9, a^22b^24, 否], [13, 3, 8, a^21b^24, 否], [13, 4, 7, a^20b^24, 否], [13, 5, 6, a^19b^24, 否], [13, 6, 5, a^18b^24, 否], [13, 7, 4, a^17b^24, 否], [13, 8, 3, a^16b^24, 否], [13, 9, 2, a^15b^24, 否], [13, 10, 1, a^14b^24, 否], [13, 11, 0, a^13b^24, 否], [14, 1, 9, a^23b^24, 否], [14, 2, 8, a^22b^24, 否], [14, 3, 7, a^21b^24, 否], [14, 4, 6, a^20b^24, 否], [14, 5, 5, a^19b^24, 否], [14, 6, 4, a^18b^24, 否], [14, 7, 3, a^17b^24, 否], [14, 8, 2, a^16b^24, 否], [14, 9, 1, a^15b^24, 否], [14, 10, 0, a^14b^24, 否], [15, 1, 8, a^23b^24, 否], [15, 2, 7, a^22b^24, 否], [15, 3, 6, a^21b^24, 否], [15, 4, 5, a^20b^24, 否], [15, 5, 4, a^19b^24, 否], [15, 6, 3, a^18b^24, 否], [15, 7, 2, a^17b^24, 否], [15, 8, 1, a^16b^24, 否], [15, 9, 0, a^15b^24, 否], [16, 1, 7, a^23b^24, 否], [16, 2, 6, a^22b^24, 否], [16, 3, 5, a^21b^24, 否], [16, 4, 4, a^20b^24, 否], [16, 5, 3, a^19b^24, 否], [16, 6, 2, a^18b^24, 否], [16, 7, 1, a^17b^24, 否], [16, 8, 0, a^16b^24, 否], [17, 1, 6, a^23b^24, 否], [17, 2, 5, a^22b^24, 否], [17, 3, 4, a^21b^24, 否], [17, 4, 3, a^20b^24, 否], [17, 5, 2, a^19b^24, 否], [17, 6, 1, a^18b^24, 否], [17, 7, 0, a^17b^24, 否], [18, 1, 5, a^23b^24, 否], [18, 2, 4, a^22b^24, 否], [18, 3, 3, a^21b^24, 否], [18, 4, 2, a^20b^24, 否], [18, 5, 1, a^19b^24, 否], [18, 6, 0, a^18b^24, 否], [19, 1, 4, a^23b^24, 否], [19, 2, 3, a^22b^24, 否], [19, 3, 2, a^21b^24, 否], [19, 4, 1, a^20b^24, 否], [19, 5, 0, a^19b^24, 否], [20, 1, 3, a^23b^24, 否], [20, 2, 2, a^22b^24, 否], [20, 3, 1, a^21b^24, 否], [20, 4, 0, a^20b^24, 否], [21, 1, 2, a^23b^24, 否], [21, 2, 1, a^22b^24, 否], [21, 3, 0, a^21b^24, 否], [22, 1, 1, a^23b^24, 否], [22, 2, 0, a^22b^24, 否], [23, 1, 0, a^23b^24, 否]], 300, 300, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(24, a^24b^24, [[0, 1, 23, a^23b^24, 否], [0, 2, 22, a^22b^24, 否], [0, 3, 21, a^21b^24, 否], [0, 4, 20, a^20b^24, 否], [0, 5, 19, a^19b^24, 否], [0, 6, 18, a^18b^24, 否], [0, 7, 17, a^17b^24, 否], [0, 8, 16, a^16b^24, 否], [0, 9, 15, a^15b^24, 否], [0, 10, 14, a^14b^24, 否], [0, 11, 13, a^13b^24, 否], [0, 12, 12, a^12b^24, 否], [0, 13, 11, a^11b^24, 否], [0, 14, 10, a^10b^24, 否], [0, 15, 9, a^9b^24, 否], [0, 16, 8, a^8b^24, 否], [0, 17, 7, a^7b^24, 否], [0, 18, 6, a^6b^24, 否], [0, 19, 5, a^5b^24, 否], [0, 20, 4, a^4b^24, 否], [0, 21, 3, a^3b^24, 否], [0, 22, 2, a^2b^24, 否], [0, 23, 1, a^1b^24, 否], [0, 24, 0, a^0b^24, 否], [1, 1, 22, a^23b^24, 否], [1, 2, 21, a^22b^24, 否], [1, 3, 20, a^21b^24, 否], [1, 4, 19, a^20b^24, 否], [1, 5, 18, a^19b^24, 否], [1, 6, 17, a^18b^24, 否], [1, 7, 16, a^17b^24, 否], [1, 8, 15, a^16b^24, 否], [1, 9, 14, a^15b^24, 否], [1, 10, 13, a^14b^24, 否], [1, 11, 12, a^13b^24, 否], [1, 12, 11, a^12b^24, 否], [1, 13, 10, a^11b^24, 否], [1, 14, 9, a^10b^24, 否], [1, 15, 8, a^9b^24, 否], [1, 16, 7, a^8b^24, 否], [1, 17, 6, a^7b^24, 否], [1, 18, 5, a^6b^24, 否], [1, 19, 4, a^5b^24, 否], [1, 20, 3, a^4b^24, 否], [1, 21, 2, a^3b^24, 否], [1, 22, 1, a^2b^24, 否], [1, 23, 0, a^1b^24, 否], [2, 1, 21, a^23b^24, 否], [2, 2, 20, a^22b^24, 否], [2, 3, 19, a^21b^24, 否], [2, 4, 18, a^20b^24, 否], [2, 5, 17, a^19b^24, 否], [2, 6, 16, a^18b^24, 否], [2, 7, 15, a^17b^24, 否], [2, 8, 14, a^16b^24, 否], [2, 9, 13, a^15b^24, 否], [2, 10, 12, a^14b^24, 否], [2, 11, 11, a^13b^24, 否], [2, 12, 10, a^12b^24, 否], [2, 13, 9, a^11b^24, 否], [2, 14, 8, a^10b^24, 否], [2, 15, 7, a^9b^24, 否], [2, 16, 6, a^8b^24, 否], [2, 17, 5, a^7b^24, 否], [2, 18, 4, a^6b^24, 否], [2, 19, 3, a^5b^24, 否], [2, 20, 2, a^4b^24, 否], [2, 21, 1, a^3b^24, 否], [2, 22, 0, a^2b^24, 否], [3, 1, 20, a^23b^24, 否], [3, 2, 19, a^22b^24, 否], [3, 3, 18, a^21b^24, 否], [3, 4, 17, a^20b^24, 否], [3, 5, 16, a^19b^24, 否], [3, 6, 15, a^18b^24, 否], [3, 7, 14, a^17b^24, 否], [3, 8, 13, a^16b^24, 否], [3, 9, 12, a^15b^24, 否], [3, 10, 11, a^14b^24, 否], [3, 11, 10, a^13b^24, 否], [3, 12, 9, a^12b^24, 否], [3, 13, 8, a^11b^24, 否], [3, 14, 7, a^10b^24, 否], [3, 15, 6, a^9b^24, 否], [3, 16, 5, a^8b^24, 否], [3, 17, 4, a^7b^24, 否], [3, 18, 3, a^6b^24, 否], [3, 19, 2, a^5b^24, 否], [3, 20, 1, a^4b^24, 否], [3, 21, 0, a^3b^24, 否], [4, 1, 19, a^23b^24, 否], [4, 2, 18, a^22b^24, 否], [4, 3, 17, a^21b^24, 否], [4, 4, 16, a^20b^24, 否], [4, 5, 15, a^19b^24, 否], [4, 6, 14, a^18b^24, 否], [4, 7, 13, a^17b^24, 否], [4, 8, 12, a^16b^24, 否], [4, 9, 11, a^15b^24, 否], [4, 10, 10, a^14b^24, 否], [4, 11, 9, a^13b^24, 否], [4, 12, 8, a^12b^24, 否], [4, 13, 7, a^11b^24, 否], [4, 14, 6, a^10b^24, 否], [4, 15, 5, a^9b^24, 否], [4, 16, 4, a^8b^24, 否], [4, 17, 3, a^7b^24, 否], [4, 18, 2, a^6b^24, 否], [4, 19, 1, a^5b^24, 否], [4, 20, 0, a^4b^24, 否], [5, 1, 18, a^23b^24, 否], [5, 2, 17, a^22b^24, 否], [5, 3, 16, a^21b^24, 否], [5, 4, 15, a^20b^24, 否], [5, 5, 14, a^19b^24, 否], [5, 6, 13, a^18b^24, 否], [5, 7, 12, a^17b^24, 否], [5, 8, 11, a^16b^24, 否], [5, 9, 10, a^15b^24, 否], [5, 10, 9, a^14b^24, 否], [5, 11, 8, a^13b^24, 否], [5, 12, 7, a^12b^24, 否], [5, 13, 6, a^11b^24, 否], [5, 14, 5, a^10b^24, 否], [5, 15, 4, a^9b^24, 否], [5, 16, 3, a^8b^24, 否], [5, 17, 2, a^7b^24, 否], [5, 18, 1, a^6b^24, 否], [5, 19, 0, a^5b^24, 否], [6, 1, 17, a^23b^24, 否], [6, 2, 16, a^22b^24, 否], [6, 3, 15, a^21b^24, 否], [6, 4, 14, a^20b^24, 否], [6, 5, 13, a^19b^24, 否], [6, 6, 12, a^18b^24, 否], [6, 7, 11, a^17b^24, 否], [6, 8, 10, a^16b^24, 否], [6, 9, 9, a^15b^24, 否], [6, 10, 8, a^14b^24, 否], [6, 11, 7, a^13b^24, 否], [6, 12, 6, a^12b^24, 否], [6, 13, 5, a^11b^24, 否], [6, 14, 4, a^10b^24, 否], [6, 15, 3, a^9b^24, 否], [6, 16, 2, a^8b^24, 否], [6, 17, 1, a^7b^24, 否], [6, 18, 0, a^6b^24, 否], [7, 1, 16, a^23b^24, 否], [7, 2, 15, a^22b^24, 否], [7, 3, 14, a^21b^24, 否], [7, 4, 13, a^20b^24, 否], [7, 5, 12, a^19b^24, 否], [7, 6, 11, a^18b^24, 否], [7, 7, 10, a^17b^24, 否], [7, 8, 9, a^16b^24, 否], [7, 9, 8, a^15b^24, 否], [7, 10, 7, a^14b^24, 否], [7, 11, 6, a^13b^24, 否], [7, 12, 5, a^12b^24, 否], [7, 13, 4, a^11b^24, 否], [7, 14, 3, a^10b^24, 否], [7, 15, 2, a^9b^24, 否], [7, 16, 1, a^8b^24, 否], [7, 17, 0, a^7b^24, 否], [8, 1, 15, a^23b^24, 否], [8, 2, 14, a^22b^24, 否], [8, 3, 13, a^21b^24, 否], [8, 4, 12, a^20b^24, 否], [8, 5, 11, a^19b^24, 否], [8, 6, 10, a^18b^24, 否], [8, 7, 9, a^17b^24, 否], [8, 8, 8, a^16b^24, 否], [8, 9, 7, a^15b^24, 否], [8, 10, 6, a^14b^24, 否], [8, 11, 5, a^13b^24, 否], [8, 12, 4, a^12b^24, 否], [8, 13, 3, a^11b^24, 否], [8, 14, 2, a^10b^24, 否], [8, 15, 1, a^9b^24, 否], [8, 16, 0, a^8b^24, 否], [9, 1, 14, a^23b^24, 否], [9, 2, 13, a^22b^24, 否], [9, 3, 12, a^21b^24, 否], [9, 4, 11, a^20b^24, 否], [9, 5, 10, a^19b^24, 否], [9, 6, 9, a^18b^24, 否], [9, 7, 8, a^17b^24, 否], [9, 8, 7, a^16b^24, 否], [9, 9, 6, a^15b^24, 否], [9, 10, 5, a^14b^24, 否], [9, 11, 4, a^13b^24, 否], [9, 12, 3, a^12b^24, 否], [9, 13, 2, a^11b^24, 否], [9, 14, 1, a^10b^24, 否], [9, 15, 0, a^9b^24, 否], [10, 1, 13, a^23b^24, 否], [10, 2, 12, a^22b^24, 否], [10, 3, 11, a^21b^24, 否], [10, 4, 10, a^20b^24, 否], [10, 5, 9, a^19b^24, 否], [10, 6, 8, a^18b^24, 否], [10, 7, 7, a^17b^24, 否], [10, 8, 6, a^16b^24, 否], [10, 9, 5, a^15b^24, 否], [10, 10, 4, a^14b^24, 否], [10, 11, 3, a^13b^24, 否], [10, 12, 2, a^12b^24, 否], [10, 13, 1, a^11b^24, 否], [10, 14, 0, a^10b^24, 否], [11, 1, 12, a^23b^24, 否], [11, 2, 11, a^22b^24, 否], [11, 3, 10, a^21b^24, 否], [11, 4, 9, a^20b^24, 否], [11, 5, 8, a^19b^24, 否], [11, 6, 7, a^18b^24, 否], [11, 7, 6, a^17b^24, 否], [11, 8, 5, a^16b^24, 否], [11, 9, 4, a^15b^24, 否], [11, 10, 3, a^14b^24, 否], [11, 11, 2, a^13b^24, 否], [11, 12, 1, a^12b^24, 否], [11, 13, 0, a^11b^24, 否], [12, 1, 11, a^23b^24, 否], [12, 2, 10, a^22b^24, 否], [12, 3, 9, a^21b^24, 否], [12, 4, 8, a^20b^24, 否], [12, 5, 7, a^19b^24, 否], [12, 6, 6, a^18b^24, 否], [12, 7, 5, a^17b^24, 否], [12, 8, 4, a^16b^24, 否], [12, 9, 3, a^15b^24, 否], [12, 10, 2, a^14b^24, 否], [12, 11, 1, a^13b^24, 否], [12, 12, 0, a^12b^24, 否], [13, 1, 10, a^23b^24, 否], [13, 2, 9, a^22b^24, 否], [13, 3, 8, a^21b^24, 否], [13, 4, 7, a^20b^24, 否], [13, 5, 6, a^19b^24, 否], [13, 6, 5, a^18b^24, 否], [13, 7, 4, a^17b^24, 否], [13, 8, 3, a^16b^24, 否], [13, 9, 2, a^15b^24, 否], [13, 10, 1, a^14b^24, 否], [13, 11, 0, a^13b^24, 否], [14, 1, 9, a^23b^24, 否], [14, 2, 8, a^22b^24, 否], [14, 3, 7, a^21b^24, 否], [14, 4, 6, a^20b^24, 否], [14, 5, 5, a^19b^24, 否], [14, 6, 4, a^18b^24, 否], [14, 7, 3, a^17b^24, 否], [14, 8, 2, a^16b^24, 否], [14, 9, 1, a^15b^24, 否], [14, 10, 0, a^14b^24, 否], [15, 1, 8, a^23b^24, 否], [15, 2, 7, a^22b^24, 否], [15, 3, 6, a^21b^24, 否], [15, 4, 5, a^20b^24, 否], [15, 5, 4, a^19b^24, 否], [15, 6, 3, a^18b^24, 否], [15, 7, 2, a^17b^24, 否], [15, 8, 1, a^16b^24, 否], [15, 9, 0, a^15b^24, 否], [16, 1, 7, a^23b^24, 否], [16, 2, 6, a^22b^24, 否], [16, 3, 5, a^21b^24, 否], [16, 4, 4, a^20b^24, 否], [16, 5, 3, a^19b^24, 否], [16, 6, 2, a^18b^24, 否], [16, 7, 1, a^17b^24, 否], [16, 8, 0, a^16b^24, 否], [17, 1, 6, a^23b^24, 否], [17, 2, 5, a^22b^24, 否], [17, 3, 4, a^21b^24, 否], [17, 4, 3, a^20b^24, 否], [17, 5, 2, a^19b^24, 否], [17, 6, 1, a^18b^24, 否], [17, 7, 0, a^17b^24, 否], [18, 1, 5, a^23b^24, 否], [18, 2, 4, a^22b^24, 否], [18, 3, 3, a^21b^24, 否], [18, 4, 2, a^20b^24, 否], [18, 5, 1, a^19b^24, 否], [18, 6, 0, a^18b^24, 否], [19, 1, 4, a^23b^24, 否], [19, 2, 3, a^22b^24, 否], [19, 3, 2, a^21b^24, 否], [19, 4, 1, a^20b^24, 否], [19, 5, 0, a^19b^24, 否], [20, 1, 3, a^23b^24, 否], [20, 2, 2, a^22b^24, 否], [20, 3, 1, a^21b^24, 否], [20, 4, 0, a^20b^24, 否], [21, 1, 2, a^23b^24, 否], [21, 2, 1, a^22b^24, 否], [21, 3, 0, a^21b^24, 否], [22, 1, 1, a^23b^24, 否], [22, 2, 0, a^22b^24, 否], [23, 1, 0, a^23b^24, 否]], 300, 300, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

75. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={"p":25}。請依 [p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope] 欄序回答。

split row依序是|x|、|y|、剩餘a數p−|x|−|y|、i=0後word、是否仍在a^n b^n；第三欄不是完整|z|，z尚含全部p個b。

查看答案與解釋

(25, a^25b^25, [[0, 1, 24, a^24b^25, 否], [0, 2, 23, a^23b^25, 否], [0, 3, 22, a^22b^25, 否], [0, 4, 21, a^21b^25, 否], [0, 5, 20, a^20b^25, 否], [0, 6, 19, a^19b^25, 否], [0, 7, 18, a^18b^25, 否], [0, 8, 17, a^17b^25, 否], [0, 9, 16, a^16b^25, 否], [0, 10, 15, a^15b^25, 否], [0, 11, 14, a^14b^25, 否], [0, 12, 13, a^13b^25, 否], [0, 13, 12, a^12b^25, 否], [0, 14, 11, a^11b^25, 否], [0, 15, 10, a^10b^25, 否], [0, 16, 9, a^9b^25, 否], [0, 17, 8, a^8b^25, 否], [0, 18, 7, a^7b^25, 否], [0, 19, 6, a^6b^25, 否], [0, 20, 5, a^5b^25, 否], [0, 21, 4, a^4b^25, 否], [0, 22, 3, a^3b^25, 否], [0, 23, 2, a^2b^25, 否], [0, 24, 1, a^1b^25, 否], [0, 25, 0, a^0b^25, 否], [1, 1, 23, a^24b^25, 否], [1, 2, 22, a^23b^25, 否], [1, 3, 21, a^22b^25, 否], [1, 4, 20, a^21b^25, 否], [1, 5, 19, a^20b^25, 否], [1, 6, 18, a^19b^25, 否], [1, 7, 17, a^18b^25, 否], [1, 8, 16, a^17b^25, 否], [1, 9, 15, a^16b^25, 否], [1, 10, 14, a^15b^25, 否], [1, 11, 13, a^14b^25, 否], [1, 12, 12, a^13b^25, 否], [1, 13, 11, a^12b^25, 否], [1, 14, 10, a^11b^25, 否], [1, 15, 9, a^10b^25, 否], [1, 16, 8, a^9b^25, 否], [1, 17, 7, a^8b^25, 否], [1, 18, 6, a^7b^25, 否], [1, 19, 5, a^6b^25, 否], [1, 20, 4, a^5b^25, 否], [1, 21, 3, a^4b^25, 否], [1, 22, 2, a^3b^25, 否], [1, 23, 1, a^2b^25, 否], [1, 24, 0, a^1b^25, 否], [2, 1, 22, a^24b^25, 否], [2, 2, 21, a^23b^25, 否], [2, 3, 20, a^22b^25, 否], [2, 4, 19, a^21b^25, 否], [2, 5, 18, a^20b^25, 否], [2, 6, 17, a^19b^25, 否], [2, 7, 16, a^18b^25, 否], [2, 8, 15, a^17b^25, 否], [2, 9, 14, a^16b^25, 否], [2, 10, 13, a^15b^25, 否], [2, 11, 12, a^14b^25, 否], [2, 12, 11, a^13b^25, 否], [2, 13, 10, a^12b^25, 否], [2, 14, 9, a^11b^25, 否], [2, 15, 8, a^10b^25, 否], [2, 16, 7, a^9b^25, 否], [2, 17, 6, a^8b^25, 否], [2, 18, 5, a^7b^25, 否], [2, 19, 4, a^6b^25, 否], [2, 20, 3, a^5b^25, 否], [2, 21, 2, a^4b^25, 否], [2, 22, 1, a^3b^25, 否], [2, 23, 0, a^2b^25, 否], [3, 1, 21, a^24b^25, 否], [3, 2, 20, a^23b^25, 否], [3, 3, 19, a^22b^25, 否], [3, 4, 18, a^21b^25, 否], [3, 5, 17, a^20b^25, 否], [3, 6, 16, a^19b^25, 否], [3, 7, 15, a^18b^25, 否], [3, 8, 14, a^17b^25, 否], [3, 9, 13, a^16b^25, 否], [3, 10, 12, a^15b^25, 否], [3, 11, 11, a^14b^25, 否], [3, 12, 10, a^13b^25, 否], [3, 13, 9, a^12b^25, 否], [3, 14, 8, a^11b^25, 否], [3, 15, 7, a^10b^25, 否], [3, 16, 6, a^9b^25, 否], [3, 17, 5, a^8b^25, 否], [3, 18, 4, a^7b^25, 否], [3, 19, 3, a^6b^25, 否], [3, 20, 2, a^5b^25, 否], [3, 21, 1, a^4b^25, 否], [3, 22, 0, a^3b^25, 否], [4, 1, 20, a^24b^25, 否], [4, 2, 19, a^23b^25, 否], [4, 3, 18, a^22b^25, 否], [4, 4, 17, a^21b^25, 否], [4, 5, 16, a^20b^25, 否], [4, 6, 15, a^19b^25, 否], [4, 7, 14, a^18b^25, 否], [4, 8, 13, a^17b^25, 否], [4, 9, 12, a^16b^25, 否], [4, 10, 11, a^15b^25, 否], [4, 11, 10, a^14b^25, 否], [4, 12, 9, a^13b^25, 否], [4, 13, 8, a^12b^25, 否], [4, 14, 7, a^11b^25, 否], [4, 15, 6, a^10b^25, 否], [4, 16, 5, a^9b^25, 否], [4, 17, 4, a^8b^25, 否], [4, 18, 3, a^7b^25, 否], [4, 19, 2, a^6b^25, 否], [4, 20, 1, a^5b^25, 否], [4, 21, 0, a^4b^25, 否], [5, 1, 19, a^24b^25, 否], [5, 2, 18, a^23b^25, 否], [5, 3, 17, a^22b^25, 否], [5, 4, 16, a^21b^25, 否], [5, 5, 15, a^20b^25, 否], [5, 6, 14, a^19b^25, 否], [5, 7, 13, a^18b^25, 否], [5, 8, 12, a^17b^25, 否], [5, 9, 11, a^16b^25, 否], [5, 10, 10, a^15b^25, 否], [5, 11, 9, a^14b^25, 否], [5, 12, 8, a^13b^25, 否], [5, 13, 7, a^12b^25, 否], [5, 14, 6, a^11b^25, 否], [5, 15, 5, a^10b^25, 否], [5, 16, 4, a^9b^25, 否], [5, 17, 3, a^8b^25, 否], [5, 18, 2, a^7b^25, 否], [5, 19, 1, a^6b^25, 否], [5, 20, 0, a^5b^25, 否], [6, 1, 18, a^24b^25, 否], [6, 2, 17, a^23b^25, 否], [6, 3, 16, a^22b^25, 否], [6, 4, 15, a^21b^25, 否], [6, 5, 14, a^20b^25, 否], [6, 6, 13, a^19b^25, 否], [6, 7, 12, a^18b^25, 否], [6, 8, 11, a^17b^25, 否], [6, 9, 10, a^16b^25, 否], [6, 10, 9, a^15b^25, 否], [6, 11, 8, a^14b^25, 否], [6, 12, 7, a^13b^25, 否], [6, 13, 6, a^12b^25, 否], [6, 14, 5, a^11b^25, 否], [6, 15, 4, a^10b^25, 否], [6, 16, 3, a^9b^25, 否], [6, 17, 2, a^8b^25, 否], [6, 18, 1, a^7b^25, 否], [6, 19, 0, a^6b^25, 否], [7, 1, 17, a^24b^25, 否], [7, 2, 16, a^23b^25, 否], [7, 3, 15, a^22b^25, 否], [7, 4, 14, a^21b^25, 否], [7, 5, 13, a^20b^25, 否], [7, 6, 12, a^19b^25, 否], [7, 7, 11, a^18b^25, 否], [7, 8, 10, a^17b^25, 否], [7, 9, 9, a^16b^25, 否], [7, 10, 8, a^15b^25, 否], [7, 11, 7, a^14b^25, 否], [7, 12, 6, a^13b^25, 否], [7, 13, 5, a^12b^25, 否], [7, 14, 4, a^11b^25, 否], [7, 15, 3, a^10b^25, 否], [7, 16, 2, a^9b^25, 否], [7, 17, 1, a^8b^25, 否], [7, 18, 0, a^7b^25, 否], [8, 1, 16, a^24b^25, 否], [8, 2, 15, a^23b^25, 否], [8, 3, 14, a^22b^25, 否], [8, 4, 13, a^21b^25, 否], [8, 5, 12, a^20b^25, 否], [8, 6, 11, a^19b^25, 否], [8, 7, 10, a^18b^25, 否], [8, 8, 9, a^17b^25, 否], [8, 9, 8, a^16b^25, 否], [8, 10, 7, a^15b^25, 否], [8, 11, 6, a^14b^25, 否], [8, 12, 5, a^13b^25, 否], [8, 13, 4, a^12b^25, 否], [8, 14, 3, a^11b^25, 否], [8, 15, 2, a^10b^25, 否], [8, 16, 1, a^9b^25, 否], [8, 17, 0, a^8b^25, 否], [9, 1, 15, a^24b^25, 否], [9, 2, 14, a^23b^25, 否], [9, 3, 13, a^22b^25, 否], [9, 4, 12, a^21b^25, 否], [9, 5, 11, a^20b^25, 否], [9, 6, 10, a^19b^25, 否], [9, 7, 9, a^18b^25, 否], [9, 8, 8, a^17b^25, 否], [9, 9, 7, a^16b^25, 否], [9, 10, 6, a^15b^25, 否], [9, 11, 5, a^14b^25, 否], [9, 12, 4, a^13b^25, 否], [9, 13, 3, a^12b^25, 否], [9, 14, 2, a^11b^25, 否], [9, 15, 1, a^10b^25, 否], [9, 16, 0, a^9b^25, 否], [10, 1, 14, a^24b^25, 否], [10, 2, 13, a^23b^25, 否], [10, 3, 12, a^22b^25, 否], [10, 4, 11, a^21b^25, 否], [10, 5, 10, a^20b^25, 否], [10, 6, 9, a^19b^25, 否], [10, 7, 8, a^18b^25, 否], [10, 8, 7, a^17b^25, 否], [10, 9, 6, a^16b^25, 否], [10, 10, 5, a^15b^25, 否], [10, 11, 4, a^14b^25, 否], [10, 12, 3, a^13b^25, 否], [10, 13, 2, a^12b^25, 否], [10, 14, 1, a^11b^25, 否], [10, 15, 0, a^10b^25, 否], [11, 1, 13, a^24b^25, 否], [11, 2, 12, a^23b^25, 否], [11, 3, 11, a^22b^25, 否], [11, 4, 10, a^21b^25, 否], [11, 5, 9, a^20b^25, 否], [11, 6, 8, a^19b^25, 否], [11, 7, 7, a^18b^25, 否], [11, 8, 6, a^17b^25, 否], [11, 9, 5, a^16b^25, 否], [11, 10, 4, a^15b^25, 否], [11, 11, 3, a^14b^25, 否], [11, 12, 2, a^13b^25, 否], [11, 13, 1, a^12b^25, 否], [11, 14, 0, a^11b^25, 否], [12, 1, 12, a^24b^25, 否], [12, 2, 11, a^23b^25, 否], [12, 3, 10, a^22b^25, 否], [12, 4, 9, a^21b^25, 否], [12, 5, 8, a^20b^25, 否], [12, 6, 7, a^19b^25, 否], [12, 7, 6, a^18b^25, 否], [12, 8, 5, a^17b^25, 否], [12, 9, 4, a^16b^25, 否], [12, 10, 3, a^15b^25, 否], [12, 11, 2, a^14b^25, 否], [12, 12, 1, a^13b^25, 否], [12, 13, 0, a^12b^25, 否], [13, 1, 11, a^24b^25, 否], [13, 2, 10, a^23b^25, 否], [13, 3, 9, a^22b^25, 否], [13, 4, 8, a^21b^25, 否], [13, 5, 7, a^20b^25, 否], [13, 6, 6, a^19b^25, 否], [13, 7, 5, a^18b^25, 否], [13, 8, 4, a^17b^25, 否], [13, 9, 3, a^16b^25, 否], [13, 10, 2, a^15b^25, 否], [13, 11, 1, a^14b^25, 否], [13, 12, 0, a^13b^25, 否], [14, 1, 10, a^24b^25, 否], [14, 2, 9, a^23b^25, 否], [14, 3, 8, a^22b^25, 否], [14, 4, 7, a^21b^25, 否], [14, 5, 6, a^20b^25, 否], [14, 6, 5, a^19b^25, 否], [14, 7, 4, a^18b^25, 否], [14, 8, 3, a^17b^25, 否], [14, 9, 2, a^16b^25, 否], [14, 10, 1, a^15b^25, 否], [14, 11, 0, a^14b^25, 否], [15, 1, 9, a^24b^25, 否], [15, 2, 8, a^23b^25, 否], [15, 3, 7, a^22b^25, 否], [15, 4, 6, a^21b^25, 否], [15, 5, 5, a^20b^25, 否], [15, 6, 4, a^19b^25, 否], [15, 7, 3, a^18b^25, 否], [15, 8, 2, a^17b^25, 否], [15, 9, 1, a^16b^25, 否], [15, 10, 0, a^15b^25, 否], [16, 1, 8, a^24b^25, 否], [16, 2, 7, a^23b^25, 否], [16, 3, 6, a^22b^25, 否], [16, 4, 5, a^21b^25, 否], [16, 5, 4, a^20b^25, 否], [16, 6, 3, a^19b^25, 否], [16, 7, 2, a^18b^25, 否], [16, 8, 1, a^17b^25, 否], [16, 9, 0, a^16b^25, 否], [17, 1, 7, a^24b^25, 否], [17, 2, 6, a^23b^25, 否], [17, 3, 5, a^22b^25, 否], [17, 4, 4, a^21b^25, 否], [17, 5, 3, a^20b^25, 否], [17, 6, 2, a^19b^25, 否], [17, 7, 1, a^18b^25, 否], [17, 8, 0, a^17b^25, 否], [18, 1, 6, a^24b^25, 否], [18, 2, 5, a^23b^25, 否], [18, 3, 4, a^22b^25, 否], [18, 4, 3, a^21b^25, 否], [18, 5, 2, a^20b^25, 否], [18, 6, 1, a^19b^25, 否], [18, 7, 0, a^18b^25, 否], [19, 1, 5, a^24b^25, 否], [19, 2, 4, a^23b^25, 否], [19, 3, 3, a^22b^25, 否], [19, 4, 2, a^21b^25, 否], [19, 5, 1, a^20b^25, 否], [19, 6, 0, a^19b^25, 否], [20, 1, 4, a^24b^25, 否], [20, 2, 3, a^23b^25, 否], [20, 3, 2, a^22b^25, 否], [20, 4, 1, a^21b^25, 否], [20, 5, 0, a^20b^25, 否], [21, 1, 3, a^24b^25, 否], [21, 2, 2, a^23b^25, 否], [21, 3, 1, a^22b^25, 否], [21, 4, 0, a^21b^25, 否], [22, 1, 2, a^24b^25, 否], [22, 2, 1, a^23b^25, 否], [22, 3, 0, a^22b^25, 否], [23, 1, 1, a^24b^25, 否], [23, 2, 0, a^23b^25, 否], [24, 1, 0, a^24b^25, 否]], 325, 325, 是, every_legal_split_is_defeated_by_i=0)

Wave6第十二層核對=(25, a^25b^25, [[0, 1, 24, a^24b^25, 否], [0, 2, 23, a^23b^25, 否], [0, 3, 22, a^22b^25, 否], [0, 4, 21, a^21b^25, 否], [0, 5, 20, a^20b^25, 否], [0, 6, 19, a^19b^25, 否], [0, 7, 18, a^18b^25, 否], [0, 8, 17, a^17b^25, 否], [0, 9, 16, a^16b^25, 否], [0, 10, 15, a^15b^25, 否], [0, 11, 14, a^14b^25, 否], [0, 12, 13, a^13b^25, 否], [0, 13, 12, a^12b^25, 否], [0, 14, 11, a^11b^25, 否], [0, 15, 10, a^10b^25, 否], [0, 16, 9, a^9b^25, 否], [0, 17, 8, a^8b^25, 否], [0, 18, 7, a^7b^25, 否], [0, 19, 6, a^6b^25, 否], [0, 20, 5, a^5b^25, 否], [0, 21, 4, a^4b^25, 否], [0, 22, 3, a^3b^25, 否], [0, 23, 2, a^2b^25, 否], [0, 24, 1, a^1b^25, 否], [0, 25, 0, a^0b^25, 否], [1, 1, 23, a^24b^25, 否], [1, 2, 22, a^23b^25, 否], [1, 3, 21, a^22b^25, 否], [1, 4, 20, a^21b^25, 否], [1, 5, 19, a^20b^25, 否], [1, 6, 18, a^19b^25, 否], [1, 7, 17, a^18b^25, 否], [1, 8, 16, a^17b^25, 否], [1, 9, 15, a^16b^25, 否], [1, 10, 14, a^15b^25, 否], [1, 11, 13, a^14b^25, 否], [1, 12, 12, a^13b^25, 否], [1, 13, 11, a^12b^25, 否], [1, 14, 10, a^11b^25, 否], [1, 15, 9, a^10b^25, 否], [1, 16, 8, a^9b^25, 否], [1, 17, 7, a^8b^25, 否], [1, 18, 6, a^7b^25, 否], [1, 19, 5, a^6b^25, 否], [1, 20, 4, a^5b^25, 否], [1, 21, 3, a^4b^25, 否], [1, 22, 2, a^3b^25, 否], [1, 23, 1, a^2b^25, 否], [1, 24, 0, a^1b^25, 否], [2, 1, 22, a^24b^25, 否], [2, 2, 21, a^23b^25, 否], [2, 3, 20, a^22b^25, 否], [2, 4, 19, a^21b^25, 否], [2, 5, 18, a^20b^25, 否], [2, 6, 17, a^19b^25, 否], [2, 7, 16, a^18b^25, 否], [2, 8, 15, a^17b^25, 否], [2, 9, 14, a^16b^25, 否], [2, 10, 13, a^15b^25, 否], [2, 11, 12, a^14b^25, 否], [2, 12, 11, a^13b^25, 否], [2, 13, 10, a^12b^25, 否], [2, 14, 9, a^11b^25, 否], [2, 15, 8, a^10b^25, 否], [2, 16, 7, a^9b^25, 否], [2, 17, 6, a^8b^25, 否], [2, 18, 5, a^7b^25, 否], [2, 19, 4, a^6b^25, 否], [2, 20, 3, a^5b^25, 否], [2, 21, 2, a^4b^25, 否], [2, 22, 1, a^3b^25, 否], [2, 23, 0, a^2b^25, 否], [3, 1, 21, a^24b^25, 否], [3, 2, 20, a^23b^25, 否], [3, 3, 19, a^22b^25, 否], [3, 4, 18, a^21b^25, 否], [3, 5, 17, a^20b^25, 否], [3, 6, 16, a^19b^25, 否], [3, 7, 15, a^18b^25, 否], [3, 8, 14, a^17b^25, 否], [3, 9, 13, a^16b^25, 否], [3, 10, 12, a^15b^25, 否], [3, 11, 11, a^14b^25, 否], [3, 12, 10, a^13b^25, 否], [3, 13, 9, a^12b^25, 否], [3, 14, 8, a^11b^25, 否], [3, 15, 7, a^10b^25, 否], [3, 16, 6, a^9b^25, 否], [3, 17, 5, a^8b^25, 否], [3, 18, 4, a^7b^25, 否], [3, 19, 3, a^6b^25, 否], [3, 20, 2, a^5b^25, 否], [3, 21, 1, a^4b^25, 否], [3, 22, 0, a^3b^25, 否], [4, 1, 20, a^24b^25, 否], [4, 2, 19, a^23b^25, 否], [4, 3, 18, a^22b^25, 否], [4, 4, 17, a^21b^25, 否], [4, 5, 16, a^20b^25, 否], [4, 6, 15, a^19b^25, 否], [4, 7, 14, a^18b^25, 否], [4, 8, 13, a^17b^25, 否], [4, 9, 12, a^16b^25, 否], [4, 10, 11, a^15b^25, 否], [4, 11, 10, a^14b^25, 否], [4, 12, 9, a^13b^25, 否], [4, 13, 8, a^12b^25, 否], [4, 14, 7, a^11b^25, 否], [4, 15, 6, a^10b^25, 否], [4, 16, 5, a^9b^25, 否], [4, 17, 4, a^8b^25, 否], [4, 18, 3, a^7b^25, 否], [4, 19, 2, a^6b^25, 否], [4, 20, 1, a^5b^25, 否], [4, 21, 0, a^4b^25, 否], [5, 1, 19, a^24b^25, 否], [5, 2, 18, a^23b^25, 否], [5, 3, 17, a^22b^25, 否], [5, 4, 16, a^21b^25, 否], [5, 5, 15, a^20b^25, 否], [5, 6, 14, a^19b^25, 否], [5, 7, 13, a^18b^25, 否], [5, 8, 12, a^17b^25, 否], [5, 9, 11, a^16b^25, 否], [5, 10, 10, a^15b^25, 否], [5, 11, 9, a^14b^25, 否], [5, 12, 8, a^13b^25, 否], [5, 13, 7, a^12b^25, 否], [5, 14, 6, a^11b^25, 否], [5, 15, 5, a^10b^25, 否], [5, 16, 4, a^9b^25, 否], [5, 17, 3, a^8b^25, 否], [5, 18, 2, a^7b^25, 否], [5, 19, 1, a^6b^25, 否], [5, 20, 0, a^5b^25, 否], [6, 1, 18, a^24b^25, 否], [6, 2, 17, a^23b^25, 否], [6, 3, 16, a^22b^25, 否], [6, 4, 15, a^21b^25, 否], [6, 5, 14, a^20b^25, 否], [6, 6, 13, a^19b^25, 否], [6, 7, 12, a^18b^25, 否], [6, 8, 11, a^17b^25, 否], [6, 9, 10, a^16b^25, 否], [6, 10, 9, a^15b^25, 否], [6, 11, 8, a^14b^25, 否], [6, 12, 7, a^13b^25, 否], [6, 13, 6, a^12b^25, 否], [6, 14, 5, a^11b^25, 否], [6, 15, 4, a^10b^25, 否], [6, 16, 3, a^9b^25, 否], [6, 17, 2, a^8b^25, 否], [6, 18, 1, a^7b^25, 否], [6, 19, 0, a^6b^25, 否], [7, 1, 17, a^24b^25, 否], [7, 2, 16, a^23b^25, 否], [7, 3, 15, a^22b^25, 否], [7, 4, 14, a^21b^25, 否], [7, 5, 13, a^20b^25, 否], [7, 6, 12, a^19b^25, 否], [7, 7, 11, a^18b^25, 否], [7, 8, 10, a^17b^25, 否], [7, 9, 9, a^16b^25, 否], [7, 10, 8, a^15b^25, 否], [7, 11, 7, a^14b^25, 否], [7, 12, 6, a^13b^25, 否], [7, 13, 5, a^12b^25, 否], [7, 14, 4, a^11b^25, 否], [7, 15, 3, a^10b^25, 否], [7, 16, 2, a^9b^25, 否], [7, 17, 1, a^8b^25, 否], [7, 18, 0, a^7b^25, 否], [8, 1, 16, a^24b^25, 否], [8, 2, 15, a^23b^25, 否], [8, 3, 14, a^22b^25, 否], [8, 4, 13, a^21b^25, 否], [8, 5, 12, a^20b^25, 否], [8, 6, 11, a^19b^25, 否], [8, 7, 10, a^18b^25, 否], [8, 8, 9, a^17b^25, 否], [8, 9, 8, a^16b^25, 否], [8, 10, 7, a^15b^25, 否], [8, 11, 6, a^14b^25, 否], [8, 12, 5, a^13b^25, 否], [8, 13, 4, a^12b^25, 否], [8, 14, 3, a^11b^25, 否], [8, 15, 2, a^10b^25, 否], [8, 16, 1, a^9b^25, 否], [8, 17, 0, a^8b^25, 否], [9, 1, 15, a^24b^25, 否], [9, 2, 14, a^23b^25, 否], [9, 3, 13, a^22b^25, 否], [9, 4, 12, a^21b^25, 否], [9, 5, 11, a^20b^25, 否], [9, 6, 10, a^19b^25, 否], [9, 7, 9, a^18b^25, 否], [9, 8, 8, a^17b^25, 否], [9, 9, 7, a^16b^25, 否], [9, 10, 6, a^15b^25, 否], [9, 11, 5, a^14b^25, 否], [9, 12, 4, a^13b^25, 否], [9, 13, 3, a^12b^25, 否], [9, 14, 2, a^11b^25, 否], [9, 15, 1, a^10b^25, 否], [9, 16, 0, a^9b^25, 否], [10, 1, 14, a^24b^25, 否], [10, 2, 13, a^23b^25, 否], [10, 3, 12, a^22b^25, 否], [10, 4, 11, a^21b^25, 否], [10, 5, 10, a^20b^25, 否], [10, 6, 9, a^19b^25, 否], [10, 7, 8, a^18b^25, 否], [10, 8, 7, a^17b^25, 否], [10, 9, 6, a^16b^25, 否], [10, 10, 5, a^15b^25, 否], [10, 11, 4, a^14b^25, 否], [10, 12, 3, a^13b^25, 否], [10, 13, 2, a^12b^25, 否], [10, 14, 1, a^11b^25, 否], [10, 15, 0, a^10b^25, 否], [11, 1, 13, a^24b^25, 否], [11, 2, 12, a^23b^25, 否], [11, 3, 11, a^22b^25, 否], [11, 4, 10, a^21b^25, 否], [11, 5, 9, a^20b^25, 否], [11, 6, 8, a^19b^25, 否], [11, 7, 7, a^18b^25, 否], [11, 8, 6, a^17b^25, 否], [11, 9, 5, a^16b^25, 否], [11, 10, 4, a^15b^25, 否], [11, 11, 3, a^14b^25, 否], [11, 12, 2, a^13b^25, 否], [11, 13, 1, a^12b^25, 否], [11, 14, 0, a^11b^25, 否], [12, 1, 12, a^24b^25, 否], [12, 2, 11, a^23b^25, 否], [12, 3, 10, a^22b^25, 否], [12, 4, 9, a^21b^25, 否], [12, 5, 8, a^20b^25, 否], [12, 6, 7, a^19b^25, 否], [12, 7, 6, a^18b^25, 否], [12, 8, 5, a^17b^25, 否], [12, 9, 4, a^16b^25, 否], [12, 10, 3, a^15b^25, 否], [12, 11, 2, a^14b^25, 否], [12, 12, 1, a^13b^25, 否], [12, 13, 0, a^12b^25, 否], [13, 1, 11, a^24b^25, 否], [13, 2, 10, a^23b^25, 否], [13, 3, 9, a^22b^25, 否], [13, 4, 8, a^21b^25, 否], [13, 5, 7, a^20b^25, 否], [13, 6, 6, a^19b^25, 否], [13, 7, 5, a^18b^25, 否], [13, 8, 4, a^17b^25, 否], [13, 9, 3, a^16b^25, 否], [13, 10, 2, a^15b^25, 否], [13, 11, 1, a^14b^25, 否], [13, 12, 0, a^13b^25, 否], [14, 1, 10, a^24b^25, 否], [14, 2, 9, a^23b^25, 否], [14, 3, 8, a^22b^25, 否], [14, 4, 7, a^21b^25, 否], [14, 5, 6, a^20b^25, 否], [14, 6, 5, a^19b^25, 否], [14, 7, 4, a^18b^25, 否], [14, 8, 3, a^17b^25, 否], [14, 9, 2, a^16b^25, 否], [14, 10, 1, a^15b^25, 否], [14, 11, 0, a^14b^25, 否], [15, 1, 9, a^24b^25, 否], [15, 2, 8, a^23b^25, 否], [15, 3, 7, a^22b^25, 否], [15, 4, 6, a^21b^25, 否], [15, 5, 5, a^20b^25, 否], [15, 6, 4, a^19b^25, 否], [15, 7, 3, a^18b^25, 否], [15, 8, 2, a^17b^25, 否], [15, 9, 1, a^16b^25, 否], [15, 10, 0, a^15b^25, 否], [16, 1, 8, a^24b^25, 否], [16, 2, 7, a^23b^25, 否], [16, 3, 6, a^22b^25, 否], [16, 4, 5, a^21b^25, 否], [16, 5, 4, a^20b^25, 否], [16, 6, 3, a^19b^25, 否], [16, 7, 2, a^18b^25, 否], [16, 8, 1, a^17b^25, 否], [16, 9, 0, a^16b^25, 否], [17, 1, 7, a^24b^25, 否], [17, 2, 6, a^23b^25, 否], [17, 3, 5, a^22b^25, 否], [17, 4, 4, a^21b^25, 否], [17, 5, 3, a^20b^25, 否], [17, 6, 2, a^19b^25, 否], [17, 7, 1, a^18b^25, 否], [17, 8, 0, a^17b^25, 否], [18, 1, 6, a^24b^25, 否], [18, 2, 5, a^23b^25, 否], [18, 3, 4, a^22b^25, 否], [18, 4, 3, a^21b^25, 否], [18, 5, 2, a^20b^25, 否], [18, 6, 1, a^19b^25, 否], [18, 7, 0, a^18b^25, 否], [19, 1, 5, a^24b^25, 否], [19, 2, 4, a^23b^25, 否], [19, 3, 3, a^22b^25, 否], [19, 4, 2, a^21b^25, 否], [19, 5, 1, a^20b^25, 否], [19, 6, 0, a^19b^25, 否], [20, 1, 4, a^24b^25, 否], [20, 2, 3, a^23b^25, 否], [20, 3, 2, a^22b^25, 否], [20, 4, 1, a^21b^25, 否], [20, 5, 0, a^20b^25, 否], [21, 1, 3, a^24b^25, 否], [21, 2, 2, a^23b^25, 否], [21, 3, 1, a^22b^25, 否], [21, 4, 0, a^21b^25, 否], [22, 1, 2, a^24b^25, 否], [22, 2, 1, a^23b^25, 否], [22, 3, 0, a^22b^25, 否], [23, 1, 1, a^24b^25, 否], [23, 2, 0, a^23b^25, 否], [24, 1, 0, a^24b^25, 否]], 325, 325, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

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正規語言Pumping引理 · qf-mod49-pump-anbn-all-splits · candidate

76. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"single_split","model":{"p":1,"word":["a","b"],"x_length":0,"y_length":1}}。請依 [mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(single_split, 1, [a, b], 0, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)

Wave6第十二層核對=(single_split, 1, [a, b], 0, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

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正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

77. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"wrong_quantifiers","model":{"p":2,"word":["a","a","b"]}}。請依 [mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(wrong_quantifiers, 2, [a, a, b], 3, adversary_word_then_prover_split_then_adversary_i, 否)

Wave6第十二層核對=(wrong_quantifiers, 2, [a, a, b], 3, adversary_word_then_prover_split_then_adversary_i, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

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正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

78. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"bounded_words","model":{"alphabet":["a","b","c"],"max_checked_length":3}}。請依 [mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(bounded_words, [a, b, c], 3, 40, 否, bounded_words_do_not_prove_regular)

Wave6第十二層核對=(bounded_words, [a, b, c], 3, 40, 否, bounded_words_do_not_prove_regular)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

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正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

79. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"decomposition_order","model":{"p":4,"word":["a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(decomposition_order, 4, [a, a, a, a, b], 10, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)

Wave6第十二層核對=(decomposition_order, 4, [a, a, a, a, b], 10, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

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正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

80. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"single_split","model":{"p":5,"word":["a","a","a","a","a","b"],"x_length":4,"y_length":1}}。請依 [mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(single_split, 5, [a, a, a, a, a, b], 4, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)

Wave6第十二層核對=(single_split, 5, [a, a, a, a, a, b], 4, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

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正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

81. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"wrong_quantifiers","model":{"p":6,"word":["a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(wrong_quantifiers, 6, [a, a, a, a, a, a, b], 21, adversary_word_then_prover_split_then_adversary_i, 否)

Wave6第十二層核對=(wrong_quantifiers, 6, [a, a, a, a, a, a, b], 21, adversary_word_then_prover_split_then_adversary_i, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

82. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"bounded_words","model":{"alphabet":["a"],"max_checked_length":2}}。請依 [mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(bounded_words, [a], 2, 3, 否, bounded_words_do_not_prove_regular)

Wave6第十二層核對=(bounded_words, [a], 2, 3, 否, bounded_words_do_not_prove_regular)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

83. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"decomposition_order","model":{"p":8,"word":["a","a","a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(decomposition_order, 8, [a, a, a, a, a, a, a, a, b], 36, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)

Wave6第十二層核對=(decomposition_order, 8, [a, a, a, a, a, a, a, a, b], 36, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

84. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"single_split","model":{"p":9,"word":["a","a","a","a","a","a","a","a","a","b"],"x_length":8,"y_length":1}}。請依 [mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(single_split, 9, [a, a, a, a, a, a, a, a, a, b], 8, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)

Wave6第十二層核對=(single_split, 9, [a, a, a, a, a, a, a, a, a, b], 8, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

85. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"wrong_quantifiers","model":{"p":10,"word":["a","a","a","a","a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(wrong_quantifiers, 10, [a, a, a, a, a, a, a, a, a, a, b], 55, adversary_word_then_prover_split_then_adversary_i, 否)

Wave6第十二層核對=(wrong_quantifiers, 10, [a, a, a, a, a, a, a, a, a, a, b], 55, adversary_word_then_prover_split_then_adversary_i, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

86. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"bounded_words","model":{"alphabet":["a","b"],"max_checked_length":1}}。請依 [mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(bounded_words, [a, b], 1, 3, 否, bounded_words_do_not_prove_regular)

Wave6第十二層核對=(bounded_words, [a, b], 1, 3, 否, bounded_words_do_not_prove_regular)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

87. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"decomposition_order","model":{"p":12,"word":["a","a","a","a","a","a","a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(decomposition_order, 12, [a, a, a, a, a, a, a, a, a, a, a, a, b], 78, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)

Wave6第十二層核對=(decomposition_order, 12, [a, a, a, a, a, a, a, a, a, a, a, a, b], 78, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

88. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"single_split","model":{"p":13,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":12,"y_length":1}}。請依 [mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(single_split, 13, [a, a, a, a, a, a, a, a, a, a, a, a, a, b], 12, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)

Wave6第十二層核對=(single_split, 13, [a, a, a, a, a, a, a, a, a, a, a, a, a, b], 12, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

89. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"wrong_quantifiers","model":{"p":14,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(wrong_quantifiers, 14, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 105, adversary_word_then_prover_split_then_adversary_i, 否)

Wave6第十二層核對=(wrong_quantifiers, 14, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 105, adversary_word_then_prover_split_then_adversary_i, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

90. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"bounded_words","model":{"alphabet":["a","b","c"],"max_checked_length":5}}。請依 [mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(bounded_words, [a, b, c], 5, 364, 否, bounded_words_do_not_prove_regular)

Wave6第十二層核對=(bounded_words, [a, b, c], 5, 364, 否, bounded_words_do_not_prove_regular)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

91. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"decomposition_order","model":{"p":16,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(decomposition_order, 16, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 136, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)

Wave6第十二層核對=(decomposition_order, 16, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 136, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

92. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"single_split","model":{"p":17,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":16,"y_length":1}}。請依 [mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(single_split, 17, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 16, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)

Wave6第十二層核對=(single_split, 17, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 16, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

93. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"wrong_quantifiers","model":{"p":18,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(wrong_quantifiers, 18, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 171, adversary_word_then_prover_split_then_adversary_i, 否)

Wave6第十二層核對=(wrong_quantifiers, 18, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 171, adversary_word_then_prover_split_then_adversary_i, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

94. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"bounded_words","model":{"alphabet":["a"],"max_checked_length":4}}。請依 [mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(bounded_words, [a], 4, 5, 否, bounded_words_do_not_prove_regular)

Wave6第十二層核對=(bounded_words, [a], 4, 5, 否, bounded_words_do_not_prove_regular)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

95. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"decomposition_order","model":{"p":20,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(decomposition_order, 20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 210, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)

Wave6第十二層核對=(decomposition_order, 20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 210, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

96. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"single_split","model":{"p":21,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":20,"y_length":1}}。請依 [mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(single_split, 21, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 20, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)

Wave6第十二層核對=(single_split, 21, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 20, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

97. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"wrong_quantifiers","model":{"p":22,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(wrong_quantifiers, 22, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 253, adversary_word_then_prover_split_then_adversary_i, 否)

Wave6第十二層核對=(wrong_quantifiers, 22, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 253, adversary_word_then_prover_split_then_adversary_i, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

98. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"bounded_words","model":{"alphabet":["a","b"],"max_checked_length":3}}。請依 [mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

查看答案與解釋

(bounded_words, [a, b], 3, 15, 否, bounded_words_do_not_prove_regular)

Wave6第十二層核對=(bounded_words, [a, b], 3, 15, 否, bounded_words_do_not_prove_regular)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

99. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"decomposition_order","model":{"p":24,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"]}}。請依 [mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

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(decomposition_order, 24, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 300, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)

Wave6第十二層核對=(decomposition_order, 24, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 300, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

100. 正規語言Pumping引理

exists p forall w exists split forall i量詞與split constraints責任。操作=pump-boundary；資料={"mode":"single_split","model":{"p":25,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":24,"y_length":1}}。請依 [mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary] 欄序回答。

四種mode為9/6/6/6欄；pumping property是regular必要非充分條件。否針對量詞或方法推論，不否認單一split可能合法。

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(single_split, 25, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 24, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)

Wave6第十二層核對=(single_split, 25, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 24, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。

本題逐步詳解（AI 二次覆核）

正規語言Pumping引理 · qf-mod49-pump-boundary · candidate

原始題目 JSONL · 驗收報告