{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0001","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [a, b], 2, 0, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187001,"parameters":{"p":1,"word":["a","b"],"x_length":0,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:1,word:[\"a\",\"b\"],x_length:0,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0002","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [a, a, b], 3, 1, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187002,"parameters":{"p":2,"word":["a","a","b"],"x_length":1,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:2,word:[\"a\",\"a\",\"b\"],x_length:1,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0003","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [a, a, a, b], 4, 2, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187003,"parameters":{"p":3,"word":["a","a","a","b"],"x_length":2,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:3,word:[\"a\",\"a\",\"a\",\"b\"],x_length:2,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0004","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [a, a, a, a, b], 5, 3, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187004,"parameters":{"p":4,"word":["a","a","a","a","b"],"x_length":3,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:4,word:[\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:3,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0005","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [a, a, a, a, a, b], 6, 4, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187005,"parameters":{"p":5,"word":["a","a","a","a","a","b"],"x_length":4,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:5,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:4,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0006","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [a, a, a, a, a, a, b], 7, 5, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187006,"parameters":{"p":6,"word":["a","a","a","a","a","a","b"],"x_length":5,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:6,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:5,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0007","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [a, a, a, a, a, a, a, b], 8, 6, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187007,"parameters":{"p":7,"word":["a","a","a","a","a","a","a","b"],"x_length":6,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:7,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:6,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0009","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187009,"parameters":{"p":9,"word":["a","a","a","a","a","a","a","a","a","b"],"x_length":8,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:9,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:8,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0010","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187010,"parameters":{"p":10,"word":["a","a","a","a","a","a","a","a","a","a","b"],"x_length":9,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:10,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:9,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0011","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187011,"parameters":{"p":11,"word":["a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":10,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:11,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:10,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0012","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187012,"parameters":{"p":12,"word":["a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":11,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:12,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:11,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0013","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187013,"parameters":{"p":13,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":12,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:13,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:12,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0014","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187014,"parameters":{"p":14,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"],"x_length":13,"y_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{p:14,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:13,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0015","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187015,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0016","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187016,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0017","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187017,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0018","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187018,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0019","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187019,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0020","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187020,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0021","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187021,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0022","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187022,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0023","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187023,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0024","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187024,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0025","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-quantifier-audit","seed":202611187025,"parameters":{"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,"domain":"Exact_regular_pumping_lemma","operation":"pump-quantifier-audit","semantic_template":"pump-quantifier-audit|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0026","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, [0, 0], [0, 1, 0], 0, 1, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187026,"parameters":{"modulus":1,"word_length":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:1,word_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0027","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 4, [0, 1, 0], [0, 2, 0], 0, 2, 2, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187027,"parameters":{"modulus":2,"word_length":4,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:2,word_length:4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0028","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 9, [0, 1, 2, 0], [0, 3, 0], 0, 3, 6, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, 是)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187028,"parameters":{"modulus":3,"word_length":9,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:3,word_length:9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0029","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187029,"parameters":{"modulus":4,"word_length":16,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:4,word_length:16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0030","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187030,"parameters":{"modulus":5,"word_length":5,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:5,word_length:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0031","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187031,"parameters":{"modulus":6,"word_length":12,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:6,word_length:12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0032","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187032,"parameters":{"modulus":7,"word_length":21,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:7,word_length:21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0033","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187033,"parameters":{"modulus":8,"word_length":32,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:8,word_length:32}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0034","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187034,"parameters":{"modulus":9,"word_length":9,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:9,word_length:9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0035","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187035,"parameters":{"modulus":10,"word_length":20,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:10,word_length:20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0036","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187036,"parameters":{"modulus":11,"word_length":33,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:11,word_length:33}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0037","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187037,"parameters":{"modulus":12,"word_length":48,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:12,word_length:48}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0038","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187038,"parameters":{"modulus":13,"word_length":13,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:13,word_length:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0039","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187039,"parameters":{"modulus":14,"word_length":28,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:14,word_length:28}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0040","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187040,"parameters":{"modulus":15,"word_length":45,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:15,word_length:45}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0041","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187041,"parameters":{"modulus":16,"word_length":64,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:16,word_length:64}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0042","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187042,"parameters":{"modulus":17,"word_length":17,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:17,word_length:17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0043","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187043,"parameters":{"modulus":18,"word_length":36,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:18,word_length:36}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0044","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187044,"parameters":{"modulus":19,"word_length":57,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:19,word_length:57}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0045","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187045,"parameters":{"modulus":20,"word_length":80,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:20,word_length:80}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0046","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187046,"parameters":{"modulus":21,"word_length":21,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:21,word_length:21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0047","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187047,"parameters":{"modulus":22,"word_length":44,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:22,word_length:44}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0048","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187048,"parameters":{"modulus":23,"word_length":69,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:23,word_length:69}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0049","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187049,"parameters":{"modulus":24,"word_length":96,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:24,word_length:96}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0050","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187050,"parameters":{"modulus":25,"word_length":25,"domain":"Exact_regular_pumping_lemma","operation":"pump-dfa-loop-witness","semantic_template":"pump-dfa-loop-witness|{modulus:25,word_length:25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0051","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, a^1b^1, [[0, 1, 0, a^0b^1, 否]], 1, 1, 是, every_legal_split_is_defeated_by_i=0)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187051,"parameters":{"p":1,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0052","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187052,"parameters":{"p":2,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0053","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187053,"parameters":{"p":3,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0054","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187054,"parameters":{"p":4,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0055","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187055,"parameters":{"p":5,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0056","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187056,"parameters":{"p":6,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:6}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0057","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187057,"parameters":{"p":7,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:7}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0058","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187058,"parameters":{"p":8,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:8}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0059","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187059,"parameters":{"p":9,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:9}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0060","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187060,"parameters":{"p":10,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:10}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0061","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187061,"parameters":{"p":11,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:11}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0062","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187062,"parameters":{"p":12,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:12}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0063","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187063,"parameters":{"p":13,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:13}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0064","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187064,"parameters":{"p":14,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:14}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0065","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187065,"parameters":{"p":15,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:15}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0066","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187066,"parameters":{"p":16,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:16}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0067","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187067,"parameters":{"p":17,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:17}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0068","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187068,"parameters":{"p":18,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:18}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0069","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187069,"parameters":{"p":19,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:19}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0070","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187070,"parameters":{"p":20,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:20}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0071","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187071,"parameters":{"p":21,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:21}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0072","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187072,"parameters":{"p":22,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:22}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0073","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187073,"parameters":{"p":23,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:23}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0074","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187074,"parameters":{"p":24,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:24}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0075","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187075,"parameters":{"p":25,"domain":"Exact_regular_pumping_lemma","operation":"pump-anbn-all-splits","semantic_template":"pump-anbn-all-splits|{p:25}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0076","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(single_split, 1, [a, b], 0, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187076,"parameters":{"mode":"single_split","model":{"p":1,"word":["a","b"],"x_length":0,"y_length":1},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:1,word:[\"a\",\"b\"],x_length:0,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0077","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_quantifiers, 2, [a, a, b], 3, adversary_word_then_prover_split_then_adversary_i, 否)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187077,"parameters":{"mode":"wrong_quantifiers","model":{"p":2,"word":["a","a","b"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:2,word:[\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0078","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_words, [a, b, c], 3, 40, 否, bounded_words_do_not_prove_regular)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187078,"parameters":{"mode":"bounded_words","model":{"alphabet":["a","b","c"],"max_checked_length":3},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{alphabet:[\"a\",\"b\",\"c\"],max_checked_length:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0079","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decomposition_order, 4, [a, a, a, a, b], 10, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187079,"parameters":{"mode":"decomposition_order","model":{"p":4,"word":["a","a","a","a","b"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:4,word:[\"a\",\"a\",\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0080","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(single_split, 5, [a, a, a, a, a, b], 4, 1, 1, 是, 否, one_decomposition_does_not_resolve_exists_split_forall_i)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187080,"parameters":{"mode":"single_split","model":{"p":5,"word":["a","a","a","a","a","b"],"x_length":4,"y_length":1},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:5,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:4,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0081","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_quantifiers, 6, [a, a, a, a, a, a, b], 21, adversary_word_then_prover_split_then_adversary_i, 否)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187081,"parameters":{"mode":"wrong_quantifiers","model":{"p":6,"word":["a","a","a","a","a","a","b"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:6,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0082","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_words, [a], 2, 3, 否, bounded_words_do_not_prove_regular)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187082,"parameters":{"mode":"bounded_words","model":{"alphabet":["a"],"max_checked_length":2},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{alphabet:[\"a\"],max_checked_length:2}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0083","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(decomposition_order, 8, [a, a, a, a, a, a, a, a, b], 36, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, 否)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187083,"parameters":{"mode":"decomposition_order","model":{"p":8,"word":["a","a","a","a","a","a","a","a","b"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:8,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0084","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187084,"parameters":{"mode":"single_split","model":{"p":9,"word":["a","a","a","a","a","a","a","a","a","b"],"x_length":8,"y_length":1},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:9,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:8,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0085","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(wrong_quantifiers, 10, [a, a, a, a, a, a, a, a, a, a, b], 55, adversary_word_then_prover_split_then_adversary_i, 否)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187085,"parameters":{"mode":"wrong_quantifiers","model":{"p":10,"word":["a","a","a","a","a","a","a","a","a","a","b"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:10,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0086","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_words, [a, b], 1, 3, 否, bounded_words_do_not_prove_regular)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187086,"parameters":{"mode":"bounded_words","model":{"alphabet":["a","b"],"max_checked_length":1},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{alphabet:[\"a\",\"b\"],max_checked_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0087","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187087,"parameters":{"mode":"decomposition_order","model":{"p":12,"word":["a","a","a","a","a","a","a","a","a","a","a","a","b"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:12,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0088","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187088,"parameters":{"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},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:13,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:12,y_length:1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0089","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187089,"parameters":{"mode":"wrong_quantifiers","model":{"p":14,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:14,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0090","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_words, [a, b, c], 5, 364, 否, bounded_words_do_not_prove_regular)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187090,"parameters":{"mode":"bounded_words","model":{"alphabet":["a","b","c"],"max_checked_length":5},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{alphabet:[\"a\",\"b\",\"c\"],max_checked_length:5}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0091","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187091,"parameters":{"mode":"decomposition_order","model":{"p":16,"word":["a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","a","b"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:16,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0092","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187092,"parameters":{"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},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0093","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187093,"parameters":{"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"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:18,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0094","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_words, [a], 4, 5, 否, bounded_words_do_not_prove_regular)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187094,"parameters":{"mode":"bounded_words","model":{"alphabet":["a"],"max_checked_length":4},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{alphabet:[\"a\"],max_checked_length:4}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0095","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187095,"parameters":{"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"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{p:20,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0096","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187096,"parameters":{"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},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0097","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187097,"parameters":{"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"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{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\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0098","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(bounded_words, [a, b], 3, 15, 否, bounded_words_do_not_prove_regular)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187098,"parameters":{"mode":"bounded_words","model":{"alphabet":["a","b"],"max_checked_length":3},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{alphabet:[\"a\",\"b\"],max_checked_length:3}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0099","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187099,"parameters":{"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"]},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{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\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0100","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","language":"zh-Hant","source_mko_ids":["mko-pumping-lemma-regular-languages"],"title_zh":"正規語言Pumping引理","stem_zh":"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 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(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)","explanation_zh":"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。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正規語言Pumping引理"]},"variant":{"family_id":"qf-mod49-pump-boundary","seed":202611187100,"parameters":{"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},"domain":"Exact_regular_pumping_lemma","operation":"pump-boundary","semantic_template":"pump-boundary|{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}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}