{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0001","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9df2261e12b3b2a007cd52123348a297c5267c4aa68f0992cd71ea25332471c5","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187001,"semantic_template":"pump-quantifier-audit|{p:1,word:[\"a\",\"b\"],x_length:0,y_length:1}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=1, word=[a, b], |x|=0, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(1, [a, b], 2, 0, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (1, [a, b], 2, 0, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0002","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da3ccff7ce46ed59f6f38e744f88bcab7d54328099f921c4caecba09cc9447c4","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187002,"semantic_template":"pump-quantifier-audit|{p:2,word:[\"a\",\"a\",\"b\"],x_length:1,y_length:1}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=2, word=[a, a, b], |x|=1, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(2, [a, a, b], 3, 1, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (2, [a, a, b], 3, 1, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0003","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a196f7c953a5fa3c394df8d3a84254d791725b4c69c0ea901816bc510f6ddb07","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187003,"semantic_template":"pump-quantifier-audit|{p:3,word:[\"a\",\"a\",\"a\",\"b\"],x_length:2,y_length:1}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=3, word=[a, a, a, b], |x|=2, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(3, [a, a, a, b], 4, 2, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (3, [a, a, a, b], 4, 2, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0004","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4858bcf5cde0204cf4b950409c3f3c01c73dd35df3f17deabc7a9d1c3c28420","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187004,"semantic_template":"pump-quantifier-audit|{p:4,word:[\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:3,y_length:1}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=4, word=[a, a, a, a, b], |x|=3, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(4, [a, a, a, a, b], 5, 3, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (4, [a, a, a, a, b], 5, 3, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0005","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caa9438ad175dde6069410ac4e397fd7794c94485fe2c3ca7886349caf112058","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187005,"semantic_template":"pump-quantifier-audit|{p:5,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:4,y_length:1}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=5, word=[a, a, a, a, a, b], |x|=4, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(5, [a, a, a, a, a, b], 6, 4, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (5, [a, a, a, a, a, b], 6, 4, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0006","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bf48a7399c7ca8a1f2522aa11d86569ae5a3258498cf8b0e77616c314ad7ca5","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187006,"semantic_template":"pump-quantifier-audit|{p:6,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:5,y_length:1}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=6, word=[a, a, a, a, a, a, b], |x|=5, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(6, [a, a, a, a, a, a, b], 7, 5, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (6, [a, a, a, a, a, a, b], 7, 5, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0007","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4d4fd3730baf2f5d18bb96e77cb88dc432fb02176476cd734a72f8ee151904c","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187007,"semantic_template":"pump-quantifier-audit|{p:7,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:6,y_length:1}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=7, word=[a, a, a, a, a, a, a, b], |x|=6, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(7, [a, a, a, a, a, a, a, b], 8, 6, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (7, [a, a, a, a, a, a, a, b], 8, 6, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0008","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdc7bb97405723803731b5aa27f7b83c2f8899b5e0ae27c280ab6e963139396f","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187008,"semantic_template":"pump-quantifier-audit|{p:8,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:7,y_length:1}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=8, word=[a, a, a, a, a, a, a, a, b], |x|=7, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(8, [a, a, a, a, a, a, a, a, b], 9, 7, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (8, [a, a, a, a, a, a, a, a, b], 9, 7, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0009","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60020a05b50928c795ed136d219c6e9b3a5a4ca89a3be5f23a9281099ed37bf0","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187009,"semantic_template":"pump-quantifier-audit|{p:9,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:8,y_length:1}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=9, word=[a, a, a, a, a, a, a, a, a, b], |x|=8, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(9, [a, a, a, a, a, a, a, a, a, b], 10, 8, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (9, [a, a, a, a, a, a, a, a, a, b], 10, 8, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0010","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5884bc44100a0785c68dd767c473023876140ddcfa61dfd67dee26c4577191c8","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187010,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=10, word=[a, a, a, a, a, a, a, a, a, a, b], |x|=9, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(10, [a, a, a, a, a, a, a, a, a, a, b], 11, 9, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (10, [a, a, a, a, a, a, a, a, a, a, b], 11, 9, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0011","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dfb5a10afb92ee934423ecf12d908233d36e5de0e3991807d392567a8e9eaec","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187011,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=11, word=[a, a, a, a, a, a, a, a, a, a, a, b], |x|=10, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(11, [a, a, a, a, a, a, a, a, a, a, a, b], 12, 10, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (11, [a, a, a, a, a, a, a, a, a, a, a, b], 12, 10, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0012","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4de27d6fcfff642ba171e18c7deacda570503e99dfe617797bf2dcb81e5f2fbb","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187012,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=12, word=[a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=11, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(12, [a, a, a, a, a, a, a, a, a, a, a, a, b], 13, 11, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (12, [a, a, a, a, a, a, a, a, a, a, a, a, b], 13, 11, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0013","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"792ac7102651d5c9b4a1984571e5a64da578ffeccb7e50c397b139f923ee15fd","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187013,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=13, word=[a, a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=12, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(13, [a, a, a, a, a, a, a, a, a, a, a, a, a, b], 14, 12, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (13, [a, a, a, a, a, a, a, a, a, a, a, a, a, b], 14, 12, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0014","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f218e39d6d06bb2bcc446b1df187c4d57fbfadfda5dc3e65c65c8fa03e7f53b1","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187014,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=14, word=[a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=13, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","answer":{"kind":"expression","value":"(14, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 15, 13, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (14, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 15, 13, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0015","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f738c5460c3eb63426f61e4c142cdc719bc45c12554c3bd39276eb569efaa1e1","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187015,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=15, word=[a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=14, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (15, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 16, 14, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0016","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d577c20b63fdf13998994c63c73e7c03181ffe58d9eaa473c4662e28a65f08ed","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187016,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=16, word=[a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=15, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (16, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 17, 15, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0017","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f79364a16003049625c49016dd4b8741d18946eb7a2445c36612ea65d235c61","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187017,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=17, word=[a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=16, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (17, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 18, 16, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0018","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9492d9e730278c4a122d6f3a89b2258cd7dc9985f743ad519d2565edee16e312","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187018,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=18, word=[a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=17, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (18, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 19, 17, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0019","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08aa890709089c7c79508c6c6cac17a233e31c35eb3f3c751b9e5b43c0f961c8","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187019,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive p=19, word=[a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=18, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0020","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"126839ac1e999b5082932482a0268d6e5eb215a45df9024d56e069a5218dfd0d","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187020,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive 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|=19, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0021","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a2997b9170d2a4ffa41292f981b1a927b32d98f384119d6739412b54880112c","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187021,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive 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|=20, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0022","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ec2f5a7831bad0813bc57f416fc2f1eb7cdb759b283ea82f7a79ae2ada360c2","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187022,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive 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|=21, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0023","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2036234182fa7d21bc8c010b316b5176d189570b0fdeca5b5b7aa315d5970f1e","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187023,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive 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|=22, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0024","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccaae6081f0f669afab1628ba478f8b413c8b56b59d1a17810adcf45b830818f","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187024,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive 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|=23, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0025","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35754d819a20a8eabfc0d32e236d87c85d3cf240829cdbd2602bdbbf7d54364f","family_id":"qf-mod49-pump-quantifier-audit","seed":202611187025,"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}"},"fields":{"title":"Pumping Quantifiers Place the Word before the Split and the Pump Count after It","stem":"Use positive 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|=24, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).","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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)","explanation":"Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (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, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)."},"skills":["Pumping Quantifiers Place the Word before the Split and the Pump Count after It"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0026","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2c23e48a19739c6a7b79808b800014314001a7eb15a36f9b38df8d913f6c2ba","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187026,"semantic_template":"pump-dfa-loop-witness|{modulus:1,word_length:1}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=1, with accepted word length=1. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (1, 1, [0, 0], [0, 1, 0], 0, 1, 0, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0027","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3b58918306345f8ed39ccd201f93ed5fab015a3a45d38bb598eb4a23104e0fb","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187027,"semantic_template":"pump-dfa-loop-witness|{modulus:2,word_length:4}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=2, with accepted word length=4. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (2, 4, [0, 1, 0], [0, 2, 0], 0, 2, 2, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0028","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"690d1b7cbfb9483ad1e427d68346c481ea2ee024ae997afadbad52765fb8c45f","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187028,"semantic_template":"pump-dfa-loop-witness|{modulus:3,word_length:9}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=3, with accepted word length=9. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (3, 9, [0, 1, 2, 0], [0, 3, 0], 0, 3, 6, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0029","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f3296e6fe2f2bd173e89dd90f5dc57fe217cb654d7fff2cef1c1e76fa0a0f93","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187029,"semantic_template":"pump-dfa-loop-witness|{modulus:4,word_length:16}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=4, with accepted word length=16. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (4, 16, [0, 1, 2, 3, 0], [0, 4, 0], 0, 4, 12, for_all_i_length=(i*modulus)+z_is_0_mod_modulus, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0030","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b122e390a55d9955ba1ebeda91a0836b77951dd165cf585f52f3c35ba1c63eb","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187030,"semantic_template":"pump-dfa-loop-witness|{modulus:5,word_length:5}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=5, with accepted word length=5. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0031","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a14aff3864c1bc8294166763e0b9b27e73ef724f407d165dac033d0698e9b0a","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187031,"semantic_template":"pump-dfa-loop-witness|{modulus:6,word_length:12}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=6, with accepted word length=12. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0032","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b5681345b9f8dbb2a3a382b5f83e685fca0c5156b7f8cc0064bdf15b3807574","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187032,"semantic_template":"pump-dfa-loop-witness|{modulus:7,word_length:21}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=7, with accepted word length=21. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0033","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abd6de1a228d8203bedc02cf72222e3cad672999ddf090d7f8a895dba67cbad2","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187033,"semantic_template":"pump-dfa-loop-witness|{modulus:8,word_length:32}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=8, with accepted word length=32. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0034","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61aed1c90587e47a84e0a1f40439592f399ee5a83becdcdb5f28eee32ea242f0","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187034,"semantic_template":"pump-dfa-loop-witness|{modulus:9,word_length:9}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=9, with accepted word length=9. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0035","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"939e531f4268db79020face03e8db72427e3bf61b7b46b9154eeb0703459e126","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187035,"semantic_template":"pump-dfa-loop-witness|{modulus:10,word_length:20}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=10, with accepted word length=20. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0036","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f753adbcefc6c12756c5091679d14f42a467a1b10dcb116a5780770c424036bf","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187036,"semantic_template":"pump-dfa-loop-witness|{modulus:11,word_length:33}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=11, with accepted word length=33. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0037","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57828975303a76c0fc0039277fd078c88ea573453f8eabb370a6bbc1dc168f1f","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187037,"semantic_template":"pump-dfa-loop-witness|{modulus:12,word_length:48}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=12, with accepted word length=48. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0038","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27b0d6d1d87239bf870fd1233170cd3bbaaca1d93ae236d28aa1c7cb345efb34","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187038,"semantic_template":"pump-dfa-loop-witness|{modulus:13,word_length:13}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=13, with accepted word length=13. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0039","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0a58033a4a56ab93cdbda38fd64e2db79a9d92f8087a8ae488a7d24756678fd","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187039,"semantic_template":"pump-dfa-loop-witness|{modulus:14,word_length:28}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=14, with accepted word length=28. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0040","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ebcbde9ed2bd002e7b26b641a2d25b09ee0a3b721ced693a435d56a1824173b","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187040,"semantic_template":"pump-dfa-loop-witness|{modulus:15,word_length:45}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=15, with accepted word length=45. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0041","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b3759562e6c692e62473d98b4a09f1bea5d98b947f8ca3d71db6599a94eb852","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187041,"semantic_template":"pump-dfa-loop-witness|{modulus:16,word_length:64}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=16, with accepted word length=64. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0042","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8198ad965c552aa2132b42ff0b31db706640d3592a4ea578267f77bfb61f781","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187042,"semantic_template":"pump-dfa-loop-witness|{modulus:17,word_length:17}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=17, with accepted word length=17. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0043","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e12572e10c35216aefadbec68f0b39cc4af5da473a69218206d1580cc251227","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187043,"semantic_template":"pump-dfa-loop-witness|{modulus:18,word_length:36}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=18, with accepted word length=36. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0044","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d3e94e4491934db6ef5ef61cf8d7d5a51fad448ab58d84a9554202f1330d74c","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187044,"semantic_template":"pump-dfa-loop-witness|{modulus:19,word_length:57}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=19, with accepted word length=57. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0045","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7dacbd2eda10f965f7859857d6d7bdf9bd9a0d3bc8a2678441354d282c385fb","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187045,"semantic_template":"pump-dfa-loop-witness|{modulus:20,word_length:80}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=20, with accepted word length=80. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0046","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f96142b498eb6960189df4fd35d87e8f0f246bfbf795aa3f873dfd04456a1e9","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187046,"semantic_template":"pump-dfa-loop-witness|{modulus:21,word_length:21}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=21, with accepted word length=21. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0047","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca4ff23a2c35c980eccd62f761bb352b077b6d35a005e12afc2c2723cab41830","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187047,"semantic_template":"pump-dfa-loop-witness|{modulus:22,word_length:44}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=22, with accepted word length=44. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0048","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a8bf6e20930d473f2d4b29e00442440260eb33e00d44d92b67df2b7a244f75e","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187048,"semantic_template":"pump-dfa-loop-witness|{modulus:23,word_length:69}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=23, with accepted word length=69. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0049","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e72265fe815ba60d1d13b6836fb8149dc96db1ac52def3556f20369cb9cf12","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187049,"semantic_template":"pump-dfa-loop-witness|{modulus:24,word_length:96}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=24, with accepted word length=96. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0050","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"830de970f0c13e94182101e574c3866151e1cac841daad79cbb7504755d41a40","family_id":"qf-mod49-pump-dfa-loop-witness","seed":202611187050,"semantic_template":"pump-dfa-loop-witness|{modulus:25,word_length:25}"},"fields":{"title":"A Complete Unary DFA Loop Gives an Invariant for Every Pump Count","stem":"Use the unary DFA for lengths divisible by modulus=25, with accepted word length=25. Give (modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified).","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, Yes)","explanation":"Follow the first modulus+1 states, repeat state0 and choose x empty, y of length modulus, and z the remainder. For every i≥0 the pumped length is i*modulus+|z|, still divisible by the modulus. This symbolic loop invariant, not finitely many pump tests, supports the whole family. The exact result, retaining the requested order, is (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, Yes)."},"skills":["A Complete Unary DFA Loop Gives an Invariant for Every Pump Count"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0051","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08137bddb3e722d3accdf7346ad57d409634a295ef626515d77fa010a1cf916f","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187051,"semantic_template":"pump-anbn-all-splits|{p:1}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=1 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(1, a^1b^1, [[0, 1, 0, a^0b^1, No]], 1, 1, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (1, a^1b^1, [[0, 1, 0, a^0b^1, No]], 1, 1, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0052","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebf3105853eba6bc57641959548a0bb27705ca87ffdff151359276ddcf50247d","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187052,"semantic_template":"pump-anbn-all-splits|{p:2}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=2 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(2, a^2b^2, [[0, 1, 1, a^1b^2, No], [0, 2, 0, a^0b^2, No], [1, 1, 0, a^1b^2, No]], 3, 3, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (2, a^2b^2, [[0, 1, 1, a^1b^2, No], [0, 2, 0, a^0b^2, No], [1, 1, 0, a^1b^2, No]], 3, 3, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0053","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5502baf84ea6f8f6b7ece2e14d0589debc7b8aeb39a848cc53447e6ab697bbc","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187053,"semantic_template":"pump-anbn-all-splits|{p:3}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=3 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(3, a^3b^3, [[0, 1, 2, a^2b^3, No], [0, 2, 1, a^1b^3, No], [0, 3, 0, a^0b^3, No], [1, 1, 1, a^2b^3, No], [1, 2, 0, a^1b^3, No], [2, 1, 0, a^2b^3, No]], 6, 6, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (3, a^3b^3, [[0, 1, 2, a^2b^3, No], [0, 2, 1, a^1b^3, No], [0, 3, 0, a^0b^3, No], [1, 1, 1, a^2b^3, No], [1, 2, 0, a^1b^3, No], [2, 1, 0, a^2b^3, No]], 6, 6, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0054","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94634c70e5a83cdd79c5b4194851a93d9f89e196b03a9857940125881072f021","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187054,"semantic_template":"pump-anbn-all-splits|{p:4}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=4 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(4, a^4b^4, [[0, 1, 3, a^3b^4, No], [0, 2, 2, a^2b^4, No], [0, 3, 1, a^1b^4, No], [0, 4, 0, a^0b^4, No], [1, 1, 2, a^3b^4, No], [1, 2, 1, a^2b^4, No], [1, 3, 0, a^1b^4, No], [2, 1, 1, a^3b^4, No], [2, 2, 0, a^2b^4, No], [3, 1, 0, a^3b^4, No]], 10, 10, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (4, a^4b^4, [[0, 1, 3, a^3b^4, No], [0, 2, 2, a^2b^4, No], [0, 3, 1, a^1b^4, No], [0, 4, 0, a^0b^4, No], [1, 1, 2, a^3b^4, No], [1, 2, 1, a^2b^4, No], [1, 3, 0, a^1b^4, No], [2, 1, 1, a^3b^4, No], [2, 2, 0, a^2b^4, No], [3, 1, 0, a^3b^4, No]], 10, 10, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0055","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d3a0b9b08b713c6cd3c93225adfb039c2c4424970bb3bef47354defd5d3176c","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187055,"semantic_template":"pump-anbn-all-splits|{p:5}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=5 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(5, a^5b^5, [[0, 1, 4, a^4b^5, No], [0, 2, 3, a^3b^5, No], [0, 3, 2, a^2b^5, No], [0, 4, 1, a^1b^5, No], [0, 5, 0, a^0b^5, No], [1, 1, 3, a^4b^5, No], [1, 2, 2, a^3b^5, No], [1, 3, 1, a^2b^5, No], [1, 4, 0, a^1b^5, No], [2, 1, 2, a^4b^5, No], [2, 2, 1, a^3b^5, No], [2, 3, 0, a^2b^5, No], [3, 1, 1, a^4b^5, No], [3, 2, 0, a^3b^5, No], [4, 1, 0, a^4b^5, No]], 15, 15, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (5, a^5b^5, [[0, 1, 4, a^4b^5, No], [0, 2, 3, a^3b^5, No], [0, 3, 2, a^2b^5, No], [0, 4, 1, a^1b^5, No], [0, 5, 0, a^0b^5, No], [1, 1, 3, a^4b^5, No], [1, 2, 2, a^3b^5, No], [1, 3, 1, a^2b^5, No], [1, 4, 0, a^1b^5, No], [2, 1, 2, a^4b^5, No], [2, 2, 1, a^3b^5, No], [2, 3, 0, a^2b^5, No], [3, 1, 1, a^4b^5, No], [3, 2, 0, a^3b^5, No], [4, 1, 0, a^4b^5, No]], 15, 15, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0056","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fd1653949aa12324a0b86dec5a6ec52f5570d934ad0d08bf1a39c47b4077d67","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187056,"semantic_template":"pump-anbn-all-splits|{p:6}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=6 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(6, a^6b^6, [[0, 1, 5, a^5b^6, No], [0, 2, 4, a^4b^6, No], [0, 3, 3, a^3b^6, No], [0, 4, 2, a^2b^6, No], [0, 5, 1, a^1b^6, No], [0, 6, 0, a^0b^6, No], [1, 1, 4, a^5b^6, No], [1, 2, 3, a^4b^6, No], [1, 3, 2, a^3b^6, No], [1, 4, 1, a^2b^6, No], [1, 5, 0, a^1b^6, No], [2, 1, 3, a^5b^6, No], [2, 2, 2, a^4b^6, No], [2, 3, 1, a^3b^6, No], [2, 4, 0, a^2b^6, No], [3, 1, 2, a^5b^6, No], [3, 2, 1, a^4b^6, No], [3, 3, 0, a^3b^6, No], [4, 1, 1, a^5b^6, No], [4, 2, 0, a^4b^6, No], [5, 1, 0, a^5b^6, No]], 21, 21, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (6, a^6b^6, [[0, 1, 5, a^5b^6, No], [0, 2, 4, a^4b^6, No], [0, 3, 3, a^3b^6, No], [0, 4, 2, a^2b^6, No], [0, 5, 1, a^1b^6, No], [0, 6, 0, a^0b^6, No], [1, 1, 4, a^5b^6, No], [1, 2, 3, a^4b^6, No], [1, 3, 2, a^3b^6, No], [1, 4, 1, a^2b^6, No], [1, 5, 0, a^1b^6, No], [2, 1, 3, a^5b^6, No], [2, 2, 2, a^4b^6, No], [2, 3, 1, a^3b^6, No], [2, 4, 0, a^2b^6, No], [3, 1, 2, a^5b^6, No], [3, 2, 1, a^4b^6, No], [3, 3, 0, a^3b^6, No], [4, 1, 1, a^5b^6, No], [4, 2, 0, a^4b^6, No], [5, 1, 0, a^5b^6, No]], 21, 21, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0057","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3825e4ecde67670fed5b1752bd5361a75f9fcd61d931f90384748ed94bc86880","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187057,"semantic_template":"pump-anbn-all-splits|{p:7}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=7 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(7, a^7b^7, [[0, 1, 6, a^6b^7, No], [0, 2, 5, a^5b^7, No], [0, 3, 4, a^4b^7, No], [0, 4, 3, a^3b^7, No], [0, 5, 2, a^2b^7, No], [0, 6, 1, a^1b^7, No], [0, 7, 0, a^0b^7, No], [1, 1, 5, a^6b^7, No], [1, 2, 4, a^5b^7, No], [1, 3, 3, a^4b^7, No], [1, 4, 2, a^3b^7, No], [1, 5, 1, a^2b^7, No], [1, 6, 0, a^1b^7, No], [2, 1, 4, a^6b^7, No], [2, 2, 3, a^5b^7, No], [2, 3, 2, a^4b^7, No], [2, 4, 1, a^3b^7, No], [2, 5, 0, a^2b^7, No], [3, 1, 3, a^6b^7, No], [3, 2, 2, a^5b^7, No], [3, 3, 1, a^4b^7, No], [3, 4, 0, a^3b^7, No], [4, 1, 2, a^6b^7, No], [4, 2, 1, a^5b^7, No], [4, 3, 0, a^4b^7, No], [5, 1, 1, a^6b^7, No], [5, 2, 0, a^5b^7, No], [6, 1, 0, a^6b^7, No]], 28, 28, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (7, a^7b^7, [[0, 1, 6, a^6b^7, No], [0, 2, 5, a^5b^7, No], [0, 3, 4, a^4b^7, No], [0, 4, 3, a^3b^7, No], [0, 5, 2, a^2b^7, No], [0, 6, 1, a^1b^7, No], [0, 7, 0, a^0b^7, No], [1, 1, 5, a^6b^7, No], [1, 2, 4, a^5b^7, No], [1, 3, 3, a^4b^7, No], [1, 4, 2, a^3b^7, No], [1, 5, 1, a^2b^7, No], [1, 6, 0, a^1b^7, No], [2, 1, 4, a^6b^7, No], [2, 2, 3, a^5b^7, No], [2, 3, 2, a^4b^7, No], [2, 4, 1, a^3b^7, No], [2, 5, 0, a^2b^7, No], [3, 1, 3, a^6b^7, No], [3, 2, 2, a^5b^7, No], [3, 3, 1, a^4b^7, No], [3, 4, 0, a^3b^7, No], [4, 1, 2, a^6b^7, No], [4, 2, 1, a^5b^7, No], [4, 3, 0, a^4b^7, No], [5, 1, 1, a^6b^7, No], [5, 2, 0, a^5b^7, No], [6, 1, 0, a^6b^7, No]], 28, 28, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0058","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54ca70608caa6c6e184fe8010e6f96cc47ee7493284519871938e38730d7ed4d","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187058,"semantic_template":"pump-anbn-all-splits|{p:8}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=8 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(8, a^8b^8, [[0, 1, 7, a^7b^8, No], [0, 2, 6, a^6b^8, No], [0, 3, 5, a^5b^8, No], [0, 4, 4, a^4b^8, No], [0, 5, 3, a^3b^8, No], [0, 6, 2, a^2b^8, No], [0, 7, 1, a^1b^8, No], [0, 8, 0, a^0b^8, No], [1, 1, 6, a^7b^8, No], [1, 2, 5, a^6b^8, No], [1, 3, 4, a^5b^8, No], [1, 4, 3, a^4b^8, No], [1, 5, 2, a^3b^8, No], [1, 6, 1, a^2b^8, No], [1, 7, 0, a^1b^8, No], [2, 1, 5, a^7b^8, No], [2, 2, 4, a^6b^8, No], [2, 3, 3, a^5b^8, No], [2, 4, 2, a^4b^8, No], [2, 5, 1, a^3b^8, No], [2, 6, 0, a^2b^8, No], [3, 1, 4, a^7b^8, No], [3, 2, 3, a^6b^8, No], [3, 3, 2, a^5b^8, No], [3, 4, 1, a^4b^8, No], [3, 5, 0, a^3b^8, No], [4, 1, 3, a^7b^8, No], [4, 2, 2, a^6b^8, No], [4, 3, 1, a^5b^8, No], [4, 4, 0, a^4b^8, No], [5, 1, 2, a^7b^8, No], [5, 2, 1, a^6b^8, No], [5, 3, 0, a^5b^8, No], [6, 1, 1, a^7b^8, No], [6, 2, 0, a^6b^8, No], [7, 1, 0, a^7b^8, No]], 36, 36, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (8, a^8b^8, [[0, 1, 7, a^7b^8, No], [0, 2, 6, a^6b^8, No], [0, 3, 5, a^5b^8, No], [0, 4, 4, a^4b^8, No], [0, 5, 3, a^3b^8, No], [0, 6, 2, a^2b^8, No], [0, 7, 1, a^1b^8, No], [0, 8, 0, a^0b^8, No], [1, 1, 6, a^7b^8, No], [1, 2, 5, a^6b^8, No], [1, 3, 4, a^5b^8, No], [1, 4, 3, a^4b^8, No], [1, 5, 2, a^3b^8, No], [1, 6, 1, a^2b^8, No], [1, 7, 0, a^1b^8, No], [2, 1, 5, a^7b^8, No], [2, 2, 4, a^6b^8, No], [2, 3, 3, a^5b^8, No], [2, 4, 2, a^4b^8, No], [2, 5, 1, a^3b^8, No], [2, 6, 0, a^2b^8, No], [3, 1, 4, a^7b^8, No], [3, 2, 3, a^6b^8, No], [3, 3, 2, a^5b^8, No], [3, 4, 1, a^4b^8, No], [3, 5, 0, a^3b^8, No], [4, 1, 3, a^7b^8, No], [4, 2, 2, a^6b^8, No], [4, 3, 1, a^5b^8, No], [4, 4, 0, a^4b^8, No], [5, 1, 2, a^7b^8, No], [5, 2, 1, a^6b^8, No], [5, 3, 0, a^5b^8, No], [6, 1, 1, a^7b^8, No], [6, 2, 0, a^6b^8, No], [7, 1, 0, a^7b^8, No]], 36, 36, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0059","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db370f9f549452c5b57fb993e22d7f234c86c4a82f7c1b04aeda6cb93607df36","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187059,"semantic_template":"pump-anbn-all-splits|{p:9}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=9 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(9, a^9b^9, [[0, 1, 8, a^8b^9, No], [0, 2, 7, a^7b^9, No], [0, 3, 6, a^6b^9, No], [0, 4, 5, a^5b^9, No], [0, 5, 4, a^4b^9, No], [0, 6, 3, a^3b^9, No], [0, 7, 2, a^2b^9, No], [0, 8, 1, a^1b^9, No], [0, 9, 0, a^0b^9, No], [1, 1, 7, a^8b^9, No], [1, 2, 6, a^7b^9, No], [1, 3, 5, a^6b^9, No], [1, 4, 4, a^5b^9, No], [1, 5, 3, a^4b^9, No], [1, 6, 2, a^3b^9, No], [1, 7, 1, a^2b^9, No], [1, 8, 0, a^1b^9, No], [2, 1, 6, a^8b^9, No], [2, 2, 5, a^7b^9, No], [2, 3, 4, a^6b^9, No], [2, 4, 3, a^5b^9, No], [2, 5, 2, a^4b^9, No], [2, 6, 1, a^3b^9, No], [2, 7, 0, a^2b^9, No], [3, 1, 5, a^8b^9, No], [3, 2, 4, a^7b^9, No], [3, 3, 3, a^6b^9, No], [3, 4, 2, a^5b^9, No], [3, 5, 1, a^4b^9, No], [3, 6, 0, a^3b^9, No], [4, 1, 4, a^8b^9, No], [4, 2, 3, a^7b^9, No], [4, 3, 2, a^6b^9, No], [4, 4, 1, a^5b^9, No], [4, 5, 0, a^4b^9, No], [5, 1, 3, a^8b^9, No], [5, 2, 2, a^7b^9, No], [5, 3, 1, a^6b^9, No], [5, 4, 0, a^5b^9, No], [6, 1, 2, a^8b^9, No], [6, 2, 1, a^7b^9, No], [6, 3, 0, a^6b^9, No], [7, 1, 1, a^8b^9, No], [7, 2, 0, a^7b^9, No], [8, 1, 0, a^8b^9, No]], 45, 45, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (9, a^9b^9, [[0, 1, 8, a^8b^9, No], [0, 2, 7, a^7b^9, No], [0, 3, 6, a^6b^9, No], [0, 4, 5, a^5b^9, No], [0, 5, 4, a^4b^9, No], [0, 6, 3, a^3b^9, No], [0, 7, 2, a^2b^9, No], [0, 8, 1, a^1b^9, No], [0, 9, 0, a^0b^9, No], [1, 1, 7, a^8b^9, No], [1, 2, 6, a^7b^9, No], [1, 3, 5, a^6b^9, No], [1, 4, 4, a^5b^9, No], [1, 5, 3, a^4b^9, No], [1, 6, 2, a^3b^9, No], [1, 7, 1, a^2b^9, No], [1, 8, 0, a^1b^9, No], [2, 1, 6, a^8b^9, No], [2, 2, 5, a^7b^9, No], [2, 3, 4, a^6b^9, No], [2, 4, 3, a^5b^9, No], [2, 5, 2, a^4b^9, No], [2, 6, 1, a^3b^9, No], [2, 7, 0, a^2b^9, No], [3, 1, 5, a^8b^9, No], [3, 2, 4, a^7b^9, No], [3, 3, 3, a^6b^9, No], [3, 4, 2, a^5b^9, No], [3, 5, 1, a^4b^9, No], [3, 6, 0, a^3b^9, No], [4, 1, 4, a^8b^9, No], [4, 2, 3, a^7b^9, No], [4, 3, 2, a^6b^9, No], [4, 4, 1, a^5b^9, No], [4, 5, 0, a^4b^9, No], [5, 1, 3, a^8b^9, No], [5, 2, 2, a^7b^9, No], [5, 3, 1, a^6b^9, No], [5, 4, 0, a^5b^9, No], [6, 1, 2, a^8b^9, No], [6, 2, 1, a^7b^9, No], [6, 3, 0, a^6b^9, No], [7, 1, 1, a^8b^9, No], [7, 2, 0, a^7b^9, No], [8, 1, 0, a^8b^9, No]], 45, 45, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0060","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3215f83b888de40fbf5321862181a67bf5bc66f6360dda962840943b99e35d69","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187060,"semantic_template":"pump-anbn-all-splits|{p:10}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=10 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(10, a^10b^10, [[0, 1, 9, a^9b^10, No], [0, 2, 8, a^8b^10, No], [0, 3, 7, a^7b^10, No], [0, 4, 6, a^6b^10, No], [0, 5, 5, a^5b^10, No], [0, 6, 4, a^4b^10, No], [0, 7, 3, a^3b^10, No], [0, 8, 2, a^2b^10, No], [0, 9, 1, a^1b^10, No], [0, 10, 0, a^0b^10, No], [1, 1, 8, a^9b^10, No], [1, 2, 7, a^8b^10, No], [1, 3, 6, a^7b^10, No], [1, 4, 5, a^6b^10, No], [1, 5, 4, a^5b^10, No], [1, 6, 3, a^4b^10, No], [1, 7, 2, a^3b^10, No], [1, 8, 1, a^2b^10, No], [1, 9, 0, a^1b^10, No], [2, 1, 7, a^9b^10, No], [2, 2, 6, a^8b^10, No], [2, 3, 5, a^7b^10, No], [2, 4, 4, a^6b^10, No], [2, 5, 3, a^5b^10, No], [2, 6, 2, a^4b^10, No], [2, 7, 1, a^3b^10, No], [2, 8, 0, a^2b^10, No], [3, 1, 6, a^9b^10, No], [3, 2, 5, a^8b^10, No], [3, 3, 4, a^7b^10, No], [3, 4, 3, a^6b^10, No], [3, 5, 2, a^5b^10, No], [3, 6, 1, a^4b^10, No], [3, 7, 0, a^3b^10, No], [4, 1, 5, a^9b^10, No], [4, 2, 4, a^8b^10, No], [4, 3, 3, a^7b^10, No], [4, 4, 2, a^6b^10, No], [4, 5, 1, a^5b^10, No], [4, 6, 0, a^4b^10, No], [5, 1, 4, a^9b^10, No], [5, 2, 3, a^8b^10, No], [5, 3, 2, a^7b^10, No], [5, 4, 1, a^6b^10, No], [5, 5, 0, a^5b^10, No], [6, 1, 3, a^9b^10, No], [6, 2, 2, a^8b^10, No], [6, 3, 1, a^7b^10, No], [6, 4, 0, a^6b^10, No], [7, 1, 2, a^9b^10, No], [7, 2, 1, a^8b^10, No], [7, 3, 0, a^7b^10, No], [8, 1, 1, a^9b^10, No], [8, 2, 0, a^8b^10, No], [9, 1, 0, a^9b^10, No]], 55, 55, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (10, a^10b^10, [[0, 1, 9, a^9b^10, No], [0, 2, 8, a^8b^10, No], [0, 3, 7, a^7b^10, No], [0, 4, 6, a^6b^10, No], [0, 5, 5, a^5b^10, No], [0, 6, 4, a^4b^10, No], [0, 7, 3, a^3b^10, No], [0, 8, 2, a^2b^10, No], [0, 9, 1, a^1b^10, No], [0, 10, 0, a^0b^10, No], [1, 1, 8, a^9b^10, No], [1, 2, 7, a^8b^10, No], [1, 3, 6, a^7b^10, No], [1, 4, 5, a^6b^10, No], [1, 5, 4, a^5b^10, No], [1, 6, 3, a^4b^10, No], [1, 7, 2, a^3b^10, No], [1, 8, 1, a^2b^10, No], [1, 9, 0, a^1b^10, No], [2, 1, 7, a^9b^10, No], [2, 2, 6, a^8b^10, No], [2, 3, 5, a^7b^10, No], [2, 4, 4, a^6b^10, No], [2, 5, 3, a^5b^10, No], [2, 6, 2, a^4b^10, No], [2, 7, 1, a^3b^10, No], [2, 8, 0, a^2b^10, No], [3, 1, 6, a^9b^10, No], [3, 2, 5, a^8b^10, No], [3, 3, 4, a^7b^10, No], [3, 4, 3, a^6b^10, No], [3, 5, 2, a^5b^10, No], [3, 6, 1, a^4b^10, No], [3, 7, 0, a^3b^10, No], [4, 1, 5, a^9b^10, No], [4, 2, 4, a^8b^10, No], [4, 3, 3, a^7b^10, No], [4, 4, 2, a^6b^10, No], [4, 5, 1, a^5b^10, No], [4, 6, 0, a^4b^10, No], [5, 1, 4, a^9b^10, No], [5, 2, 3, a^8b^10, No], [5, 3, 2, a^7b^10, No], [5, 4, 1, a^6b^10, No], [5, 5, 0, a^5b^10, No], [6, 1, 3, a^9b^10, No], [6, 2, 2, a^8b^10, No], [6, 3, 1, a^7b^10, No], [6, 4, 0, a^6b^10, No], [7, 1, 2, a^9b^10, No], [7, 2, 1, a^8b^10, No], [7, 3, 0, a^7b^10, No], [8, 1, 1, a^9b^10, No], [8, 2, 0, a^8b^10, No], [9, 1, 0, a^9b^10, No]], 55, 55, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0061","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33ccbdd10927719a77028b365df41173e0c7253f2ce72232edba2c9cf6197b2f","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187061,"semantic_template":"pump-anbn-all-splits|{p:11}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=11 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(11, a^11b^11, [[0, 1, 10, a^10b^11, No], [0, 2, 9, a^9b^11, No], [0, 3, 8, a^8b^11, No], [0, 4, 7, a^7b^11, No], [0, 5, 6, a^6b^11, No], [0, 6, 5, a^5b^11, No], [0, 7, 4, a^4b^11, No], [0, 8, 3, a^3b^11, No], [0, 9, 2, a^2b^11, No], [0, 10, 1, a^1b^11, No], [0, 11, 0, a^0b^11, No], [1, 1, 9, a^10b^11, No], [1, 2, 8, a^9b^11, No], [1, 3, 7, a^8b^11, No], [1, 4, 6, a^7b^11, No], [1, 5, 5, a^6b^11, No], [1, 6, 4, a^5b^11, No], [1, 7, 3, a^4b^11, No], [1, 8, 2, a^3b^11, No], [1, 9, 1, a^2b^11, No], [1, 10, 0, a^1b^11, No], [2, 1, 8, a^10b^11, No], [2, 2, 7, a^9b^11, No], [2, 3, 6, a^8b^11, No], [2, 4, 5, a^7b^11, No], [2, 5, 4, a^6b^11, No], [2, 6, 3, a^5b^11, No], [2, 7, 2, a^4b^11, No], [2, 8, 1, a^3b^11, No], [2, 9, 0, a^2b^11, No], [3, 1, 7, a^10b^11, No], [3, 2, 6, a^9b^11, No], [3, 3, 5, a^8b^11, No], [3, 4, 4, a^7b^11, No], [3, 5, 3, a^6b^11, No], [3, 6, 2, a^5b^11, No], [3, 7, 1, a^4b^11, No], [3, 8, 0, a^3b^11, No], [4, 1, 6, a^10b^11, No], [4, 2, 5, a^9b^11, No], [4, 3, 4, a^8b^11, No], [4, 4, 3, a^7b^11, No], [4, 5, 2, a^6b^11, No], [4, 6, 1, a^5b^11, No], [4, 7, 0, a^4b^11, No], [5, 1, 5, a^10b^11, No], [5, 2, 4, a^9b^11, No], [5, 3, 3, a^8b^11, No], [5, 4, 2, a^7b^11, No], [5, 5, 1, a^6b^11, No], [5, 6, 0, a^5b^11, No], [6, 1, 4, a^10b^11, No], [6, 2, 3, a^9b^11, No], [6, 3, 2, a^8b^11, No], [6, 4, 1, a^7b^11, No], [6, 5, 0, a^6b^11, No], [7, 1, 3, a^10b^11, No], [7, 2, 2, a^9b^11, No], [7, 3, 1, a^8b^11, No], [7, 4, 0, a^7b^11, No], [8, 1, 2, a^10b^11, No], [8, 2, 1, a^9b^11, No], [8, 3, 0, a^8b^11, No], [9, 1, 1, a^10b^11, No], [9, 2, 0, a^9b^11, No], [10, 1, 0, a^10b^11, No]], 66, 66, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (11, a^11b^11, [[0, 1, 10, a^10b^11, No], [0, 2, 9, a^9b^11, No], [0, 3, 8, a^8b^11, No], [0, 4, 7, a^7b^11, No], [0, 5, 6, a^6b^11, No], [0, 6, 5, a^5b^11, No], [0, 7, 4, a^4b^11, No], [0, 8, 3, a^3b^11, No], [0, 9, 2, a^2b^11, No], [0, 10, 1, a^1b^11, No], [0, 11, 0, a^0b^11, No], [1, 1, 9, a^10b^11, No], [1, 2, 8, a^9b^11, No], [1, 3, 7, a^8b^11, No], [1, 4, 6, a^7b^11, No], [1, 5, 5, a^6b^11, No], [1, 6, 4, a^5b^11, No], [1, 7, 3, a^4b^11, No], [1, 8, 2, a^3b^11, No], [1, 9, 1, a^2b^11, No], [1, 10, 0, a^1b^11, No], [2, 1, 8, a^10b^11, No], [2, 2, 7, a^9b^11, No], [2, 3, 6, a^8b^11, No], [2, 4, 5, a^7b^11, No], [2, 5, 4, a^6b^11, No], [2, 6, 3, a^5b^11, No], [2, 7, 2, a^4b^11, No], [2, 8, 1, a^3b^11, No], [2, 9, 0, a^2b^11, No], [3, 1, 7, a^10b^11, No], [3, 2, 6, a^9b^11, No], [3, 3, 5, a^8b^11, No], [3, 4, 4, a^7b^11, No], [3, 5, 3, a^6b^11, No], [3, 6, 2, a^5b^11, No], [3, 7, 1, a^4b^11, No], [3, 8, 0, a^3b^11, No], [4, 1, 6, a^10b^11, No], [4, 2, 5, a^9b^11, No], [4, 3, 4, a^8b^11, No], [4, 4, 3, a^7b^11, No], [4, 5, 2, a^6b^11, No], [4, 6, 1, a^5b^11, No], [4, 7, 0, a^4b^11, No], [5, 1, 5, a^10b^11, No], [5, 2, 4, a^9b^11, No], [5, 3, 3, a^8b^11, No], [5, 4, 2, a^7b^11, No], [5, 5, 1, a^6b^11, No], [5, 6, 0, a^5b^11, No], [6, 1, 4, a^10b^11, No], [6, 2, 3, a^9b^11, No], [6, 3, 2, a^8b^11, No], [6, 4, 1, a^7b^11, No], [6, 5, 0, a^6b^11, No], [7, 1, 3, a^10b^11, No], [7, 2, 2, a^9b^11, No], [7, 3, 1, a^8b^11, No], [7, 4, 0, a^7b^11, No], [8, 1, 2, a^10b^11, No], [8, 2, 1, a^9b^11, No], [8, 3, 0, a^8b^11, No], [9, 1, 1, a^10b^11, No], [9, 2, 0, a^9b^11, No], [10, 1, 0, a^10b^11, No]], 66, 66, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0062","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a61d3b71d6676f1c5b0064d7e9328c65883ab93e9de7b24783d412c62bc4a3b3","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187062,"semantic_template":"pump-anbn-all-splits|{p:12}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=12 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(12, a^12b^12, [[0, 1, 11, a^11b^12, No], [0, 2, 10, a^10b^12, No], [0, 3, 9, a^9b^12, No], [0, 4, 8, a^8b^12, No], [0, 5, 7, a^7b^12, No], [0, 6, 6, a^6b^12, No], [0, 7, 5, a^5b^12, No], [0, 8, 4, a^4b^12, No], [0, 9, 3, a^3b^12, No], [0, 10, 2, a^2b^12, No], [0, 11, 1, a^1b^12, No], [0, 12, 0, a^0b^12, No], [1, 1, 10, a^11b^12, No], [1, 2, 9, a^10b^12, No], [1, 3, 8, a^9b^12, No], [1, 4, 7, a^8b^12, No], [1, 5, 6, a^7b^12, No], [1, 6, 5, a^6b^12, No], [1, 7, 4, a^5b^12, No], [1, 8, 3, a^4b^12, No], [1, 9, 2, a^3b^12, No], [1, 10, 1, a^2b^12, No], [1, 11, 0, a^1b^12, No], [2, 1, 9, a^11b^12, No], [2, 2, 8, a^10b^12, No], [2, 3, 7, a^9b^12, No], [2, 4, 6, a^8b^12, No], [2, 5, 5, a^7b^12, No], [2, 6, 4, a^6b^12, No], [2, 7, 3, a^5b^12, No], [2, 8, 2, a^4b^12, No], [2, 9, 1, a^3b^12, No], [2, 10, 0, a^2b^12, No], [3, 1, 8, a^11b^12, No], [3, 2, 7, a^10b^12, No], [3, 3, 6, a^9b^12, No], [3, 4, 5, a^8b^12, No], [3, 5, 4, a^7b^12, No], [3, 6, 3, a^6b^12, No], [3, 7, 2, a^5b^12, No], [3, 8, 1, a^4b^12, No], [3, 9, 0, a^3b^12, No], [4, 1, 7, a^11b^12, No], [4, 2, 6, a^10b^12, No], [4, 3, 5, a^9b^12, No], [4, 4, 4, a^8b^12, No], [4, 5, 3, a^7b^12, No], [4, 6, 2, a^6b^12, No], [4, 7, 1, a^5b^12, No], [4, 8, 0, a^4b^12, No], [5, 1, 6, a^11b^12, No], [5, 2, 5, a^10b^12, No], [5, 3, 4, a^9b^12, No], [5, 4, 3, a^8b^12, No], [5, 5, 2, a^7b^12, No], [5, 6, 1, a^6b^12, No], [5, 7, 0, a^5b^12, No], [6, 1, 5, a^11b^12, No], [6, 2, 4, a^10b^12, No], [6, 3, 3, a^9b^12, No], [6, 4, 2, a^8b^12, No], [6, 5, 1, a^7b^12, No], [6, 6, 0, a^6b^12, No], [7, 1, 4, a^11b^12, No], [7, 2, 3, a^10b^12, No], [7, 3, 2, a^9b^12, No], [7, 4, 1, a^8b^12, No], [7, 5, 0, a^7b^12, No], [8, 1, 3, a^11b^12, No], [8, 2, 2, a^10b^12, No], [8, 3, 1, a^9b^12, No], [8, 4, 0, a^8b^12, No], [9, 1, 2, a^11b^12, No], [9, 2, 1, a^10b^12, No], [9, 3, 0, a^9b^12, No], [10, 1, 1, a^11b^12, No], [10, 2, 0, a^10b^12, No], [11, 1, 0, a^11b^12, No]], 78, 78, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (12, a^12b^12, [[0, 1, 11, a^11b^12, No], [0, 2, 10, a^10b^12, No], [0, 3, 9, a^9b^12, No], [0, 4, 8, a^8b^12, No], [0, 5, 7, a^7b^12, No], [0, 6, 6, a^6b^12, No], [0, 7, 5, a^5b^12, No], [0, 8, 4, a^4b^12, No], [0, 9, 3, a^3b^12, No], [0, 10, 2, a^2b^12, No], [0, 11, 1, a^1b^12, No], [0, 12, 0, a^0b^12, No], [1, 1, 10, a^11b^12, No], [1, 2, 9, a^10b^12, No], [1, 3, 8, a^9b^12, No], [1, 4, 7, a^8b^12, No], [1, 5, 6, a^7b^12, No], [1, 6, 5, a^6b^12, No], [1, 7, 4, a^5b^12, No], [1, 8, 3, a^4b^12, No], [1, 9, 2, a^3b^12, No], [1, 10, 1, a^2b^12, No], [1, 11, 0, a^1b^12, No], [2, 1, 9, a^11b^12, No], [2, 2, 8, a^10b^12, No], [2, 3, 7, a^9b^12, No], [2, 4, 6, a^8b^12, No], [2, 5, 5, a^7b^12, No], [2, 6, 4, a^6b^12, No], [2, 7, 3, a^5b^12, No], [2, 8, 2, a^4b^12, No], [2, 9, 1, a^3b^12, No], [2, 10, 0, a^2b^12, No], [3, 1, 8, a^11b^12, No], [3, 2, 7, a^10b^12, No], [3, 3, 6, a^9b^12, No], [3, 4, 5, a^8b^12, No], [3, 5, 4, a^7b^12, No], [3, 6, 3, a^6b^12, No], [3, 7, 2, a^5b^12, No], [3, 8, 1, a^4b^12, No], [3, 9, 0, a^3b^12, No], [4, 1, 7, a^11b^12, No], [4, 2, 6, a^10b^12, No], [4, 3, 5, a^9b^12, No], [4, 4, 4, a^8b^12, No], [4, 5, 3, a^7b^12, No], [4, 6, 2, a^6b^12, No], [4, 7, 1, a^5b^12, No], [4, 8, 0, a^4b^12, No], [5, 1, 6, a^11b^12, No], [5, 2, 5, a^10b^12, No], [5, 3, 4, a^9b^12, No], [5, 4, 3, a^8b^12, No], [5, 5, 2, a^7b^12, No], [5, 6, 1, a^6b^12, No], [5, 7, 0, a^5b^12, No], [6, 1, 5, a^11b^12, No], [6, 2, 4, a^10b^12, No], [6, 3, 3, a^9b^12, No], [6, 4, 2, a^8b^12, No], [6, 5, 1, a^7b^12, No], [6, 6, 0, a^6b^12, No], [7, 1, 4, a^11b^12, No], [7, 2, 3, a^10b^12, No], [7, 3, 2, a^9b^12, No], [7, 4, 1, a^8b^12, No], [7, 5, 0, a^7b^12, No], [8, 1, 3, a^11b^12, No], [8, 2, 2, a^10b^12, No], [8, 3, 1, a^9b^12, No], [8, 4, 0, a^8b^12, No], [9, 1, 2, a^11b^12, No], [9, 2, 1, a^10b^12, No], [9, 3, 0, a^9b^12, No], [10, 1, 1, a^11b^12, No], [10, 2, 0, a^10b^12, No], [11, 1, 0, a^11b^12, No]], 78, 78, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0063","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e8dacaa34685d9f25b925f55ea930d7f5c8cbc7978379dceddafdb4ae9ae9d3","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187063,"semantic_template":"pump-anbn-all-splits|{p:13}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=13 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(13, a^13b^13, [[0, 1, 12, a^12b^13, No], [0, 2, 11, a^11b^13, No], [0, 3, 10, a^10b^13, No], [0, 4, 9, a^9b^13, No], [0, 5, 8, a^8b^13, No], [0, 6, 7, a^7b^13, No], [0, 7, 6, a^6b^13, No], [0, 8, 5, a^5b^13, No], [0, 9, 4, a^4b^13, No], [0, 10, 3, a^3b^13, No], [0, 11, 2, a^2b^13, No], [0, 12, 1, a^1b^13, No], [0, 13, 0, a^0b^13, No], [1, 1, 11, a^12b^13, No], [1, 2, 10, a^11b^13, No], [1, 3, 9, a^10b^13, No], [1, 4, 8, a^9b^13, No], [1, 5, 7, a^8b^13, No], [1, 6, 6, a^7b^13, No], [1, 7, 5, a^6b^13, No], [1, 8, 4, a^5b^13, No], [1, 9, 3, a^4b^13, No], [1, 10, 2, a^3b^13, No], [1, 11, 1, a^2b^13, No], [1, 12, 0, a^1b^13, No], [2, 1, 10, a^12b^13, No], [2, 2, 9, a^11b^13, No], [2, 3, 8, a^10b^13, No], [2, 4, 7, a^9b^13, No], [2, 5, 6, a^8b^13, No], [2, 6, 5, a^7b^13, No], [2, 7, 4, a^6b^13, No], [2, 8, 3, a^5b^13, No], [2, 9, 2, a^4b^13, No], [2, 10, 1, a^3b^13, No], [2, 11, 0, a^2b^13, No], [3, 1, 9, a^12b^13, No], [3, 2, 8, a^11b^13, No], [3, 3, 7, a^10b^13, No], [3, 4, 6, a^9b^13, No], [3, 5, 5, a^8b^13, No], [3, 6, 4, a^7b^13, No], [3, 7, 3, a^6b^13, No], [3, 8, 2, a^5b^13, No], [3, 9, 1, a^4b^13, No], [3, 10, 0, a^3b^13, No], [4, 1, 8, a^12b^13, No], [4, 2, 7, a^11b^13, No], [4, 3, 6, a^10b^13, No], [4, 4, 5, a^9b^13, No], [4, 5, 4, a^8b^13, No], [4, 6, 3, a^7b^13, No], [4, 7, 2, a^6b^13, No], [4, 8, 1, a^5b^13, No], [4, 9, 0, a^4b^13, No], [5, 1, 7, a^12b^13, No], [5, 2, 6, a^11b^13, No], [5, 3, 5, a^10b^13, No], [5, 4, 4, a^9b^13, No], [5, 5, 3, a^8b^13, No], [5, 6, 2, a^7b^13, No], [5, 7, 1, a^6b^13, No], [5, 8, 0, a^5b^13, No], [6, 1, 6, a^12b^13, No], [6, 2, 5, a^11b^13, No], [6, 3, 4, a^10b^13, No], [6, 4, 3, a^9b^13, No], [6, 5, 2, a^8b^13, No], [6, 6, 1, a^7b^13, No], [6, 7, 0, a^6b^13, No], [7, 1, 5, a^12b^13, No], [7, 2, 4, a^11b^13, No], [7, 3, 3, a^10b^13, No], [7, 4, 2, a^9b^13, No], [7, 5, 1, a^8b^13, No], [7, 6, 0, a^7b^13, No], [8, 1, 4, a^12b^13, No], [8, 2, 3, a^11b^13, No], [8, 3, 2, a^10b^13, No], [8, 4, 1, a^9b^13, No], [8, 5, 0, a^8b^13, No], [9, 1, 3, a^12b^13, No], [9, 2, 2, a^11b^13, No], [9, 3, 1, a^10b^13, No], [9, 4, 0, a^9b^13, No], [10, 1, 2, a^12b^13, No], [10, 2, 1, a^11b^13, No], [10, 3, 0, a^10b^13, No], [11, 1, 1, a^12b^13, No], [11, 2, 0, a^11b^13, No], [12, 1, 0, a^12b^13, No]], 91, 91, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (13, a^13b^13, [[0, 1, 12, a^12b^13, No], [0, 2, 11, a^11b^13, No], [0, 3, 10, a^10b^13, No], [0, 4, 9, a^9b^13, No], [0, 5, 8, a^8b^13, No], [0, 6, 7, a^7b^13, No], [0, 7, 6, a^6b^13, No], [0, 8, 5, a^5b^13, No], [0, 9, 4, a^4b^13, No], [0, 10, 3, a^3b^13, No], [0, 11, 2, a^2b^13, No], [0, 12, 1, a^1b^13, No], [0, 13, 0, a^0b^13, No], [1, 1, 11, a^12b^13, No], [1, 2, 10, a^11b^13, No], [1, 3, 9, a^10b^13, No], [1, 4, 8, a^9b^13, No], [1, 5, 7, a^8b^13, No], [1, 6, 6, a^7b^13, No], [1, 7, 5, a^6b^13, No], [1, 8, 4, a^5b^13, No], [1, 9, 3, a^4b^13, No], [1, 10, 2, a^3b^13, No], [1, 11, 1, a^2b^13, No], [1, 12, 0, a^1b^13, No], [2, 1, 10, a^12b^13, No], [2, 2, 9, a^11b^13, No], [2, 3, 8, a^10b^13, No], [2, 4, 7, a^9b^13, No], [2, 5, 6, a^8b^13, No], [2, 6, 5, a^7b^13, No], [2, 7, 4, a^6b^13, No], [2, 8, 3, a^5b^13, No], [2, 9, 2, a^4b^13, No], [2, 10, 1, a^3b^13, No], [2, 11, 0, a^2b^13, No], [3, 1, 9, a^12b^13, No], [3, 2, 8, a^11b^13, No], [3, 3, 7, a^10b^13, No], [3, 4, 6, a^9b^13, No], [3, 5, 5, a^8b^13, No], [3, 6, 4, a^7b^13, No], [3, 7, 3, a^6b^13, No], [3, 8, 2, a^5b^13, No], [3, 9, 1, a^4b^13, No], [3, 10, 0, a^3b^13, No], [4, 1, 8, a^12b^13, No], [4, 2, 7, a^11b^13, No], [4, 3, 6, a^10b^13, No], [4, 4, 5, a^9b^13, No], [4, 5, 4, a^8b^13, No], [4, 6, 3, a^7b^13, No], [4, 7, 2, a^6b^13, No], [4, 8, 1, a^5b^13, No], [4, 9, 0, a^4b^13, No], [5, 1, 7, a^12b^13, No], [5, 2, 6, a^11b^13, No], [5, 3, 5, a^10b^13, No], [5, 4, 4, a^9b^13, No], [5, 5, 3, a^8b^13, No], [5, 6, 2, a^7b^13, No], [5, 7, 1, a^6b^13, No], [5, 8, 0, a^5b^13, No], [6, 1, 6, a^12b^13, No], [6, 2, 5, a^11b^13, No], [6, 3, 4, a^10b^13, No], [6, 4, 3, a^9b^13, No], [6, 5, 2, a^8b^13, No], [6, 6, 1, a^7b^13, No], [6, 7, 0, a^6b^13, No], [7, 1, 5, a^12b^13, No], [7, 2, 4, a^11b^13, No], [7, 3, 3, a^10b^13, No], [7, 4, 2, a^9b^13, No], [7, 5, 1, a^8b^13, No], [7, 6, 0, a^7b^13, No], [8, 1, 4, a^12b^13, No], [8, 2, 3, a^11b^13, No], [8, 3, 2, a^10b^13, No], [8, 4, 1, a^9b^13, No], [8, 5, 0, a^8b^13, No], [9, 1, 3, a^12b^13, No], [9, 2, 2, a^11b^13, No], [9, 3, 1, a^10b^13, No], [9, 4, 0, a^9b^13, No], [10, 1, 2, a^12b^13, No], [10, 2, 1, a^11b^13, No], [10, 3, 0, a^10b^13, No], [11, 1, 1, a^12b^13, No], [11, 2, 0, a^11b^13, No], [12, 1, 0, a^12b^13, No]], 91, 91, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0064","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57d70f8d9bbd534a8503655eae2ec254f8b78baea96576bc0c29d8ed37a5bcb1","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187064,"semantic_template":"pump-anbn-all-splits|{p:14}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=14 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(14, a^14b^14, [[0, 1, 13, a^13b^14, No], [0, 2, 12, a^12b^14, No], [0, 3, 11, a^11b^14, No], [0, 4, 10, a^10b^14, No], [0, 5, 9, a^9b^14, No], [0, 6, 8, a^8b^14, No], [0, 7, 7, a^7b^14, No], [0, 8, 6, a^6b^14, No], [0, 9, 5, a^5b^14, No], [0, 10, 4, a^4b^14, No], [0, 11, 3, a^3b^14, No], [0, 12, 2, a^2b^14, No], [0, 13, 1, a^1b^14, No], [0, 14, 0, a^0b^14, No], [1, 1, 12, a^13b^14, No], [1, 2, 11, a^12b^14, No], [1, 3, 10, a^11b^14, No], [1, 4, 9, a^10b^14, No], [1, 5, 8, a^9b^14, No], [1, 6, 7, a^8b^14, No], [1, 7, 6, a^7b^14, No], [1, 8, 5, a^6b^14, No], [1, 9, 4, a^5b^14, No], [1, 10, 3, a^4b^14, No], [1, 11, 2, a^3b^14, No], [1, 12, 1, a^2b^14, No], [1, 13, 0, a^1b^14, No], [2, 1, 11, a^13b^14, No], [2, 2, 10, a^12b^14, No], [2, 3, 9, a^11b^14, No], [2, 4, 8, a^10b^14, No], [2, 5, 7, a^9b^14, No], [2, 6, 6, a^8b^14, No], [2, 7, 5, a^7b^14, No], [2, 8, 4, a^6b^14, No], [2, 9, 3, a^5b^14, No], [2, 10, 2, a^4b^14, No], [2, 11, 1, a^3b^14, No], [2, 12, 0, a^2b^14, No], [3, 1, 10, a^13b^14, No], [3, 2, 9, a^12b^14, No], [3, 3, 8, a^11b^14, No], [3, 4, 7, a^10b^14, No], [3, 5, 6, a^9b^14, No], [3, 6, 5, a^8b^14, No], [3, 7, 4, a^7b^14, No], [3, 8, 3, a^6b^14, No], [3, 9, 2, a^5b^14, No], [3, 10, 1, a^4b^14, No], [3, 11, 0, a^3b^14, No], [4, 1, 9, a^13b^14, No], [4, 2, 8, a^12b^14, No], [4, 3, 7, a^11b^14, No], [4, 4, 6, a^10b^14, No], [4, 5, 5, a^9b^14, No], [4, 6, 4, a^8b^14, No], [4, 7, 3, a^7b^14, No], [4, 8, 2, a^6b^14, No], [4, 9, 1, a^5b^14, No], [4, 10, 0, a^4b^14, No], [5, 1, 8, a^13b^14, No], [5, 2, 7, a^12b^14, No], [5, 3, 6, a^11b^14, No], [5, 4, 5, a^10b^14, No], [5, 5, 4, a^9b^14, No], [5, 6, 3, a^8b^14, No], [5, 7, 2, a^7b^14, No], [5, 8, 1, a^6b^14, No], [5, 9, 0, a^5b^14, No], [6, 1, 7, a^13b^14, No], [6, 2, 6, a^12b^14, No], [6, 3, 5, a^11b^14, No], [6, 4, 4, a^10b^14, No], [6, 5, 3, a^9b^14, No], [6, 6, 2, a^8b^14, No], [6, 7, 1, a^7b^14, No], [6, 8, 0, a^6b^14, No], [7, 1, 6, a^13b^14, No], [7, 2, 5, a^12b^14, No], [7, 3, 4, a^11b^14, No], [7, 4, 3, a^10b^14, No], [7, 5, 2, a^9b^14, No], [7, 6, 1, a^8b^14, No], [7, 7, 0, a^7b^14, No], [8, 1, 5, a^13b^14, No], [8, 2, 4, a^12b^14, No], [8, 3, 3, a^11b^14, No], [8, 4, 2, a^10b^14, No], [8, 5, 1, a^9b^14, No], [8, 6, 0, a^8b^14, No], [9, 1, 4, a^13b^14, No], [9, 2, 3, a^12b^14, No], [9, 3, 2, a^11b^14, No], [9, 4, 1, a^10b^14, No], [9, 5, 0, a^9b^14, No], [10, 1, 3, a^13b^14, No], [10, 2, 2, a^12b^14, No], [10, 3, 1, a^11b^14, No], [10, 4, 0, a^10b^14, No], [11, 1, 2, a^13b^14, No], [11, 2, 1, a^12b^14, No], [11, 3, 0, a^11b^14, No], [12, 1, 1, a^13b^14, No], [12, 2, 0, a^12b^14, No], [13, 1, 0, a^13b^14, No]], 105, 105, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (14, a^14b^14, [[0, 1, 13, a^13b^14, No], [0, 2, 12, a^12b^14, No], [0, 3, 11, a^11b^14, No], [0, 4, 10, a^10b^14, No], [0, 5, 9, a^9b^14, No], [0, 6, 8, a^8b^14, No], [0, 7, 7, a^7b^14, No], [0, 8, 6, a^6b^14, No], [0, 9, 5, a^5b^14, No], [0, 10, 4, a^4b^14, No], [0, 11, 3, a^3b^14, No], [0, 12, 2, a^2b^14, No], [0, 13, 1, a^1b^14, No], [0, 14, 0, a^0b^14, No], [1, 1, 12, a^13b^14, No], [1, 2, 11, a^12b^14, No], [1, 3, 10, a^11b^14, No], [1, 4, 9, a^10b^14, No], [1, 5, 8, a^9b^14, No], [1, 6, 7, a^8b^14, No], [1, 7, 6, a^7b^14, No], [1, 8, 5, a^6b^14, No], [1, 9, 4, a^5b^14, No], [1, 10, 3, a^4b^14, No], [1, 11, 2, a^3b^14, No], [1, 12, 1, a^2b^14, No], [1, 13, 0, a^1b^14, No], [2, 1, 11, a^13b^14, No], [2, 2, 10, a^12b^14, No], [2, 3, 9, a^11b^14, No], [2, 4, 8, a^10b^14, No], [2, 5, 7, a^9b^14, No], [2, 6, 6, a^8b^14, No], [2, 7, 5, a^7b^14, No], [2, 8, 4, a^6b^14, No], [2, 9, 3, a^5b^14, No], [2, 10, 2, a^4b^14, No], [2, 11, 1, a^3b^14, No], [2, 12, 0, a^2b^14, No], [3, 1, 10, a^13b^14, No], [3, 2, 9, a^12b^14, No], [3, 3, 8, a^11b^14, No], [3, 4, 7, a^10b^14, No], [3, 5, 6, a^9b^14, No], [3, 6, 5, a^8b^14, No], [3, 7, 4, a^7b^14, No], [3, 8, 3, a^6b^14, No], [3, 9, 2, a^5b^14, No], [3, 10, 1, a^4b^14, No], [3, 11, 0, a^3b^14, No], [4, 1, 9, a^13b^14, No], [4, 2, 8, a^12b^14, No], [4, 3, 7, a^11b^14, No], [4, 4, 6, a^10b^14, No], [4, 5, 5, a^9b^14, No], [4, 6, 4, a^8b^14, No], [4, 7, 3, a^7b^14, No], [4, 8, 2, a^6b^14, No], [4, 9, 1, a^5b^14, No], [4, 10, 0, a^4b^14, No], [5, 1, 8, a^13b^14, No], [5, 2, 7, a^12b^14, No], [5, 3, 6, a^11b^14, No], [5, 4, 5, a^10b^14, No], [5, 5, 4, a^9b^14, No], [5, 6, 3, a^8b^14, No], [5, 7, 2, a^7b^14, No], [5, 8, 1, a^6b^14, No], [5, 9, 0, a^5b^14, No], [6, 1, 7, a^13b^14, No], [6, 2, 6, a^12b^14, No], [6, 3, 5, a^11b^14, No], [6, 4, 4, a^10b^14, No], [6, 5, 3, a^9b^14, No], [6, 6, 2, a^8b^14, No], [6, 7, 1, a^7b^14, No], [6, 8, 0, a^6b^14, No], [7, 1, 6, a^13b^14, No], [7, 2, 5, a^12b^14, No], [7, 3, 4, a^11b^14, No], [7, 4, 3, a^10b^14, No], [7, 5, 2, a^9b^14, No], [7, 6, 1, a^8b^14, No], [7, 7, 0, a^7b^14, No], [8, 1, 5, a^13b^14, No], [8, 2, 4, a^12b^14, No], [8, 3, 3, a^11b^14, No], [8, 4, 2, a^10b^14, No], [8, 5, 1, a^9b^14, No], [8, 6, 0, a^8b^14, No], [9, 1, 4, a^13b^14, No], [9, 2, 3, a^12b^14, No], [9, 3, 2, a^11b^14, No], [9, 4, 1, a^10b^14, No], [9, 5, 0, a^9b^14, No], [10, 1, 3, a^13b^14, No], [10, 2, 2, a^12b^14, No], [10, 3, 1, a^11b^14, No], [10, 4, 0, a^10b^14, No], [11, 1, 2, a^13b^14, No], [11, 2, 1, a^12b^14, No], [11, 3, 0, a^11b^14, No], [12, 1, 1, a^13b^14, No], [12, 2, 0, a^12b^14, No], [13, 1, 0, a^13b^14, No]], 105, 105, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0065","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a41a745d013ee86ba7b9b36625d21df06d46c8caf4617770b9cca861a1c31f1","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187065,"semantic_template":"pump-anbn-all-splits|{p:15}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=15 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(15, a^15b^15, [[0, 1, 14, a^14b^15, No], [0, 2, 13, a^13b^15, No], [0, 3, 12, a^12b^15, No], [0, 4, 11, a^11b^15, No], [0, 5, 10, a^10b^15, No], [0, 6, 9, a^9b^15, No], [0, 7, 8, a^8b^15, No], [0, 8, 7, a^7b^15, No], [0, 9, 6, a^6b^15, No], [0, 10, 5, a^5b^15, No], [0, 11, 4, a^4b^15, No], [0, 12, 3, a^3b^15, No], [0, 13, 2, a^2b^15, No], [0, 14, 1, a^1b^15, No], [0, 15, 0, a^0b^15, No], [1, 1, 13, a^14b^15, No], [1, 2, 12, a^13b^15, No], [1, 3, 11, a^12b^15, No], [1, 4, 10, a^11b^15, No], [1, 5, 9, a^10b^15, No], [1, 6, 8, a^9b^15, No], [1, 7, 7, a^8b^15, No], [1, 8, 6, a^7b^15, No], [1, 9, 5, a^6b^15, No], [1, 10, 4, a^5b^15, No], [1, 11, 3, a^4b^15, No], [1, 12, 2, a^3b^15, No], [1, 13, 1, a^2b^15, No], [1, 14, 0, a^1b^15, No], [2, 1, 12, a^14b^15, No], [2, 2, 11, a^13b^15, No], [2, 3, 10, a^12b^15, No], [2, 4, 9, a^11b^15, No], [2, 5, 8, a^10b^15, No], [2, 6, 7, a^9b^15, No], [2, 7, 6, a^8b^15, No], [2, 8, 5, a^7b^15, No], [2, 9, 4, a^6b^15, No], [2, 10, 3, a^5b^15, No], [2, 11, 2, a^4b^15, No], [2, 12, 1, a^3b^15, No], [2, 13, 0, a^2b^15, No], [3, 1, 11, a^14b^15, No], [3, 2, 10, a^13b^15, No], [3, 3, 9, a^12b^15, No], [3, 4, 8, a^11b^15, No], [3, 5, 7, a^10b^15, No], [3, 6, 6, a^9b^15, No], [3, 7, 5, a^8b^15, No], [3, 8, 4, a^7b^15, No], [3, 9, 3, a^6b^15, No], [3, 10, 2, a^5b^15, No], [3, 11, 1, a^4b^15, No], [3, 12, 0, a^3b^15, No], [4, 1, 10, a^14b^15, No], [4, 2, 9, a^13b^15, No], [4, 3, 8, a^12b^15, No], [4, 4, 7, a^11b^15, No], [4, 5, 6, a^10b^15, No], [4, 6, 5, a^9b^15, No], [4, 7, 4, a^8b^15, No], [4, 8, 3, a^7b^15, No], [4, 9, 2, a^6b^15, No], [4, 10, 1, a^5b^15, No], [4, 11, 0, a^4b^15, No], [5, 1, 9, a^14b^15, No], [5, 2, 8, a^13b^15, No], [5, 3, 7, a^12b^15, No], [5, 4, 6, a^11b^15, No], [5, 5, 5, a^10b^15, No], [5, 6, 4, a^9b^15, No], [5, 7, 3, a^8b^15, No], [5, 8, 2, a^7b^15, No], [5, 9, 1, a^6b^15, No], [5, 10, 0, a^5b^15, No], [6, 1, 8, a^14b^15, No], [6, 2, 7, a^13b^15, No], [6, 3, 6, a^12b^15, No], [6, 4, 5, a^11b^15, No], [6, 5, 4, a^10b^15, No], [6, 6, 3, a^9b^15, No], [6, 7, 2, a^8b^15, No], [6, 8, 1, a^7b^15, No], [6, 9, 0, a^6b^15, No], [7, 1, 7, a^14b^15, No], [7, 2, 6, a^13b^15, No], [7, 3, 5, a^12b^15, No], [7, 4, 4, a^11b^15, No], [7, 5, 3, a^10b^15, No], [7, 6, 2, a^9b^15, No], [7, 7, 1, a^8b^15, No], [7, 8, 0, a^7b^15, No], [8, 1, 6, a^14b^15, No], [8, 2, 5, a^13b^15, No], [8, 3, 4, a^12b^15, No], [8, 4, 3, a^11b^15, No], [8, 5, 2, a^10b^15, No], [8, 6, 1, a^9b^15, No], [8, 7, 0, a^8b^15, No], [9, 1, 5, a^14b^15, No], [9, 2, 4, a^13b^15, No], [9, 3, 3, a^12b^15, No], [9, 4, 2, a^11b^15, No], [9, 5, 1, a^10b^15, No], [9, 6, 0, a^9b^15, No], [10, 1, 4, a^14b^15, No], [10, 2, 3, a^13b^15, No], [10, 3, 2, a^12b^15, No], [10, 4, 1, a^11b^15, No], [10, 5, 0, a^10b^15, No], [11, 1, 3, a^14b^15, No], [11, 2, 2, a^13b^15, No], [11, 3, 1, a^12b^15, No], [11, 4, 0, a^11b^15, No], [12, 1, 2, a^14b^15, No], [12, 2, 1, a^13b^15, No], [12, 3, 0, a^12b^15, No], [13, 1, 1, a^14b^15, No], [13, 2, 0, a^13b^15, No], [14, 1, 0, a^14b^15, No]], 120, 120, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (15, a^15b^15, [[0, 1, 14, a^14b^15, No], [0, 2, 13, a^13b^15, No], [0, 3, 12, a^12b^15, No], [0, 4, 11, a^11b^15, No], [0, 5, 10, a^10b^15, No], [0, 6, 9, a^9b^15, No], [0, 7, 8, a^8b^15, No], [0, 8, 7, a^7b^15, No], [0, 9, 6, a^6b^15, No], [0, 10, 5, a^5b^15, No], [0, 11, 4, a^4b^15, No], [0, 12, 3, a^3b^15, No], [0, 13, 2, a^2b^15, No], [0, 14, 1, a^1b^15, No], [0, 15, 0, a^0b^15, No], [1, 1, 13, a^14b^15, No], [1, 2, 12, a^13b^15, No], [1, 3, 11, a^12b^15, No], [1, 4, 10, a^11b^15, No], [1, 5, 9, a^10b^15, No], [1, 6, 8, a^9b^15, No], [1, 7, 7, a^8b^15, No], [1, 8, 6, a^7b^15, No], [1, 9, 5, a^6b^15, No], [1, 10, 4, a^5b^15, No], [1, 11, 3, a^4b^15, No], [1, 12, 2, a^3b^15, No], [1, 13, 1, a^2b^15, No], [1, 14, 0, a^1b^15, No], [2, 1, 12, a^14b^15, No], [2, 2, 11, a^13b^15, No], [2, 3, 10, a^12b^15, No], [2, 4, 9, a^11b^15, No], [2, 5, 8, a^10b^15, No], [2, 6, 7, a^9b^15, No], [2, 7, 6, a^8b^15, No], [2, 8, 5, a^7b^15, No], [2, 9, 4, a^6b^15, No], [2, 10, 3, a^5b^15, No], [2, 11, 2, a^4b^15, No], [2, 12, 1, a^3b^15, No], [2, 13, 0, a^2b^15, No], [3, 1, 11, a^14b^15, No], [3, 2, 10, a^13b^15, No], [3, 3, 9, a^12b^15, No], [3, 4, 8, a^11b^15, No], [3, 5, 7, a^10b^15, No], [3, 6, 6, a^9b^15, No], [3, 7, 5, a^8b^15, No], [3, 8, 4, a^7b^15, No], [3, 9, 3, a^6b^15, No], [3, 10, 2, a^5b^15, No], [3, 11, 1, a^4b^15, No], [3, 12, 0, a^3b^15, No], [4, 1, 10, a^14b^15, No], [4, 2, 9, a^13b^15, No], [4, 3, 8, a^12b^15, No], [4, 4, 7, a^11b^15, No], [4, 5, 6, a^10b^15, No], [4, 6, 5, a^9b^15, No], [4, 7, 4, a^8b^15, No], [4, 8, 3, a^7b^15, No], [4, 9, 2, a^6b^15, No], [4, 10, 1, a^5b^15, No], [4, 11, 0, a^4b^15, No], [5, 1, 9, a^14b^15, No], [5, 2, 8, a^13b^15, No], [5, 3, 7, a^12b^15, No], [5, 4, 6, a^11b^15, No], [5, 5, 5, a^10b^15, No], [5, 6, 4, a^9b^15, No], [5, 7, 3, a^8b^15, No], [5, 8, 2, a^7b^15, No], [5, 9, 1, a^6b^15, No], [5, 10, 0, a^5b^15, No], [6, 1, 8, a^14b^15, No], [6, 2, 7, a^13b^15, No], [6, 3, 6, a^12b^15, No], [6, 4, 5, a^11b^15, No], [6, 5, 4, a^10b^15, No], [6, 6, 3, a^9b^15, No], [6, 7, 2, a^8b^15, No], [6, 8, 1, a^7b^15, No], [6, 9, 0, a^6b^15, No], [7, 1, 7, a^14b^15, No], [7, 2, 6, a^13b^15, No], [7, 3, 5, a^12b^15, No], [7, 4, 4, a^11b^15, No], [7, 5, 3, a^10b^15, No], [7, 6, 2, a^9b^15, No], [7, 7, 1, a^8b^15, No], [7, 8, 0, a^7b^15, No], [8, 1, 6, a^14b^15, No], [8, 2, 5, a^13b^15, No], [8, 3, 4, a^12b^15, No], [8, 4, 3, a^11b^15, No], [8, 5, 2, a^10b^15, No], [8, 6, 1, a^9b^15, No], [8, 7, 0, a^8b^15, No], [9, 1, 5, a^14b^15, No], [9, 2, 4, a^13b^15, No], [9, 3, 3, a^12b^15, No], [9, 4, 2, a^11b^15, No], [9, 5, 1, a^10b^15, No], [9, 6, 0, a^9b^15, No], [10, 1, 4, a^14b^15, No], [10, 2, 3, a^13b^15, No], [10, 3, 2, a^12b^15, No], [10, 4, 1, a^11b^15, No], [10, 5, 0, a^10b^15, No], [11, 1, 3, a^14b^15, No], [11, 2, 2, a^13b^15, No], [11, 3, 1, a^12b^15, No], [11, 4, 0, a^11b^15, No], [12, 1, 2, a^14b^15, No], [12, 2, 1, a^13b^15, No], [12, 3, 0, a^12b^15, No], [13, 1, 1, a^14b^15, No], [13, 2, 0, a^13b^15, No], [14, 1, 0, a^14b^15, No]], 120, 120, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0066","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c16269d53e140976af6d6400357437d017fe7d3169728845b75b3dbf69d6288","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187066,"semantic_template":"pump-anbn-all-splits|{p:16}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=16 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(16, a^16b^16, [[0, 1, 15, a^15b^16, No], [0, 2, 14, a^14b^16, No], [0, 3, 13, a^13b^16, No], [0, 4, 12, a^12b^16, No], [0, 5, 11, a^11b^16, No], [0, 6, 10, a^10b^16, No], [0, 7, 9, a^9b^16, No], [0, 8, 8, a^8b^16, No], [0, 9, 7, a^7b^16, No], [0, 10, 6, a^6b^16, No], [0, 11, 5, a^5b^16, No], [0, 12, 4, a^4b^16, No], [0, 13, 3, a^3b^16, No], [0, 14, 2, a^2b^16, No], [0, 15, 1, a^1b^16, No], [0, 16, 0, a^0b^16, No], [1, 1, 14, a^15b^16, No], [1, 2, 13, a^14b^16, No], [1, 3, 12, a^13b^16, No], [1, 4, 11, a^12b^16, No], [1, 5, 10, a^11b^16, No], [1, 6, 9, a^10b^16, No], [1, 7, 8, a^9b^16, No], [1, 8, 7, a^8b^16, No], [1, 9, 6, a^7b^16, No], [1, 10, 5, a^6b^16, No], [1, 11, 4, a^5b^16, No], [1, 12, 3, a^4b^16, No], [1, 13, 2, a^3b^16, No], [1, 14, 1, a^2b^16, No], [1, 15, 0, a^1b^16, No], [2, 1, 13, a^15b^16, No], [2, 2, 12, a^14b^16, No], [2, 3, 11, a^13b^16, No], [2, 4, 10, a^12b^16, No], [2, 5, 9, a^11b^16, No], [2, 6, 8, a^10b^16, No], [2, 7, 7, a^9b^16, No], [2, 8, 6, a^8b^16, No], [2, 9, 5, a^7b^16, No], [2, 10, 4, a^6b^16, No], [2, 11, 3, a^5b^16, No], [2, 12, 2, a^4b^16, No], [2, 13, 1, a^3b^16, No], [2, 14, 0, a^2b^16, No], [3, 1, 12, a^15b^16, No], [3, 2, 11, a^14b^16, No], [3, 3, 10, a^13b^16, No], [3, 4, 9, a^12b^16, No], [3, 5, 8, a^11b^16, No], [3, 6, 7, a^10b^16, No], [3, 7, 6, a^9b^16, No], [3, 8, 5, a^8b^16, No], [3, 9, 4, a^7b^16, No], [3, 10, 3, a^6b^16, No], [3, 11, 2, a^5b^16, No], [3, 12, 1, a^4b^16, No], [3, 13, 0, a^3b^16, No], [4, 1, 11, a^15b^16, No], [4, 2, 10, a^14b^16, No], [4, 3, 9, a^13b^16, No], [4, 4, 8, a^12b^16, No], [4, 5, 7, a^11b^16, No], [4, 6, 6, a^10b^16, No], [4, 7, 5, a^9b^16, No], [4, 8, 4, a^8b^16, No], [4, 9, 3, a^7b^16, No], [4, 10, 2, a^6b^16, No], [4, 11, 1, a^5b^16, No], [4, 12, 0, a^4b^16, No], [5, 1, 10, a^15b^16, No], [5, 2, 9, a^14b^16, No], [5, 3, 8, a^13b^16, No], [5, 4, 7, a^12b^16, No], [5, 5, 6, a^11b^16, No], [5, 6, 5, a^10b^16, No], [5, 7, 4, a^9b^16, No], [5, 8, 3, a^8b^16, No], [5, 9, 2, a^7b^16, No], [5, 10, 1, a^6b^16, No], [5, 11, 0, a^5b^16, No], [6, 1, 9, a^15b^16, No], [6, 2, 8, a^14b^16, No], [6, 3, 7, a^13b^16, No], [6, 4, 6, a^12b^16, No], [6, 5, 5, a^11b^16, No], [6, 6, 4, a^10b^16, No], [6, 7, 3, a^9b^16, No], [6, 8, 2, a^8b^16, No], [6, 9, 1, a^7b^16, No], [6, 10, 0, a^6b^16, No], [7, 1, 8, a^15b^16, No], [7, 2, 7, a^14b^16, No], [7, 3, 6, a^13b^16, No], [7, 4, 5, a^12b^16, No], [7, 5, 4, a^11b^16, No], [7, 6, 3, a^10b^16, No], [7, 7, 2, a^9b^16, No], [7, 8, 1, a^8b^16, No], [7, 9, 0, a^7b^16, No], [8, 1, 7, a^15b^16, No], [8, 2, 6, a^14b^16, No], [8, 3, 5, a^13b^16, No], [8, 4, 4, a^12b^16, No], [8, 5, 3, a^11b^16, No], [8, 6, 2, a^10b^16, No], [8, 7, 1, a^9b^16, No], [8, 8, 0, a^8b^16, No], [9, 1, 6, a^15b^16, No], [9, 2, 5, a^14b^16, No], [9, 3, 4, a^13b^16, No], [9, 4, 3, a^12b^16, No], [9, 5, 2, a^11b^16, No], [9, 6, 1, a^10b^16, No], [9, 7, 0, a^9b^16, No], [10, 1, 5, a^15b^16, No], [10, 2, 4, a^14b^16, No], [10, 3, 3, a^13b^16, No], [10, 4, 2, a^12b^16, No], [10, 5, 1, a^11b^16, No], [10, 6, 0, a^10b^16, No], [11, 1, 4, a^15b^16, No], [11, 2, 3, a^14b^16, No], [11, 3, 2, a^13b^16, No], [11, 4, 1, a^12b^16, No], [11, 5, 0, a^11b^16, No], [12, 1, 3, a^15b^16, No], [12, 2, 2, a^14b^16, No], [12, 3, 1, a^13b^16, No], [12, 4, 0, a^12b^16, No], [13, 1, 2, a^15b^16, No], [13, 2, 1, a^14b^16, No], [13, 3, 0, a^13b^16, No], [14, 1, 1, a^15b^16, No], [14, 2, 0, a^14b^16, No], [15, 1, 0, a^15b^16, No]], 136, 136, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (16, a^16b^16, [[0, 1, 15, a^15b^16, No], [0, 2, 14, a^14b^16, No], [0, 3, 13, a^13b^16, No], [0, 4, 12, a^12b^16, No], [0, 5, 11, a^11b^16, No], [0, 6, 10, a^10b^16, No], [0, 7, 9, a^9b^16, No], [0, 8, 8, a^8b^16, No], [0, 9, 7, a^7b^16, No], [0, 10, 6, a^6b^16, No], [0, 11, 5, a^5b^16, No], [0, 12, 4, a^4b^16, No], [0, 13, 3, a^3b^16, No], [0, 14, 2, a^2b^16, No], [0, 15, 1, a^1b^16, No], [0, 16, 0, a^0b^16, No], [1, 1, 14, a^15b^16, No], [1, 2, 13, a^14b^16, No], [1, 3, 12, a^13b^16, No], [1, 4, 11, a^12b^16, No], [1, 5, 10, a^11b^16, No], [1, 6, 9, a^10b^16, No], [1, 7, 8, a^9b^16, No], [1, 8, 7, a^8b^16, No], [1, 9, 6, a^7b^16, No], [1, 10, 5, a^6b^16, No], [1, 11, 4, a^5b^16, No], [1, 12, 3, a^4b^16, No], [1, 13, 2, a^3b^16, No], [1, 14, 1, a^2b^16, No], [1, 15, 0, a^1b^16, No], [2, 1, 13, a^15b^16, No], [2, 2, 12, a^14b^16, No], [2, 3, 11, a^13b^16, No], [2, 4, 10, a^12b^16, No], [2, 5, 9, a^11b^16, No], [2, 6, 8, a^10b^16, No], [2, 7, 7, a^9b^16, No], [2, 8, 6, a^8b^16, No], [2, 9, 5, a^7b^16, No], [2, 10, 4, a^6b^16, No], [2, 11, 3, a^5b^16, No], [2, 12, 2, a^4b^16, No], [2, 13, 1, a^3b^16, No], [2, 14, 0, a^2b^16, No], [3, 1, 12, a^15b^16, No], [3, 2, 11, a^14b^16, No], [3, 3, 10, a^13b^16, No], [3, 4, 9, a^12b^16, No], [3, 5, 8, a^11b^16, No], [3, 6, 7, a^10b^16, No], [3, 7, 6, a^9b^16, No], [3, 8, 5, a^8b^16, No], [3, 9, 4, a^7b^16, No], [3, 10, 3, a^6b^16, No], [3, 11, 2, a^5b^16, No], [3, 12, 1, a^4b^16, No], [3, 13, 0, a^3b^16, No], [4, 1, 11, a^15b^16, No], [4, 2, 10, a^14b^16, No], [4, 3, 9, a^13b^16, No], [4, 4, 8, a^12b^16, No], [4, 5, 7, a^11b^16, No], [4, 6, 6, a^10b^16, No], [4, 7, 5, a^9b^16, No], [4, 8, 4, a^8b^16, No], [4, 9, 3, a^7b^16, No], [4, 10, 2, a^6b^16, No], [4, 11, 1, a^5b^16, No], [4, 12, 0, a^4b^16, No], [5, 1, 10, a^15b^16, No], [5, 2, 9, a^14b^16, No], [5, 3, 8, a^13b^16, No], [5, 4, 7, a^12b^16, No], [5, 5, 6, a^11b^16, No], [5, 6, 5, a^10b^16, No], [5, 7, 4, a^9b^16, No], [5, 8, 3, a^8b^16, No], [5, 9, 2, a^7b^16, No], [5, 10, 1, a^6b^16, No], [5, 11, 0, a^5b^16, No], [6, 1, 9, a^15b^16, No], [6, 2, 8, a^14b^16, No], [6, 3, 7, a^13b^16, No], [6, 4, 6, a^12b^16, No], [6, 5, 5, a^11b^16, No], [6, 6, 4, a^10b^16, No], [6, 7, 3, a^9b^16, No], [6, 8, 2, a^8b^16, No], [6, 9, 1, a^7b^16, No], [6, 10, 0, a^6b^16, No], [7, 1, 8, a^15b^16, No], [7, 2, 7, a^14b^16, No], [7, 3, 6, a^13b^16, No], [7, 4, 5, a^12b^16, No], [7, 5, 4, a^11b^16, No], [7, 6, 3, a^10b^16, No], [7, 7, 2, a^9b^16, No], [7, 8, 1, a^8b^16, No], [7, 9, 0, a^7b^16, No], [8, 1, 7, a^15b^16, No], [8, 2, 6, a^14b^16, No], [8, 3, 5, a^13b^16, No], [8, 4, 4, a^12b^16, No], [8, 5, 3, a^11b^16, No], [8, 6, 2, a^10b^16, No], [8, 7, 1, a^9b^16, No], [8, 8, 0, a^8b^16, No], [9, 1, 6, a^15b^16, No], [9, 2, 5, a^14b^16, No], [9, 3, 4, a^13b^16, No], [9, 4, 3, a^12b^16, No], [9, 5, 2, a^11b^16, No], [9, 6, 1, a^10b^16, No], [9, 7, 0, a^9b^16, No], [10, 1, 5, a^15b^16, No], [10, 2, 4, a^14b^16, No], [10, 3, 3, a^13b^16, No], [10, 4, 2, a^12b^16, No], [10, 5, 1, a^11b^16, No], [10, 6, 0, a^10b^16, No], [11, 1, 4, a^15b^16, No], [11, 2, 3, a^14b^16, No], [11, 3, 2, a^13b^16, No], [11, 4, 1, a^12b^16, No], [11, 5, 0, a^11b^16, No], [12, 1, 3, a^15b^16, No], [12, 2, 2, a^14b^16, No], [12, 3, 1, a^13b^16, No], [12, 4, 0, a^12b^16, No], [13, 1, 2, a^15b^16, No], [13, 2, 1, a^14b^16, No], [13, 3, 0, a^13b^16, No], [14, 1, 1, a^15b^16, No], [14, 2, 0, a^14b^16, No], [15, 1, 0, a^15b^16, No]], 136, 136, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0067","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ca82cbcc2242d27db5cf37a0bc128b0461ae131ed86975de742f7d7ab33df0a","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187067,"semantic_template":"pump-anbn-all-splits|{p:17}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=17 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(17, a^17b^17, [[0, 1, 16, a^16b^17, No], [0, 2, 15, a^15b^17, No], [0, 3, 14, a^14b^17, No], [0, 4, 13, a^13b^17, No], [0, 5, 12, a^12b^17, No], [0, 6, 11, a^11b^17, No], [0, 7, 10, a^10b^17, No], [0, 8, 9, a^9b^17, No], [0, 9, 8, a^8b^17, No], [0, 10, 7, a^7b^17, No], [0, 11, 6, a^6b^17, No], [0, 12, 5, a^5b^17, No], [0, 13, 4, a^4b^17, No], [0, 14, 3, a^3b^17, No], [0, 15, 2, a^2b^17, No], [0, 16, 1, a^1b^17, No], [0, 17, 0, a^0b^17, No], [1, 1, 15, a^16b^17, No], [1, 2, 14, a^15b^17, No], [1, 3, 13, a^14b^17, No], [1, 4, 12, a^13b^17, No], [1, 5, 11, a^12b^17, No], [1, 6, 10, a^11b^17, No], [1, 7, 9, a^10b^17, No], [1, 8, 8, a^9b^17, No], [1, 9, 7, a^8b^17, No], [1, 10, 6, a^7b^17, No], [1, 11, 5, a^6b^17, No], [1, 12, 4, a^5b^17, No], [1, 13, 3, a^4b^17, No], [1, 14, 2, a^3b^17, No], [1, 15, 1, a^2b^17, No], [1, 16, 0, a^1b^17, No], [2, 1, 14, a^16b^17, No], [2, 2, 13, a^15b^17, No], [2, 3, 12, a^14b^17, No], [2, 4, 11, a^13b^17, No], [2, 5, 10, a^12b^17, No], [2, 6, 9, a^11b^17, No], [2, 7, 8, a^10b^17, No], [2, 8, 7, a^9b^17, No], [2, 9, 6, a^8b^17, No], [2, 10, 5, a^7b^17, No], [2, 11, 4, a^6b^17, No], [2, 12, 3, a^5b^17, No], [2, 13, 2, a^4b^17, No], [2, 14, 1, a^3b^17, No], [2, 15, 0, a^2b^17, No], [3, 1, 13, a^16b^17, No], [3, 2, 12, a^15b^17, No], [3, 3, 11, a^14b^17, No], [3, 4, 10, a^13b^17, No], [3, 5, 9, a^12b^17, No], [3, 6, 8, a^11b^17, No], [3, 7, 7, a^10b^17, No], [3, 8, 6, a^9b^17, No], [3, 9, 5, a^8b^17, No], [3, 10, 4, a^7b^17, No], [3, 11, 3, a^6b^17, No], [3, 12, 2, a^5b^17, No], [3, 13, 1, a^4b^17, No], [3, 14, 0, a^3b^17, No], [4, 1, 12, a^16b^17, No], [4, 2, 11, a^15b^17, No], [4, 3, 10, a^14b^17, No], [4, 4, 9, a^13b^17, No], [4, 5, 8, a^12b^17, No], [4, 6, 7, a^11b^17, No], [4, 7, 6, a^10b^17, No], [4, 8, 5, a^9b^17, No], [4, 9, 4, a^8b^17, No], [4, 10, 3, a^7b^17, No], [4, 11, 2, a^6b^17, No], [4, 12, 1, a^5b^17, No], [4, 13, 0, a^4b^17, No], [5, 1, 11, a^16b^17, No], [5, 2, 10, a^15b^17, No], [5, 3, 9, a^14b^17, No], [5, 4, 8, a^13b^17, No], [5, 5, 7, a^12b^17, No], [5, 6, 6, a^11b^17, No], [5, 7, 5, a^10b^17, No], [5, 8, 4, a^9b^17, No], [5, 9, 3, a^8b^17, No], [5, 10, 2, a^7b^17, No], [5, 11, 1, a^6b^17, No], [5, 12, 0, a^5b^17, No], [6, 1, 10, a^16b^17, No], [6, 2, 9, a^15b^17, No], [6, 3, 8, a^14b^17, No], [6, 4, 7, a^13b^17, No], [6, 5, 6, a^12b^17, No], [6, 6, 5, a^11b^17, No], [6, 7, 4, a^10b^17, No], [6, 8, 3, a^9b^17, No], [6, 9, 2, a^8b^17, No], [6, 10, 1, a^7b^17, No], [6, 11, 0, a^6b^17, No], [7, 1, 9, a^16b^17, No], [7, 2, 8, a^15b^17, No], [7, 3, 7, a^14b^17, No], [7, 4, 6, a^13b^17, No], [7, 5, 5, a^12b^17, No], [7, 6, 4, a^11b^17, No], [7, 7, 3, a^10b^17, No], [7, 8, 2, a^9b^17, No], [7, 9, 1, a^8b^17, No], [7, 10, 0, a^7b^17, No], [8, 1, 8, a^16b^17, No], [8, 2, 7, a^15b^17, No], [8, 3, 6, a^14b^17, No], [8, 4, 5, a^13b^17, No], [8, 5, 4, a^12b^17, No], [8, 6, 3, a^11b^17, No], [8, 7, 2, a^10b^17, No], [8, 8, 1, a^9b^17, No], [8, 9, 0, a^8b^17, No], [9, 1, 7, a^16b^17, No], [9, 2, 6, a^15b^17, No], [9, 3, 5, a^14b^17, No], [9, 4, 4, a^13b^17, No], [9, 5, 3, a^12b^17, No], [9, 6, 2, a^11b^17, No], [9, 7, 1, a^10b^17, No], [9, 8, 0, a^9b^17, No], [10, 1, 6, a^16b^17, No], [10, 2, 5, a^15b^17, No], [10, 3, 4, a^14b^17, No], [10, 4, 3, a^13b^17, No], [10, 5, 2, a^12b^17, No], [10, 6, 1, a^11b^17, No], [10, 7, 0, a^10b^17, No], [11, 1, 5, a^16b^17, No], [11, 2, 4, a^15b^17, No], [11, 3, 3, a^14b^17, No], [11, 4, 2, a^13b^17, No], [11, 5, 1, a^12b^17, No], [11, 6, 0, a^11b^17, No], [12, 1, 4, a^16b^17, No], [12, 2, 3, a^15b^17, No], [12, 3, 2, a^14b^17, No], [12, 4, 1, a^13b^17, No], [12, 5, 0, a^12b^17, No], [13, 1, 3, a^16b^17, No], [13, 2, 2, a^15b^17, No], [13, 3, 1, a^14b^17, No], [13, 4, 0, a^13b^17, No], [14, 1, 2, a^16b^17, No], [14, 2, 1, a^15b^17, No], [14, 3, 0, a^14b^17, No], [15, 1, 1, a^16b^17, No], [15, 2, 0, a^15b^17, No], [16, 1, 0, a^16b^17, No]], 153, 153, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (17, a^17b^17, [[0, 1, 16, a^16b^17, No], [0, 2, 15, a^15b^17, No], [0, 3, 14, a^14b^17, No], [0, 4, 13, a^13b^17, No], [0, 5, 12, a^12b^17, No], [0, 6, 11, a^11b^17, No], [0, 7, 10, a^10b^17, No], [0, 8, 9, a^9b^17, No], [0, 9, 8, a^8b^17, No], [0, 10, 7, a^7b^17, No], [0, 11, 6, a^6b^17, No], [0, 12, 5, a^5b^17, No], [0, 13, 4, a^4b^17, No], [0, 14, 3, a^3b^17, No], [0, 15, 2, a^2b^17, No], [0, 16, 1, a^1b^17, No], [0, 17, 0, a^0b^17, No], [1, 1, 15, a^16b^17, No], [1, 2, 14, a^15b^17, No], [1, 3, 13, a^14b^17, No], [1, 4, 12, a^13b^17, No], [1, 5, 11, a^12b^17, No], [1, 6, 10, a^11b^17, No], [1, 7, 9, a^10b^17, No], [1, 8, 8, a^9b^17, No], [1, 9, 7, a^8b^17, No], [1, 10, 6, a^7b^17, No], [1, 11, 5, a^6b^17, No], [1, 12, 4, a^5b^17, No], [1, 13, 3, a^4b^17, No], [1, 14, 2, a^3b^17, No], [1, 15, 1, a^2b^17, No], [1, 16, 0, a^1b^17, No], [2, 1, 14, a^16b^17, No], [2, 2, 13, a^15b^17, No], [2, 3, 12, a^14b^17, No], [2, 4, 11, a^13b^17, No], [2, 5, 10, a^12b^17, No], [2, 6, 9, a^11b^17, No], [2, 7, 8, a^10b^17, No], [2, 8, 7, a^9b^17, No], [2, 9, 6, a^8b^17, No], [2, 10, 5, a^7b^17, No], [2, 11, 4, a^6b^17, No], [2, 12, 3, a^5b^17, No], [2, 13, 2, a^4b^17, No], [2, 14, 1, a^3b^17, No], [2, 15, 0, a^2b^17, No], [3, 1, 13, a^16b^17, No], [3, 2, 12, a^15b^17, No], [3, 3, 11, a^14b^17, No], [3, 4, 10, a^13b^17, No], [3, 5, 9, a^12b^17, No], [3, 6, 8, a^11b^17, No], [3, 7, 7, a^10b^17, No], [3, 8, 6, a^9b^17, No], [3, 9, 5, a^8b^17, No], [3, 10, 4, a^7b^17, No], [3, 11, 3, a^6b^17, No], [3, 12, 2, a^5b^17, No], [3, 13, 1, a^4b^17, No], [3, 14, 0, a^3b^17, No], [4, 1, 12, a^16b^17, No], [4, 2, 11, a^15b^17, No], [4, 3, 10, a^14b^17, No], [4, 4, 9, a^13b^17, No], [4, 5, 8, a^12b^17, No], [4, 6, 7, a^11b^17, No], [4, 7, 6, a^10b^17, No], [4, 8, 5, a^9b^17, No], [4, 9, 4, a^8b^17, No], [4, 10, 3, a^7b^17, No], [4, 11, 2, a^6b^17, No], [4, 12, 1, a^5b^17, No], [4, 13, 0, a^4b^17, No], [5, 1, 11, a^16b^17, No], [5, 2, 10, a^15b^17, No], [5, 3, 9, a^14b^17, No], [5, 4, 8, a^13b^17, No], [5, 5, 7, a^12b^17, No], [5, 6, 6, a^11b^17, No], [5, 7, 5, a^10b^17, No], [5, 8, 4, a^9b^17, No], [5, 9, 3, a^8b^17, No], [5, 10, 2, a^7b^17, No], [5, 11, 1, a^6b^17, No], [5, 12, 0, a^5b^17, No], [6, 1, 10, a^16b^17, No], [6, 2, 9, a^15b^17, No], [6, 3, 8, a^14b^17, No], [6, 4, 7, a^13b^17, No], [6, 5, 6, a^12b^17, No], [6, 6, 5, a^11b^17, No], [6, 7, 4, a^10b^17, No], [6, 8, 3, a^9b^17, No], [6, 9, 2, a^8b^17, No], [6, 10, 1, a^7b^17, No], [6, 11, 0, a^6b^17, No], [7, 1, 9, a^16b^17, No], [7, 2, 8, a^15b^17, No], [7, 3, 7, a^14b^17, No], [7, 4, 6, a^13b^17, No], [7, 5, 5, a^12b^17, No], [7, 6, 4, a^11b^17, No], [7, 7, 3, a^10b^17, No], [7, 8, 2, a^9b^17, No], [7, 9, 1, a^8b^17, No], [7, 10, 0, a^7b^17, No], [8, 1, 8, a^16b^17, No], [8, 2, 7, a^15b^17, No], [8, 3, 6, a^14b^17, No], [8, 4, 5, a^13b^17, No], [8, 5, 4, a^12b^17, No], [8, 6, 3, a^11b^17, No], [8, 7, 2, a^10b^17, No], [8, 8, 1, a^9b^17, No], [8, 9, 0, a^8b^17, No], [9, 1, 7, a^16b^17, No], [9, 2, 6, a^15b^17, No], [9, 3, 5, a^14b^17, No], [9, 4, 4, a^13b^17, No], [9, 5, 3, a^12b^17, No], [9, 6, 2, a^11b^17, No], [9, 7, 1, a^10b^17, No], [9, 8, 0, a^9b^17, No], [10, 1, 6, a^16b^17, No], [10, 2, 5, a^15b^17, No], [10, 3, 4, a^14b^17, No], [10, 4, 3, a^13b^17, No], [10, 5, 2, a^12b^17, No], [10, 6, 1, a^11b^17, No], [10, 7, 0, a^10b^17, No], [11, 1, 5, a^16b^17, No], [11, 2, 4, a^15b^17, No], [11, 3, 3, a^14b^17, No], [11, 4, 2, a^13b^17, No], [11, 5, 1, a^12b^17, No], [11, 6, 0, a^11b^17, No], [12, 1, 4, a^16b^17, No], [12, 2, 3, a^15b^17, No], [12, 3, 2, a^14b^17, No], [12, 4, 1, a^13b^17, No], [12, 5, 0, a^12b^17, No], [13, 1, 3, a^16b^17, No], [13, 2, 2, a^15b^17, No], [13, 3, 1, a^14b^17, No], [13, 4, 0, a^13b^17, No], [14, 1, 2, a^16b^17, No], [14, 2, 1, a^15b^17, No], [14, 3, 0, a^14b^17, No], [15, 1, 1, a^16b^17, No], [15, 2, 0, a^15b^17, No], [16, 1, 0, a^16b^17, No]], 153, 153, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0068","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"070b0ac177f6ef8c02f1589730ca8c7beda4c5a3e898aaafa7f707bec6472aea","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187068,"semantic_template":"pump-anbn-all-splits|{p:18}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=18 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(18, a^18b^18, [[0, 1, 17, a^17b^18, No], [0, 2, 16, a^16b^18, No], [0, 3, 15, a^15b^18, No], [0, 4, 14, a^14b^18, No], [0, 5, 13, a^13b^18, No], [0, 6, 12, a^12b^18, No], [0, 7, 11, a^11b^18, No], [0, 8, 10, a^10b^18, No], [0, 9, 9, a^9b^18, No], [0, 10, 8, a^8b^18, No], [0, 11, 7, a^7b^18, No], [0, 12, 6, a^6b^18, No], [0, 13, 5, a^5b^18, No], [0, 14, 4, a^4b^18, No], [0, 15, 3, a^3b^18, No], [0, 16, 2, a^2b^18, No], [0, 17, 1, a^1b^18, No], [0, 18, 0, a^0b^18, No], [1, 1, 16, a^17b^18, No], [1, 2, 15, a^16b^18, No], [1, 3, 14, a^15b^18, No], [1, 4, 13, a^14b^18, No], [1, 5, 12, a^13b^18, No], [1, 6, 11, a^12b^18, No], [1, 7, 10, a^11b^18, No], [1, 8, 9, a^10b^18, No], [1, 9, 8, a^9b^18, No], [1, 10, 7, a^8b^18, No], [1, 11, 6, a^7b^18, No], [1, 12, 5, a^6b^18, No], [1, 13, 4, a^5b^18, No], [1, 14, 3, a^4b^18, No], [1, 15, 2, a^3b^18, No], [1, 16, 1, a^2b^18, No], [1, 17, 0, a^1b^18, No], [2, 1, 15, a^17b^18, No], [2, 2, 14, a^16b^18, No], [2, 3, 13, a^15b^18, No], [2, 4, 12, a^14b^18, No], [2, 5, 11, a^13b^18, No], [2, 6, 10, a^12b^18, No], [2, 7, 9, a^11b^18, No], [2, 8, 8, a^10b^18, No], [2, 9, 7, a^9b^18, No], [2, 10, 6, a^8b^18, No], [2, 11, 5, a^7b^18, No], [2, 12, 4, a^6b^18, No], [2, 13, 3, a^5b^18, No], [2, 14, 2, a^4b^18, No], [2, 15, 1, a^3b^18, No], [2, 16, 0, a^2b^18, No], [3, 1, 14, a^17b^18, No], [3, 2, 13, a^16b^18, No], [3, 3, 12, a^15b^18, No], [3, 4, 11, a^14b^18, No], [3, 5, 10, a^13b^18, No], [3, 6, 9, a^12b^18, No], [3, 7, 8, a^11b^18, No], [3, 8, 7, a^10b^18, No], [3, 9, 6, a^9b^18, No], [3, 10, 5, a^8b^18, No], [3, 11, 4, a^7b^18, No], [3, 12, 3, a^6b^18, No], [3, 13, 2, a^5b^18, No], [3, 14, 1, a^4b^18, No], [3, 15, 0, a^3b^18, No], [4, 1, 13, a^17b^18, No], [4, 2, 12, a^16b^18, No], [4, 3, 11, a^15b^18, No], [4, 4, 10, a^14b^18, No], [4, 5, 9, a^13b^18, No], [4, 6, 8, a^12b^18, No], [4, 7, 7, a^11b^18, No], [4, 8, 6, a^10b^18, No], [4, 9, 5, a^9b^18, No], [4, 10, 4, a^8b^18, No], [4, 11, 3, a^7b^18, No], [4, 12, 2, a^6b^18, No], [4, 13, 1, a^5b^18, No], [4, 14, 0, a^4b^18, No], [5, 1, 12, a^17b^18, No], [5, 2, 11, a^16b^18, No], [5, 3, 10, a^15b^18, No], [5, 4, 9, a^14b^18, No], [5, 5, 8, a^13b^18, No], [5, 6, 7, a^12b^18, No], [5, 7, 6, a^11b^18, No], [5, 8, 5, a^10b^18, No], [5, 9, 4, a^9b^18, No], [5, 10, 3, a^8b^18, No], [5, 11, 2, a^7b^18, No], [5, 12, 1, a^6b^18, No], [5, 13, 0, a^5b^18, No], [6, 1, 11, a^17b^18, No], [6, 2, 10, a^16b^18, No], [6, 3, 9, a^15b^18, No], [6, 4, 8, a^14b^18, No], [6, 5, 7, a^13b^18, No], [6, 6, 6, a^12b^18, No], [6, 7, 5, a^11b^18, No], [6, 8, 4, a^10b^18, No], [6, 9, 3, a^9b^18, No], [6, 10, 2, a^8b^18, No], [6, 11, 1, a^7b^18, No], [6, 12, 0, a^6b^18, No], [7, 1, 10, a^17b^18, No], [7, 2, 9, a^16b^18, No], [7, 3, 8, a^15b^18, No], [7, 4, 7, a^14b^18, No], [7, 5, 6, a^13b^18, No], [7, 6, 5, a^12b^18, No], [7, 7, 4, a^11b^18, No], [7, 8, 3, a^10b^18, No], [7, 9, 2, a^9b^18, No], [7, 10, 1, a^8b^18, No], [7, 11, 0, a^7b^18, No], [8, 1, 9, a^17b^18, No], [8, 2, 8, a^16b^18, No], [8, 3, 7, a^15b^18, No], [8, 4, 6, a^14b^18, No], [8, 5, 5, a^13b^18, No], [8, 6, 4, a^12b^18, No], [8, 7, 3, a^11b^18, No], [8, 8, 2, a^10b^18, No], [8, 9, 1, a^9b^18, No], [8, 10, 0, a^8b^18, No], [9, 1, 8, a^17b^18, No], [9, 2, 7, a^16b^18, No], [9, 3, 6, a^15b^18, No], [9, 4, 5, a^14b^18, No], [9, 5, 4, a^13b^18, No], [9, 6, 3, a^12b^18, No], [9, 7, 2, a^11b^18, No], [9, 8, 1, a^10b^18, No], [9, 9, 0, a^9b^18, No], [10, 1, 7, a^17b^18, No], [10, 2, 6, a^16b^18, No], [10, 3, 5, a^15b^18, No], [10, 4, 4, a^14b^18, No], [10, 5, 3, a^13b^18, No], [10, 6, 2, a^12b^18, No], [10, 7, 1, a^11b^18, No], [10, 8, 0, a^10b^18, No], [11, 1, 6, a^17b^18, No], [11, 2, 5, a^16b^18, No], [11, 3, 4, a^15b^18, No], [11, 4, 3, a^14b^18, No], [11, 5, 2, a^13b^18, No], [11, 6, 1, a^12b^18, No], [11, 7, 0, a^11b^18, No], [12, 1, 5, a^17b^18, No], [12, 2, 4, a^16b^18, No], [12, 3, 3, a^15b^18, No], [12, 4, 2, a^14b^18, No], [12, 5, 1, a^13b^18, No], [12, 6, 0, a^12b^18, No], [13, 1, 4, a^17b^18, No], [13, 2, 3, a^16b^18, No], [13, 3, 2, a^15b^18, No], [13, 4, 1, a^14b^18, No], [13, 5, 0, a^13b^18, No], [14, 1, 3, a^17b^18, No], [14, 2, 2, a^16b^18, No], [14, 3, 1, a^15b^18, No], [14, 4, 0, a^14b^18, No], [15, 1, 2, a^17b^18, No], [15, 2, 1, a^16b^18, No], [15, 3, 0, a^15b^18, No], [16, 1, 1, a^17b^18, No], [16, 2, 0, a^16b^18, No], [17, 1, 0, a^17b^18, No]], 171, 171, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (18, a^18b^18, [[0, 1, 17, a^17b^18, No], [0, 2, 16, a^16b^18, No], [0, 3, 15, a^15b^18, No], [0, 4, 14, a^14b^18, No], [0, 5, 13, a^13b^18, No], [0, 6, 12, a^12b^18, No], [0, 7, 11, a^11b^18, No], [0, 8, 10, a^10b^18, No], [0, 9, 9, a^9b^18, No], [0, 10, 8, a^8b^18, No], [0, 11, 7, a^7b^18, No], [0, 12, 6, a^6b^18, No], [0, 13, 5, a^5b^18, No], [0, 14, 4, a^4b^18, No], [0, 15, 3, a^3b^18, No], [0, 16, 2, a^2b^18, No], [0, 17, 1, a^1b^18, No], [0, 18, 0, a^0b^18, No], [1, 1, 16, a^17b^18, No], [1, 2, 15, a^16b^18, No], [1, 3, 14, a^15b^18, No], [1, 4, 13, a^14b^18, No], [1, 5, 12, a^13b^18, No], [1, 6, 11, a^12b^18, No], [1, 7, 10, a^11b^18, No], [1, 8, 9, a^10b^18, No], [1, 9, 8, a^9b^18, No], [1, 10, 7, a^8b^18, No], [1, 11, 6, a^7b^18, No], [1, 12, 5, a^6b^18, No], [1, 13, 4, a^5b^18, No], [1, 14, 3, a^4b^18, No], [1, 15, 2, a^3b^18, No], [1, 16, 1, a^2b^18, No], [1, 17, 0, a^1b^18, No], [2, 1, 15, a^17b^18, No], [2, 2, 14, a^16b^18, No], [2, 3, 13, a^15b^18, No], [2, 4, 12, a^14b^18, No], [2, 5, 11, a^13b^18, No], [2, 6, 10, a^12b^18, No], [2, 7, 9, a^11b^18, No], [2, 8, 8, a^10b^18, No], [2, 9, 7, a^9b^18, No], [2, 10, 6, a^8b^18, No], [2, 11, 5, a^7b^18, No], [2, 12, 4, a^6b^18, No], [2, 13, 3, a^5b^18, No], [2, 14, 2, a^4b^18, No], [2, 15, 1, a^3b^18, No], [2, 16, 0, a^2b^18, No], [3, 1, 14, a^17b^18, No], [3, 2, 13, a^16b^18, No], [3, 3, 12, a^15b^18, No], [3, 4, 11, a^14b^18, No], [3, 5, 10, a^13b^18, No], [3, 6, 9, a^12b^18, No], [3, 7, 8, a^11b^18, No], [3, 8, 7, a^10b^18, No], [3, 9, 6, a^9b^18, No], [3, 10, 5, a^8b^18, No], [3, 11, 4, a^7b^18, No], [3, 12, 3, a^6b^18, No], [3, 13, 2, a^5b^18, No], [3, 14, 1, a^4b^18, No], [3, 15, 0, a^3b^18, No], [4, 1, 13, a^17b^18, No], [4, 2, 12, a^16b^18, No], [4, 3, 11, a^15b^18, No], [4, 4, 10, a^14b^18, No], [4, 5, 9, a^13b^18, No], [4, 6, 8, a^12b^18, No], [4, 7, 7, a^11b^18, No], [4, 8, 6, a^10b^18, No], [4, 9, 5, a^9b^18, No], [4, 10, 4, a^8b^18, No], [4, 11, 3, a^7b^18, No], [4, 12, 2, a^6b^18, No], [4, 13, 1, a^5b^18, No], [4, 14, 0, a^4b^18, No], [5, 1, 12, a^17b^18, No], [5, 2, 11, a^16b^18, No], [5, 3, 10, a^15b^18, No], [5, 4, 9, a^14b^18, No], [5, 5, 8, a^13b^18, No], [5, 6, 7, a^12b^18, No], [5, 7, 6, a^11b^18, No], [5, 8, 5, a^10b^18, No], [5, 9, 4, a^9b^18, No], [5, 10, 3, a^8b^18, No], [5, 11, 2, a^7b^18, No], [5, 12, 1, a^6b^18, No], [5, 13, 0, a^5b^18, No], [6, 1, 11, a^17b^18, No], [6, 2, 10, a^16b^18, No], [6, 3, 9, a^15b^18, No], [6, 4, 8, a^14b^18, No], [6, 5, 7, a^13b^18, No], [6, 6, 6, a^12b^18, No], [6, 7, 5, a^11b^18, No], [6, 8, 4, a^10b^18, No], [6, 9, 3, a^9b^18, No], [6, 10, 2, a^8b^18, No], [6, 11, 1, a^7b^18, No], [6, 12, 0, a^6b^18, No], [7, 1, 10, a^17b^18, No], [7, 2, 9, a^16b^18, No], [7, 3, 8, a^15b^18, No], [7, 4, 7, a^14b^18, No], [7, 5, 6, a^13b^18, No], [7, 6, 5, a^12b^18, No], [7, 7, 4, a^11b^18, No], [7, 8, 3, a^10b^18, No], [7, 9, 2, a^9b^18, No], [7, 10, 1, a^8b^18, No], [7, 11, 0, a^7b^18, No], [8, 1, 9, a^17b^18, No], [8, 2, 8, a^16b^18, No], [8, 3, 7, a^15b^18, No], [8, 4, 6, a^14b^18, No], [8, 5, 5, a^13b^18, No], [8, 6, 4, a^12b^18, No], [8, 7, 3, a^11b^18, No], [8, 8, 2, a^10b^18, No], [8, 9, 1, a^9b^18, No], [8, 10, 0, a^8b^18, No], [9, 1, 8, a^17b^18, No], [9, 2, 7, a^16b^18, No], [9, 3, 6, a^15b^18, No], [9, 4, 5, a^14b^18, No], [9, 5, 4, a^13b^18, No], [9, 6, 3, a^12b^18, No], [9, 7, 2, a^11b^18, No], [9, 8, 1, a^10b^18, No], [9, 9, 0, a^9b^18, No], [10, 1, 7, a^17b^18, No], [10, 2, 6, a^16b^18, No], [10, 3, 5, a^15b^18, No], [10, 4, 4, a^14b^18, No], [10, 5, 3, a^13b^18, No], [10, 6, 2, a^12b^18, No], [10, 7, 1, a^11b^18, No], [10, 8, 0, a^10b^18, No], [11, 1, 6, a^17b^18, No], [11, 2, 5, a^16b^18, No], [11, 3, 4, a^15b^18, No], [11, 4, 3, a^14b^18, No], [11, 5, 2, a^13b^18, No], [11, 6, 1, a^12b^18, No], [11, 7, 0, a^11b^18, No], [12, 1, 5, a^17b^18, No], [12, 2, 4, a^16b^18, No], [12, 3, 3, a^15b^18, No], [12, 4, 2, a^14b^18, No], [12, 5, 1, a^13b^18, No], [12, 6, 0, a^12b^18, No], [13, 1, 4, a^17b^18, No], [13, 2, 3, a^16b^18, No], [13, 3, 2, a^15b^18, No], [13, 4, 1, a^14b^18, No], [13, 5, 0, a^13b^18, No], [14, 1, 3, a^17b^18, No], [14, 2, 2, a^16b^18, No], [14, 3, 1, a^15b^18, No], [14, 4, 0, a^14b^18, No], [15, 1, 2, a^17b^18, No], [15, 2, 1, a^16b^18, No], [15, 3, 0, a^15b^18, No], [16, 1, 1, a^17b^18, No], [16, 2, 0, a^16b^18, No], [17, 1, 0, a^17b^18, No]], 171, 171, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0069","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"952944f6931c1d1563215cb3e72f7a189216710f223614eccbd23740883dfbd2","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187069,"semantic_template":"pump-anbn-all-splits|{p:19}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=19 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(19, a^19b^19, [[0, 1, 18, a^18b^19, No], [0, 2, 17, a^17b^19, No], [0, 3, 16, a^16b^19, No], [0, 4, 15, a^15b^19, No], [0, 5, 14, a^14b^19, No], [0, 6, 13, a^13b^19, No], [0, 7, 12, a^12b^19, No], [0, 8, 11, a^11b^19, No], [0, 9, 10, a^10b^19, No], [0, 10, 9, a^9b^19, No], [0, 11, 8, a^8b^19, No], [0, 12, 7, a^7b^19, No], [0, 13, 6, a^6b^19, No], [0, 14, 5, a^5b^19, No], [0, 15, 4, a^4b^19, No], [0, 16, 3, a^3b^19, No], [0, 17, 2, a^2b^19, No], [0, 18, 1, a^1b^19, No], [0, 19, 0, a^0b^19, No], [1, 1, 17, a^18b^19, No], [1, 2, 16, a^17b^19, No], [1, 3, 15, a^16b^19, No], [1, 4, 14, a^15b^19, No], [1, 5, 13, a^14b^19, No], [1, 6, 12, a^13b^19, No], [1, 7, 11, a^12b^19, No], [1, 8, 10, a^11b^19, No], [1, 9, 9, a^10b^19, No], [1, 10, 8, a^9b^19, No], [1, 11, 7, a^8b^19, No], [1, 12, 6, a^7b^19, No], [1, 13, 5, a^6b^19, No], [1, 14, 4, a^5b^19, No], [1, 15, 3, a^4b^19, No], [1, 16, 2, a^3b^19, No], [1, 17, 1, a^2b^19, No], [1, 18, 0, a^1b^19, No], [2, 1, 16, a^18b^19, No], [2, 2, 15, a^17b^19, No], [2, 3, 14, a^16b^19, No], [2, 4, 13, a^15b^19, No], [2, 5, 12, a^14b^19, No], [2, 6, 11, a^13b^19, No], [2, 7, 10, a^12b^19, No], [2, 8, 9, a^11b^19, No], [2, 9, 8, a^10b^19, No], [2, 10, 7, a^9b^19, No], [2, 11, 6, a^8b^19, No], [2, 12, 5, a^7b^19, No], [2, 13, 4, a^6b^19, No], [2, 14, 3, a^5b^19, No], [2, 15, 2, a^4b^19, No], [2, 16, 1, a^3b^19, No], [2, 17, 0, a^2b^19, No], [3, 1, 15, a^18b^19, No], [3, 2, 14, a^17b^19, No], [3, 3, 13, a^16b^19, No], [3, 4, 12, a^15b^19, No], [3, 5, 11, a^14b^19, No], [3, 6, 10, a^13b^19, No], [3, 7, 9, a^12b^19, No], [3, 8, 8, a^11b^19, No], [3, 9, 7, a^10b^19, No], [3, 10, 6, a^9b^19, No], [3, 11, 5, a^8b^19, No], [3, 12, 4, a^7b^19, No], [3, 13, 3, a^6b^19, No], [3, 14, 2, a^5b^19, No], [3, 15, 1, a^4b^19, No], [3, 16, 0, a^3b^19, No], [4, 1, 14, a^18b^19, No], [4, 2, 13, a^17b^19, No], [4, 3, 12, a^16b^19, No], [4, 4, 11, a^15b^19, No], [4, 5, 10, a^14b^19, No], [4, 6, 9, a^13b^19, No], [4, 7, 8, a^12b^19, No], [4, 8, 7, a^11b^19, No], [4, 9, 6, a^10b^19, No], [4, 10, 5, a^9b^19, No], [4, 11, 4, a^8b^19, No], [4, 12, 3, a^7b^19, No], [4, 13, 2, a^6b^19, No], [4, 14, 1, a^5b^19, No], [4, 15, 0, a^4b^19, No], [5, 1, 13, a^18b^19, No], [5, 2, 12, a^17b^19, No], [5, 3, 11, a^16b^19, No], [5, 4, 10, a^15b^19, No], [5, 5, 9, a^14b^19, No], [5, 6, 8, a^13b^19, No], [5, 7, 7, a^12b^19, No], [5, 8, 6, a^11b^19, No], [5, 9, 5, a^10b^19, No], [5, 10, 4, a^9b^19, No], [5, 11, 3, a^8b^19, No], [5, 12, 2, a^7b^19, No], [5, 13, 1, a^6b^19, No], [5, 14, 0, a^5b^19, No], [6, 1, 12, a^18b^19, No], [6, 2, 11, a^17b^19, No], [6, 3, 10, a^16b^19, No], [6, 4, 9, a^15b^19, No], [6, 5, 8, a^14b^19, No], [6, 6, 7, a^13b^19, No], [6, 7, 6, a^12b^19, No], [6, 8, 5, a^11b^19, No], [6, 9, 4, a^10b^19, No], [6, 10, 3, a^9b^19, No], [6, 11, 2, a^8b^19, No], [6, 12, 1, a^7b^19, No], [6, 13, 0, a^6b^19, No], [7, 1, 11, a^18b^19, No], [7, 2, 10, a^17b^19, No], [7, 3, 9, a^16b^19, No], [7, 4, 8, a^15b^19, No], [7, 5, 7, a^14b^19, No], [7, 6, 6, a^13b^19, No], [7, 7, 5, a^12b^19, No], [7, 8, 4, a^11b^19, No], [7, 9, 3, a^10b^19, No], [7, 10, 2, a^9b^19, No], [7, 11, 1, a^8b^19, No], [7, 12, 0, a^7b^19, No], [8, 1, 10, a^18b^19, No], [8, 2, 9, a^17b^19, No], [8, 3, 8, a^16b^19, No], [8, 4, 7, a^15b^19, No], [8, 5, 6, a^14b^19, No], [8, 6, 5, a^13b^19, No], [8, 7, 4, a^12b^19, No], [8, 8, 3, a^11b^19, No], [8, 9, 2, a^10b^19, No], [8, 10, 1, a^9b^19, No], [8, 11, 0, a^8b^19, No], [9, 1, 9, a^18b^19, No], [9, 2, 8, a^17b^19, No], [9, 3, 7, a^16b^19, No], [9, 4, 6, a^15b^19, No], [9, 5, 5, a^14b^19, No], [9, 6, 4, a^13b^19, No], [9, 7, 3, a^12b^19, No], [9, 8, 2, a^11b^19, No], [9, 9, 1, a^10b^19, No], [9, 10, 0, a^9b^19, No], [10, 1, 8, a^18b^19, No], [10, 2, 7, a^17b^19, No], [10, 3, 6, a^16b^19, No], [10, 4, 5, a^15b^19, No], [10, 5, 4, a^14b^19, No], [10, 6, 3, a^13b^19, No], [10, 7, 2, a^12b^19, No], [10, 8, 1, a^11b^19, No], [10, 9, 0, a^10b^19, No], [11, 1, 7, a^18b^19, No], [11, 2, 6, a^17b^19, No], [11, 3, 5, a^16b^19, No], [11, 4, 4, a^15b^19, No], [11, 5, 3, a^14b^19, No], [11, 6, 2, a^13b^19, No], [11, 7, 1, a^12b^19, No], [11, 8, 0, a^11b^19, No], [12, 1, 6, a^18b^19, No], [12, 2, 5, a^17b^19, No], [12, 3, 4, a^16b^19, No], [12, 4, 3, a^15b^19, No], [12, 5, 2, a^14b^19, No], [12, 6, 1, a^13b^19, No], [12, 7, 0, a^12b^19, No], [13, 1, 5, a^18b^19, No], [13, 2, 4, a^17b^19, No], [13, 3, 3, a^16b^19, No], [13, 4, 2, a^15b^19, No], [13, 5, 1, a^14b^19, No], [13, 6, 0, a^13b^19, No], [14, 1, 4, a^18b^19, No], [14, 2, 3, a^17b^19, No], [14, 3, 2, a^16b^19, No], [14, 4, 1, a^15b^19, No], [14, 5, 0, a^14b^19, No], [15, 1, 3, a^18b^19, No], [15, 2, 2, a^17b^19, No], [15, 3, 1, a^16b^19, No], [15, 4, 0, a^15b^19, No], [16, 1, 2, a^18b^19, No], [16, 2, 1, a^17b^19, No], [16, 3, 0, a^16b^19, No], [17, 1, 1, a^18b^19, No], [17, 2, 0, a^17b^19, No], [18, 1, 0, a^18b^19, No]], 190, 190, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (19, a^19b^19, [[0, 1, 18, a^18b^19, No], [0, 2, 17, a^17b^19, No], [0, 3, 16, a^16b^19, No], [0, 4, 15, a^15b^19, No], [0, 5, 14, a^14b^19, No], [0, 6, 13, a^13b^19, No], [0, 7, 12, a^12b^19, No], [0, 8, 11, a^11b^19, No], [0, 9, 10, a^10b^19, No], [0, 10, 9, a^9b^19, No], [0, 11, 8, a^8b^19, No], [0, 12, 7, a^7b^19, No], [0, 13, 6, a^6b^19, No], [0, 14, 5, a^5b^19, No], [0, 15, 4, a^4b^19, No], [0, 16, 3, a^3b^19, No], [0, 17, 2, a^2b^19, No], [0, 18, 1, a^1b^19, No], [0, 19, 0, a^0b^19, No], [1, 1, 17, a^18b^19, No], [1, 2, 16, a^17b^19, No], [1, 3, 15, a^16b^19, No], [1, 4, 14, a^15b^19, No], [1, 5, 13, a^14b^19, No], [1, 6, 12, a^13b^19, No], [1, 7, 11, a^12b^19, No], [1, 8, 10, a^11b^19, No], [1, 9, 9, a^10b^19, No], [1, 10, 8, a^9b^19, No], [1, 11, 7, a^8b^19, No], [1, 12, 6, a^7b^19, No], [1, 13, 5, a^6b^19, No], [1, 14, 4, a^5b^19, No], [1, 15, 3, a^4b^19, No], [1, 16, 2, a^3b^19, No], [1, 17, 1, a^2b^19, No], [1, 18, 0, a^1b^19, No], [2, 1, 16, a^18b^19, No], [2, 2, 15, a^17b^19, No], [2, 3, 14, a^16b^19, No], [2, 4, 13, a^15b^19, No], [2, 5, 12, a^14b^19, No], [2, 6, 11, a^13b^19, No], [2, 7, 10, a^12b^19, No], [2, 8, 9, a^11b^19, No], [2, 9, 8, a^10b^19, No], [2, 10, 7, a^9b^19, No], [2, 11, 6, a^8b^19, No], [2, 12, 5, a^7b^19, No], [2, 13, 4, a^6b^19, No], [2, 14, 3, a^5b^19, No], [2, 15, 2, a^4b^19, No], [2, 16, 1, a^3b^19, No], [2, 17, 0, a^2b^19, No], [3, 1, 15, a^18b^19, No], [3, 2, 14, a^17b^19, No], [3, 3, 13, a^16b^19, No], [3, 4, 12, a^15b^19, No], [3, 5, 11, a^14b^19, No], [3, 6, 10, a^13b^19, No], [3, 7, 9, a^12b^19, No], [3, 8, 8, a^11b^19, No], [3, 9, 7, a^10b^19, No], [3, 10, 6, a^9b^19, No], [3, 11, 5, a^8b^19, No], [3, 12, 4, a^7b^19, No], [3, 13, 3, a^6b^19, No], [3, 14, 2, a^5b^19, No], [3, 15, 1, a^4b^19, No], [3, 16, 0, a^3b^19, No], [4, 1, 14, a^18b^19, No], [4, 2, 13, a^17b^19, No], [4, 3, 12, a^16b^19, No], [4, 4, 11, a^15b^19, No], [4, 5, 10, a^14b^19, No], [4, 6, 9, a^13b^19, No], [4, 7, 8, a^12b^19, No], [4, 8, 7, a^11b^19, No], [4, 9, 6, a^10b^19, No], [4, 10, 5, a^9b^19, No], [4, 11, 4, a^8b^19, No], [4, 12, 3, a^7b^19, No], [4, 13, 2, a^6b^19, No], [4, 14, 1, a^5b^19, No], [4, 15, 0, a^4b^19, No], [5, 1, 13, a^18b^19, No], [5, 2, 12, a^17b^19, No], [5, 3, 11, a^16b^19, No], [5, 4, 10, a^15b^19, No], [5, 5, 9, a^14b^19, No], [5, 6, 8, a^13b^19, No], [5, 7, 7, a^12b^19, No], [5, 8, 6, a^11b^19, No], [5, 9, 5, a^10b^19, No], [5, 10, 4, a^9b^19, No], [5, 11, 3, a^8b^19, No], [5, 12, 2, a^7b^19, No], [5, 13, 1, a^6b^19, No], [5, 14, 0, a^5b^19, No], [6, 1, 12, a^18b^19, No], [6, 2, 11, a^17b^19, No], [6, 3, 10, a^16b^19, No], [6, 4, 9, a^15b^19, No], [6, 5, 8, a^14b^19, No], [6, 6, 7, a^13b^19, No], [6, 7, 6, a^12b^19, No], [6, 8, 5, a^11b^19, No], [6, 9, 4, a^10b^19, No], [6, 10, 3, a^9b^19, No], [6, 11, 2, a^8b^19, No], [6, 12, 1, a^7b^19, No], [6, 13, 0, a^6b^19, No], [7, 1, 11, a^18b^19, No], [7, 2, 10, a^17b^19, No], [7, 3, 9, a^16b^19, No], [7, 4, 8, a^15b^19, No], [7, 5, 7, a^14b^19, No], [7, 6, 6, a^13b^19, No], [7, 7, 5, a^12b^19, No], [7, 8, 4, a^11b^19, No], [7, 9, 3, a^10b^19, No], [7, 10, 2, a^9b^19, No], [7, 11, 1, a^8b^19, No], [7, 12, 0, a^7b^19, No], [8, 1, 10, a^18b^19, No], [8, 2, 9, a^17b^19, No], [8, 3, 8, a^16b^19, No], [8, 4, 7, a^15b^19, No], [8, 5, 6, a^14b^19, No], [8, 6, 5, a^13b^19, No], [8, 7, 4, a^12b^19, No], [8, 8, 3, a^11b^19, No], [8, 9, 2, a^10b^19, No], [8, 10, 1, a^9b^19, No], [8, 11, 0, a^8b^19, No], [9, 1, 9, a^18b^19, No], [9, 2, 8, a^17b^19, No], [9, 3, 7, a^16b^19, No], [9, 4, 6, a^15b^19, No], [9, 5, 5, a^14b^19, No], [9, 6, 4, a^13b^19, No], [9, 7, 3, a^12b^19, No], [9, 8, 2, a^11b^19, No], [9, 9, 1, a^10b^19, No], [9, 10, 0, a^9b^19, No], [10, 1, 8, a^18b^19, No], [10, 2, 7, a^17b^19, No], [10, 3, 6, a^16b^19, No], [10, 4, 5, a^15b^19, No], [10, 5, 4, a^14b^19, No], [10, 6, 3, a^13b^19, No], [10, 7, 2, a^12b^19, No], [10, 8, 1, a^11b^19, No], [10, 9, 0, a^10b^19, No], [11, 1, 7, a^18b^19, No], [11, 2, 6, a^17b^19, No], [11, 3, 5, a^16b^19, No], [11, 4, 4, a^15b^19, No], [11, 5, 3, a^14b^19, No], [11, 6, 2, a^13b^19, No], [11, 7, 1, a^12b^19, No], [11, 8, 0, a^11b^19, No], [12, 1, 6, a^18b^19, No], [12, 2, 5, a^17b^19, No], [12, 3, 4, a^16b^19, No], [12, 4, 3, a^15b^19, No], [12, 5, 2, a^14b^19, No], [12, 6, 1, a^13b^19, No], [12, 7, 0, a^12b^19, No], [13, 1, 5, a^18b^19, No], [13, 2, 4, a^17b^19, No], [13, 3, 3, a^16b^19, No], [13, 4, 2, a^15b^19, No], [13, 5, 1, a^14b^19, No], [13, 6, 0, a^13b^19, No], [14, 1, 4, a^18b^19, No], [14, 2, 3, a^17b^19, No], [14, 3, 2, a^16b^19, No], [14, 4, 1, a^15b^19, No], [14, 5, 0, a^14b^19, No], [15, 1, 3, a^18b^19, No], [15, 2, 2, a^17b^19, No], [15, 3, 1, a^16b^19, No], [15, 4, 0, a^15b^19, No], [16, 1, 2, a^18b^19, No], [16, 2, 1, a^17b^19, No], [16, 3, 0, a^16b^19, No], [17, 1, 1, a^18b^19, No], [17, 2, 0, a^17b^19, No], [18, 1, 0, a^18b^19, No]], 190, 190, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0070","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43c9c1afec90fe1f2d3862c9420cf2e13f7a30ad77b66d76b36a67726d3487b1","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187070,"semantic_template":"pump-anbn-all-splits|{p:20}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=20 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(20, a^20b^20, [[0, 1, 19, a^19b^20, No], [0, 2, 18, a^18b^20, No], [0, 3, 17, a^17b^20, No], [0, 4, 16, a^16b^20, No], [0, 5, 15, a^15b^20, No], [0, 6, 14, a^14b^20, No], [0, 7, 13, a^13b^20, No], [0, 8, 12, a^12b^20, No], [0, 9, 11, a^11b^20, No], [0, 10, 10, a^10b^20, No], [0, 11, 9, a^9b^20, No], [0, 12, 8, a^8b^20, No], [0, 13, 7, a^7b^20, No], [0, 14, 6, a^6b^20, No], [0, 15, 5, a^5b^20, No], [0, 16, 4, a^4b^20, No], [0, 17, 3, a^3b^20, No], [0, 18, 2, a^2b^20, No], [0, 19, 1, a^1b^20, No], [0, 20, 0, a^0b^20, No], [1, 1, 18, a^19b^20, No], [1, 2, 17, a^18b^20, No], [1, 3, 16, a^17b^20, No], [1, 4, 15, a^16b^20, No], [1, 5, 14, a^15b^20, No], [1, 6, 13, a^14b^20, No], [1, 7, 12, a^13b^20, No], [1, 8, 11, a^12b^20, No], [1, 9, 10, a^11b^20, No], [1, 10, 9, a^10b^20, No], [1, 11, 8, a^9b^20, No], [1, 12, 7, a^8b^20, No], [1, 13, 6, a^7b^20, No], [1, 14, 5, a^6b^20, No], [1, 15, 4, a^5b^20, No], [1, 16, 3, a^4b^20, No], [1, 17, 2, a^3b^20, No], [1, 18, 1, a^2b^20, No], [1, 19, 0, a^1b^20, No], [2, 1, 17, a^19b^20, No], [2, 2, 16, a^18b^20, No], [2, 3, 15, a^17b^20, No], [2, 4, 14, a^16b^20, No], [2, 5, 13, a^15b^20, No], [2, 6, 12, a^14b^20, No], [2, 7, 11, a^13b^20, No], [2, 8, 10, a^12b^20, No], [2, 9, 9, a^11b^20, No], [2, 10, 8, a^10b^20, No], [2, 11, 7, a^9b^20, No], [2, 12, 6, a^8b^20, No], [2, 13, 5, a^7b^20, No], [2, 14, 4, a^6b^20, No], [2, 15, 3, a^5b^20, No], [2, 16, 2, a^4b^20, No], [2, 17, 1, a^3b^20, No], [2, 18, 0, a^2b^20, No], [3, 1, 16, a^19b^20, No], [3, 2, 15, a^18b^20, No], [3, 3, 14, a^17b^20, No], [3, 4, 13, a^16b^20, No], [3, 5, 12, a^15b^20, No], [3, 6, 11, a^14b^20, No], [3, 7, 10, a^13b^20, No], [3, 8, 9, a^12b^20, No], [3, 9, 8, a^11b^20, No], [3, 10, 7, a^10b^20, No], [3, 11, 6, a^9b^20, No], [3, 12, 5, a^8b^20, No], [3, 13, 4, a^7b^20, No], [3, 14, 3, a^6b^20, No], [3, 15, 2, a^5b^20, No], [3, 16, 1, a^4b^20, No], [3, 17, 0, a^3b^20, No], [4, 1, 15, a^19b^20, No], [4, 2, 14, a^18b^20, No], [4, 3, 13, a^17b^20, No], [4, 4, 12, a^16b^20, No], [4, 5, 11, a^15b^20, No], [4, 6, 10, a^14b^20, No], [4, 7, 9, a^13b^20, No], [4, 8, 8, a^12b^20, No], [4, 9, 7, a^11b^20, No], [4, 10, 6, a^10b^20, No], [4, 11, 5, a^9b^20, No], [4, 12, 4, a^8b^20, No], [4, 13, 3, a^7b^20, No], [4, 14, 2, a^6b^20, No], [4, 15, 1, a^5b^20, No], [4, 16, 0, a^4b^20, No], [5, 1, 14, a^19b^20, No], [5, 2, 13, a^18b^20, No], [5, 3, 12, a^17b^20, No], [5, 4, 11, a^16b^20, No], [5, 5, 10, a^15b^20, No], [5, 6, 9, a^14b^20, No], [5, 7, 8, a^13b^20, No], [5, 8, 7, a^12b^20, No], [5, 9, 6, a^11b^20, No], [5, 10, 5, a^10b^20, No], [5, 11, 4, a^9b^20, No], [5, 12, 3, a^8b^20, No], [5, 13, 2, a^7b^20, No], [5, 14, 1, a^6b^20, No], [5, 15, 0, a^5b^20, No], [6, 1, 13, a^19b^20, No], [6, 2, 12, a^18b^20, No], [6, 3, 11, a^17b^20, No], [6, 4, 10, a^16b^20, No], [6, 5, 9, a^15b^20, No], [6, 6, 8, a^14b^20, No], [6, 7, 7, a^13b^20, No], [6, 8, 6, a^12b^20, No], [6, 9, 5, a^11b^20, No], [6, 10, 4, a^10b^20, No], [6, 11, 3, a^9b^20, No], [6, 12, 2, a^8b^20, No], [6, 13, 1, a^7b^20, No], [6, 14, 0, a^6b^20, No], [7, 1, 12, a^19b^20, No], [7, 2, 11, a^18b^20, No], [7, 3, 10, a^17b^20, No], [7, 4, 9, a^16b^20, No], [7, 5, 8, a^15b^20, No], [7, 6, 7, a^14b^20, No], [7, 7, 6, a^13b^20, No], [7, 8, 5, a^12b^20, No], [7, 9, 4, a^11b^20, No], [7, 10, 3, a^10b^20, No], [7, 11, 2, a^9b^20, No], [7, 12, 1, a^8b^20, No], [7, 13, 0, a^7b^20, No], [8, 1, 11, a^19b^20, No], [8, 2, 10, a^18b^20, No], [8, 3, 9, a^17b^20, No], [8, 4, 8, a^16b^20, No], [8, 5, 7, a^15b^20, No], [8, 6, 6, a^14b^20, No], [8, 7, 5, a^13b^20, No], [8, 8, 4, a^12b^20, No], [8, 9, 3, a^11b^20, No], [8, 10, 2, a^10b^20, No], [8, 11, 1, a^9b^20, No], [8, 12, 0, a^8b^20, No], [9, 1, 10, a^19b^20, No], [9, 2, 9, a^18b^20, No], [9, 3, 8, a^17b^20, No], [9, 4, 7, a^16b^20, No], [9, 5, 6, a^15b^20, No], [9, 6, 5, a^14b^20, No], [9, 7, 4, a^13b^20, No], [9, 8, 3, a^12b^20, No], [9, 9, 2, a^11b^20, No], [9, 10, 1, a^10b^20, No], [9, 11, 0, a^9b^20, No], [10, 1, 9, a^19b^20, No], [10, 2, 8, a^18b^20, No], [10, 3, 7, a^17b^20, No], [10, 4, 6, a^16b^20, No], [10, 5, 5, a^15b^20, No], [10, 6, 4, a^14b^20, No], [10, 7, 3, a^13b^20, No], [10, 8, 2, a^12b^20, No], [10, 9, 1, a^11b^20, No], [10, 10, 0, a^10b^20, No], [11, 1, 8, a^19b^20, No], [11, 2, 7, a^18b^20, No], [11, 3, 6, a^17b^20, No], [11, 4, 5, a^16b^20, No], [11, 5, 4, a^15b^20, No], [11, 6, 3, a^14b^20, No], [11, 7, 2, a^13b^20, No], [11, 8, 1, a^12b^20, No], [11, 9, 0, a^11b^20, No], [12, 1, 7, a^19b^20, No], [12, 2, 6, a^18b^20, No], [12, 3, 5, a^17b^20, No], [12, 4, 4, a^16b^20, No], [12, 5, 3, a^15b^20, No], [12, 6, 2, a^14b^20, No], [12, 7, 1, a^13b^20, No], [12, 8, 0, a^12b^20, No], [13, 1, 6, a^19b^20, No], [13, 2, 5, a^18b^20, No], [13, 3, 4, a^17b^20, No], [13, 4, 3, a^16b^20, No], [13, 5, 2, a^15b^20, No], [13, 6, 1, a^14b^20, No], [13, 7, 0, a^13b^20, No], [14, 1, 5, a^19b^20, No], [14, 2, 4, a^18b^20, No], [14, 3, 3, a^17b^20, No], [14, 4, 2, a^16b^20, No], [14, 5, 1, a^15b^20, No], [14, 6, 0, a^14b^20, No], [15, 1, 4, a^19b^20, No], [15, 2, 3, a^18b^20, No], [15, 3, 2, a^17b^20, No], [15, 4, 1, a^16b^20, No], [15, 5, 0, a^15b^20, No], [16, 1, 3, a^19b^20, No], [16, 2, 2, a^18b^20, No], [16, 3, 1, a^17b^20, No], [16, 4, 0, a^16b^20, No], [17, 1, 2, a^19b^20, No], [17, 2, 1, a^18b^20, No], [17, 3, 0, a^17b^20, No], [18, 1, 1, a^19b^20, No], [18, 2, 0, a^18b^20, No], [19, 1, 0, a^19b^20, No]], 210, 210, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (20, a^20b^20, [[0, 1, 19, a^19b^20, No], [0, 2, 18, a^18b^20, No], [0, 3, 17, a^17b^20, No], [0, 4, 16, a^16b^20, No], [0, 5, 15, a^15b^20, No], [0, 6, 14, a^14b^20, No], [0, 7, 13, a^13b^20, No], [0, 8, 12, a^12b^20, No], [0, 9, 11, a^11b^20, No], [0, 10, 10, a^10b^20, No], [0, 11, 9, a^9b^20, No], [0, 12, 8, a^8b^20, No], [0, 13, 7, a^7b^20, No], [0, 14, 6, a^6b^20, No], [0, 15, 5, a^5b^20, No], [0, 16, 4, a^4b^20, No], [0, 17, 3, a^3b^20, No], [0, 18, 2, a^2b^20, No], [0, 19, 1, a^1b^20, No], [0, 20, 0, a^0b^20, No], [1, 1, 18, a^19b^20, No], [1, 2, 17, a^18b^20, No], [1, 3, 16, a^17b^20, No], [1, 4, 15, a^16b^20, No], [1, 5, 14, a^15b^20, No], [1, 6, 13, a^14b^20, No], [1, 7, 12, a^13b^20, No], [1, 8, 11, a^12b^20, No], [1, 9, 10, a^11b^20, No], [1, 10, 9, a^10b^20, No], [1, 11, 8, a^9b^20, No], [1, 12, 7, a^8b^20, No], [1, 13, 6, a^7b^20, No], [1, 14, 5, a^6b^20, No], [1, 15, 4, a^5b^20, No], [1, 16, 3, a^4b^20, No], [1, 17, 2, a^3b^20, No], [1, 18, 1, a^2b^20, No], [1, 19, 0, a^1b^20, No], [2, 1, 17, a^19b^20, No], [2, 2, 16, a^18b^20, No], [2, 3, 15, a^17b^20, No], [2, 4, 14, a^16b^20, No], [2, 5, 13, a^15b^20, No], [2, 6, 12, a^14b^20, No], [2, 7, 11, a^13b^20, No], [2, 8, 10, a^12b^20, No], [2, 9, 9, a^11b^20, No], [2, 10, 8, a^10b^20, No], [2, 11, 7, a^9b^20, No], [2, 12, 6, a^8b^20, No], [2, 13, 5, a^7b^20, No], [2, 14, 4, a^6b^20, No], [2, 15, 3, a^5b^20, No], [2, 16, 2, a^4b^20, No], [2, 17, 1, a^3b^20, No], [2, 18, 0, a^2b^20, No], [3, 1, 16, a^19b^20, No], [3, 2, 15, a^18b^20, No], [3, 3, 14, a^17b^20, No], [3, 4, 13, a^16b^20, No], [3, 5, 12, a^15b^20, No], [3, 6, 11, a^14b^20, No], [3, 7, 10, a^13b^20, No], [3, 8, 9, a^12b^20, No], [3, 9, 8, a^11b^20, No], [3, 10, 7, a^10b^20, No], [3, 11, 6, a^9b^20, No], [3, 12, 5, a^8b^20, No], [3, 13, 4, a^7b^20, No], [3, 14, 3, a^6b^20, No], [3, 15, 2, a^5b^20, No], [3, 16, 1, a^4b^20, No], [3, 17, 0, a^3b^20, No], [4, 1, 15, a^19b^20, No], [4, 2, 14, a^18b^20, No], [4, 3, 13, a^17b^20, No], [4, 4, 12, a^16b^20, No], [4, 5, 11, a^15b^20, No], [4, 6, 10, a^14b^20, No], [4, 7, 9, a^13b^20, No], [4, 8, 8, a^12b^20, No], [4, 9, 7, a^11b^20, No], [4, 10, 6, a^10b^20, No], [4, 11, 5, a^9b^20, No], [4, 12, 4, a^8b^20, No], [4, 13, 3, a^7b^20, No], [4, 14, 2, a^6b^20, No], [4, 15, 1, a^5b^20, No], [4, 16, 0, a^4b^20, No], [5, 1, 14, a^19b^20, No], [5, 2, 13, a^18b^20, No], [5, 3, 12, a^17b^20, No], [5, 4, 11, a^16b^20, No], [5, 5, 10, a^15b^20, No], [5, 6, 9, a^14b^20, No], [5, 7, 8, a^13b^20, No], [5, 8, 7, a^12b^20, No], [5, 9, 6, a^11b^20, No], [5, 10, 5, a^10b^20, No], [5, 11, 4, a^9b^20, No], [5, 12, 3, a^8b^20, No], [5, 13, 2, a^7b^20, No], [5, 14, 1, a^6b^20, No], [5, 15, 0, a^5b^20, No], [6, 1, 13, a^19b^20, No], [6, 2, 12, a^18b^20, No], [6, 3, 11, a^17b^20, No], [6, 4, 10, a^16b^20, No], [6, 5, 9, a^15b^20, No], [6, 6, 8, a^14b^20, No], [6, 7, 7, a^13b^20, No], [6, 8, 6, a^12b^20, No], [6, 9, 5, a^11b^20, No], [6, 10, 4, a^10b^20, No], [6, 11, 3, a^9b^20, No], [6, 12, 2, a^8b^20, No], [6, 13, 1, a^7b^20, No], [6, 14, 0, a^6b^20, No], [7, 1, 12, a^19b^20, No], [7, 2, 11, a^18b^20, No], [7, 3, 10, a^17b^20, No], [7, 4, 9, a^16b^20, No], [7, 5, 8, a^15b^20, No], [7, 6, 7, a^14b^20, No], [7, 7, 6, a^13b^20, No], [7, 8, 5, a^12b^20, No], [7, 9, 4, a^11b^20, No], [7, 10, 3, a^10b^20, No], [7, 11, 2, a^9b^20, No], [7, 12, 1, a^8b^20, No], [7, 13, 0, a^7b^20, No], [8, 1, 11, a^19b^20, No], [8, 2, 10, a^18b^20, No], [8, 3, 9, a^17b^20, No], [8, 4, 8, a^16b^20, No], [8, 5, 7, a^15b^20, No], [8, 6, 6, a^14b^20, No], [8, 7, 5, a^13b^20, No], [8, 8, 4, a^12b^20, No], [8, 9, 3, a^11b^20, No], [8, 10, 2, a^10b^20, No], [8, 11, 1, a^9b^20, No], [8, 12, 0, a^8b^20, No], [9, 1, 10, a^19b^20, No], [9, 2, 9, a^18b^20, No], [9, 3, 8, a^17b^20, No], [9, 4, 7, a^16b^20, No], [9, 5, 6, a^15b^20, No], [9, 6, 5, a^14b^20, No], [9, 7, 4, a^13b^20, No], [9, 8, 3, a^12b^20, No], [9, 9, 2, a^11b^20, No], [9, 10, 1, a^10b^20, No], [9, 11, 0, a^9b^20, No], [10, 1, 9, a^19b^20, No], [10, 2, 8, a^18b^20, No], [10, 3, 7, a^17b^20, No], [10, 4, 6, a^16b^20, No], [10, 5, 5, a^15b^20, No], [10, 6, 4, a^14b^20, No], [10, 7, 3, a^13b^20, No], [10, 8, 2, a^12b^20, No], [10, 9, 1, a^11b^20, No], [10, 10, 0, a^10b^20, No], [11, 1, 8, a^19b^20, No], [11, 2, 7, a^18b^20, No], [11, 3, 6, a^17b^20, No], [11, 4, 5, a^16b^20, No], [11, 5, 4, a^15b^20, No], [11, 6, 3, a^14b^20, No], [11, 7, 2, a^13b^20, No], [11, 8, 1, a^12b^20, No], [11, 9, 0, a^11b^20, No], [12, 1, 7, a^19b^20, No], [12, 2, 6, a^18b^20, No], [12, 3, 5, a^17b^20, No], [12, 4, 4, a^16b^20, No], [12, 5, 3, a^15b^20, No], [12, 6, 2, a^14b^20, No], [12, 7, 1, a^13b^20, No], [12, 8, 0, a^12b^20, No], [13, 1, 6, a^19b^20, No], [13, 2, 5, a^18b^20, No], [13, 3, 4, a^17b^20, No], [13, 4, 3, a^16b^20, No], [13, 5, 2, a^15b^20, No], [13, 6, 1, a^14b^20, No], [13, 7, 0, a^13b^20, No], [14, 1, 5, a^19b^20, No], [14, 2, 4, a^18b^20, No], [14, 3, 3, a^17b^20, No], [14, 4, 2, a^16b^20, No], [14, 5, 1, a^15b^20, No], [14, 6, 0, a^14b^20, No], [15, 1, 4, a^19b^20, No], [15, 2, 3, a^18b^20, No], [15, 3, 2, a^17b^20, No], [15, 4, 1, a^16b^20, No], [15, 5, 0, a^15b^20, No], [16, 1, 3, a^19b^20, No], [16, 2, 2, a^18b^20, No], [16, 3, 1, a^17b^20, No], [16, 4, 0, a^16b^20, No], [17, 1, 2, a^19b^20, No], [17, 2, 1, a^18b^20, No], [17, 3, 0, a^17b^20, No], [18, 1, 1, a^19b^20, No], [18, 2, 0, a^18b^20, No], [19, 1, 0, a^19b^20, No]], 210, 210, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0071","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f423de73fcfc0a4103c8c005a8fa0c6a62986463f417f59284b92177fd519aa","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187071,"semantic_template":"pump-anbn-all-splits|{p:21}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=21 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(21, a^21b^21, [[0, 1, 20, a^20b^21, No], [0, 2, 19, a^19b^21, No], [0, 3, 18, a^18b^21, No], [0, 4, 17, a^17b^21, No], [0, 5, 16, a^16b^21, No], [0, 6, 15, a^15b^21, No], [0, 7, 14, a^14b^21, No], [0, 8, 13, a^13b^21, No], [0, 9, 12, a^12b^21, No], [0, 10, 11, a^11b^21, No], [0, 11, 10, a^10b^21, No], [0, 12, 9, a^9b^21, No], [0, 13, 8, a^8b^21, No], [0, 14, 7, a^7b^21, No], [0, 15, 6, a^6b^21, No], [0, 16, 5, a^5b^21, No], [0, 17, 4, a^4b^21, No], [0, 18, 3, a^3b^21, No], [0, 19, 2, a^2b^21, No], [0, 20, 1, a^1b^21, No], [0, 21, 0, a^0b^21, No], [1, 1, 19, a^20b^21, No], [1, 2, 18, a^19b^21, No], [1, 3, 17, a^18b^21, No], [1, 4, 16, a^17b^21, No], [1, 5, 15, a^16b^21, No], [1, 6, 14, a^15b^21, No], [1, 7, 13, a^14b^21, No], [1, 8, 12, a^13b^21, No], [1, 9, 11, a^12b^21, No], [1, 10, 10, a^11b^21, No], [1, 11, 9, a^10b^21, No], [1, 12, 8, a^9b^21, No], [1, 13, 7, a^8b^21, No], [1, 14, 6, a^7b^21, No], [1, 15, 5, a^6b^21, No], [1, 16, 4, a^5b^21, No], [1, 17, 3, a^4b^21, No], [1, 18, 2, a^3b^21, No], [1, 19, 1, a^2b^21, No], [1, 20, 0, a^1b^21, No], [2, 1, 18, a^20b^21, No], [2, 2, 17, a^19b^21, No], [2, 3, 16, a^18b^21, No], [2, 4, 15, a^17b^21, No], [2, 5, 14, a^16b^21, No], [2, 6, 13, a^15b^21, No], [2, 7, 12, a^14b^21, No], [2, 8, 11, a^13b^21, No], [2, 9, 10, a^12b^21, No], [2, 10, 9, a^11b^21, No], [2, 11, 8, a^10b^21, No], [2, 12, 7, a^9b^21, No], [2, 13, 6, a^8b^21, No], [2, 14, 5, a^7b^21, No], [2, 15, 4, a^6b^21, No], [2, 16, 3, a^5b^21, No], [2, 17, 2, a^4b^21, No], [2, 18, 1, a^3b^21, No], [2, 19, 0, a^2b^21, No], [3, 1, 17, a^20b^21, No], [3, 2, 16, a^19b^21, No], [3, 3, 15, a^18b^21, No], [3, 4, 14, a^17b^21, No], [3, 5, 13, a^16b^21, No], [3, 6, 12, a^15b^21, No], [3, 7, 11, a^14b^21, No], [3, 8, 10, a^13b^21, No], [3, 9, 9, a^12b^21, No], [3, 10, 8, a^11b^21, No], [3, 11, 7, a^10b^21, No], [3, 12, 6, a^9b^21, No], [3, 13, 5, a^8b^21, No], [3, 14, 4, a^7b^21, No], [3, 15, 3, a^6b^21, No], [3, 16, 2, a^5b^21, No], [3, 17, 1, a^4b^21, No], [3, 18, 0, a^3b^21, No], [4, 1, 16, a^20b^21, No], [4, 2, 15, a^19b^21, No], [4, 3, 14, a^18b^21, No], [4, 4, 13, a^17b^21, No], [4, 5, 12, a^16b^21, No], [4, 6, 11, a^15b^21, No], [4, 7, 10, a^14b^21, No], [4, 8, 9, a^13b^21, No], [4, 9, 8, a^12b^21, No], [4, 10, 7, a^11b^21, No], [4, 11, 6, a^10b^21, No], [4, 12, 5, a^9b^21, No], [4, 13, 4, a^8b^21, No], [4, 14, 3, a^7b^21, No], [4, 15, 2, a^6b^21, No], [4, 16, 1, a^5b^21, No], [4, 17, 0, a^4b^21, No], [5, 1, 15, a^20b^21, No], [5, 2, 14, a^19b^21, No], [5, 3, 13, a^18b^21, No], [5, 4, 12, a^17b^21, No], [5, 5, 11, a^16b^21, No], [5, 6, 10, a^15b^21, No], [5, 7, 9, a^14b^21, No], [5, 8, 8, a^13b^21, No], [5, 9, 7, a^12b^21, No], [5, 10, 6, a^11b^21, No], [5, 11, 5, a^10b^21, No], [5, 12, 4, a^9b^21, No], [5, 13, 3, a^8b^21, No], [5, 14, 2, a^7b^21, No], [5, 15, 1, a^6b^21, No], [5, 16, 0, a^5b^21, No], [6, 1, 14, a^20b^21, No], [6, 2, 13, a^19b^21, No], [6, 3, 12, a^18b^21, No], [6, 4, 11, a^17b^21, No], [6, 5, 10, a^16b^21, No], [6, 6, 9, a^15b^21, No], [6, 7, 8, a^14b^21, No], [6, 8, 7, a^13b^21, No], [6, 9, 6, a^12b^21, No], [6, 10, 5, a^11b^21, No], [6, 11, 4, a^10b^21, No], [6, 12, 3, a^9b^21, No], [6, 13, 2, a^8b^21, No], [6, 14, 1, a^7b^21, No], [6, 15, 0, a^6b^21, No], [7, 1, 13, a^20b^21, No], [7, 2, 12, a^19b^21, No], [7, 3, 11, a^18b^21, No], [7, 4, 10, a^17b^21, No], [7, 5, 9, a^16b^21, No], [7, 6, 8, a^15b^21, No], [7, 7, 7, a^14b^21, No], [7, 8, 6, a^13b^21, No], [7, 9, 5, a^12b^21, No], [7, 10, 4, a^11b^21, No], [7, 11, 3, a^10b^21, No], [7, 12, 2, a^9b^21, No], [7, 13, 1, a^8b^21, No], [7, 14, 0, a^7b^21, No], [8, 1, 12, a^20b^21, No], [8, 2, 11, a^19b^21, No], [8, 3, 10, a^18b^21, No], [8, 4, 9, a^17b^21, No], [8, 5, 8, a^16b^21, No], [8, 6, 7, a^15b^21, No], [8, 7, 6, a^14b^21, No], [8, 8, 5, a^13b^21, No], [8, 9, 4, a^12b^21, No], [8, 10, 3, a^11b^21, No], [8, 11, 2, a^10b^21, No], [8, 12, 1, a^9b^21, No], [8, 13, 0, a^8b^21, No], [9, 1, 11, a^20b^21, No], [9, 2, 10, a^19b^21, No], [9, 3, 9, a^18b^21, No], [9, 4, 8, a^17b^21, No], [9, 5, 7, a^16b^21, No], [9, 6, 6, a^15b^21, No], [9, 7, 5, a^14b^21, No], [9, 8, 4, a^13b^21, No], [9, 9, 3, a^12b^21, No], [9, 10, 2, a^11b^21, No], [9, 11, 1, a^10b^21, No], [9, 12, 0, a^9b^21, No], [10, 1, 10, a^20b^21, No], [10, 2, 9, a^19b^21, No], [10, 3, 8, a^18b^21, No], [10, 4, 7, a^17b^21, No], [10, 5, 6, a^16b^21, No], [10, 6, 5, a^15b^21, No], [10, 7, 4, a^14b^21, No], [10, 8, 3, a^13b^21, No], [10, 9, 2, a^12b^21, No], [10, 10, 1, a^11b^21, No], [10, 11, 0, a^10b^21, No], [11, 1, 9, a^20b^21, No], [11, 2, 8, a^19b^21, No], [11, 3, 7, a^18b^21, No], [11, 4, 6, a^17b^21, No], [11, 5, 5, a^16b^21, No], [11, 6, 4, a^15b^21, No], [11, 7, 3, a^14b^21, No], [11, 8, 2, a^13b^21, No], [11, 9, 1, a^12b^21, No], [11, 10, 0, a^11b^21, No], [12, 1, 8, a^20b^21, No], [12, 2, 7, a^19b^21, No], [12, 3, 6, a^18b^21, No], [12, 4, 5, a^17b^21, No], [12, 5, 4, a^16b^21, No], [12, 6, 3, a^15b^21, No], [12, 7, 2, a^14b^21, No], [12, 8, 1, a^13b^21, No], [12, 9, 0, a^12b^21, No], [13, 1, 7, a^20b^21, No], [13, 2, 6, a^19b^21, No], [13, 3, 5, a^18b^21, No], [13, 4, 4, a^17b^21, No], [13, 5, 3, a^16b^21, No], [13, 6, 2, a^15b^21, No], [13, 7, 1, a^14b^21, No], [13, 8, 0, a^13b^21, No], [14, 1, 6, a^20b^21, No], [14, 2, 5, a^19b^21, No], [14, 3, 4, a^18b^21, No], [14, 4, 3, a^17b^21, No], [14, 5, 2, a^16b^21, No], [14, 6, 1, a^15b^21, No], [14, 7, 0, a^14b^21, No], [15, 1, 5, a^20b^21, No], [15, 2, 4, a^19b^21, No], [15, 3, 3, a^18b^21, No], [15, 4, 2, a^17b^21, No], [15, 5, 1, a^16b^21, No], [15, 6, 0, a^15b^21, No], [16, 1, 4, a^20b^21, No], [16, 2, 3, a^19b^21, No], [16, 3, 2, a^18b^21, No], [16, 4, 1, a^17b^21, No], [16, 5, 0, a^16b^21, No], [17, 1, 3, a^20b^21, No], [17, 2, 2, a^19b^21, No], [17, 3, 1, a^18b^21, No], [17, 4, 0, a^17b^21, No], [18, 1, 2, a^20b^21, No], [18, 2, 1, a^19b^21, No], [18, 3, 0, a^18b^21, No], [19, 1, 1, a^20b^21, No], [19, 2, 0, a^19b^21, No], [20, 1, 0, a^20b^21, No]], 231, 231, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (21, a^21b^21, [[0, 1, 20, a^20b^21, No], [0, 2, 19, a^19b^21, No], [0, 3, 18, a^18b^21, No], [0, 4, 17, a^17b^21, No], [0, 5, 16, a^16b^21, No], [0, 6, 15, a^15b^21, No], [0, 7, 14, a^14b^21, No], [0, 8, 13, a^13b^21, No], [0, 9, 12, a^12b^21, No], [0, 10, 11, a^11b^21, No], [0, 11, 10, a^10b^21, No], [0, 12, 9, a^9b^21, No], [0, 13, 8, a^8b^21, No], [0, 14, 7, a^7b^21, No], [0, 15, 6, a^6b^21, No], [0, 16, 5, a^5b^21, No], [0, 17, 4, a^4b^21, No], [0, 18, 3, a^3b^21, No], [0, 19, 2, a^2b^21, No], [0, 20, 1, a^1b^21, No], [0, 21, 0, a^0b^21, No], [1, 1, 19, a^20b^21, No], [1, 2, 18, a^19b^21, No], [1, 3, 17, a^18b^21, No], [1, 4, 16, a^17b^21, No], [1, 5, 15, a^16b^21, No], [1, 6, 14, a^15b^21, No], [1, 7, 13, a^14b^21, No], [1, 8, 12, a^13b^21, No], [1, 9, 11, a^12b^21, No], [1, 10, 10, a^11b^21, No], [1, 11, 9, a^10b^21, No], [1, 12, 8, a^9b^21, No], [1, 13, 7, a^8b^21, No], [1, 14, 6, a^7b^21, No], [1, 15, 5, a^6b^21, No], [1, 16, 4, a^5b^21, No], [1, 17, 3, a^4b^21, No], [1, 18, 2, a^3b^21, No], [1, 19, 1, a^2b^21, No], [1, 20, 0, a^1b^21, No], [2, 1, 18, a^20b^21, No], [2, 2, 17, a^19b^21, No], [2, 3, 16, a^18b^21, No], [2, 4, 15, a^17b^21, No], [2, 5, 14, a^16b^21, No], [2, 6, 13, a^15b^21, No], [2, 7, 12, a^14b^21, No], [2, 8, 11, a^13b^21, No], [2, 9, 10, a^12b^21, No], [2, 10, 9, a^11b^21, No], [2, 11, 8, a^10b^21, No], [2, 12, 7, a^9b^21, No], [2, 13, 6, a^8b^21, No], [2, 14, 5, a^7b^21, No], [2, 15, 4, a^6b^21, No], [2, 16, 3, a^5b^21, No], [2, 17, 2, a^4b^21, No], [2, 18, 1, a^3b^21, No], [2, 19, 0, a^2b^21, No], [3, 1, 17, a^20b^21, No], [3, 2, 16, a^19b^21, No], [3, 3, 15, a^18b^21, No], [3, 4, 14, a^17b^21, No], [3, 5, 13, a^16b^21, No], [3, 6, 12, a^15b^21, No], [3, 7, 11, a^14b^21, No], [3, 8, 10, a^13b^21, No], [3, 9, 9, a^12b^21, No], [3, 10, 8, a^11b^21, No], [3, 11, 7, a^10b^21, No], [3, 12, 6, a^9b^21, No], [3, 13, 5, a^8b^21, No], [3, 14, 4, a^7b^21, No], [3, 15, 3, a^6b^21, No], [3, 16, 2, a^5b^21, No], [3, 17, 1, a^4b^21, No], [3, 18, 0, a^3b^21, No], [4, 1, 16, a^20b^21, No], [4, 2, 15, a^19b^21, No], [4, 3, 14, a^18b^21, No], [4, 4, 13, a^17b^21, No], [4, 5, 12, a^16b^21, No], [4, 6, 11, a^15b^21, No], [4, 7, 10, a^14b^21, No], [4, 8, 9, a^13b^21, No], [4, 9, 8, a^12b^21, No], [4, 10, 7, a^11b^21, No], [4, 11, 6, a^10b^21, No], [4, 12, 5, a^9b^21, No], [4, 13, 4, a^8b^21, No], [4, 14, 3, a^7b^21, No], [4, 15, 2, a^6b^21, No], [4, 16, 1, a^5b^21, No], [4, 17, 0, a^4b^21, No], [5, 1, 15, a^20b^21, No], [5, 2, 14, a^19b^21, No], [5, 3, 13, a^18b^21, No], [5, 4, 12, a^17b^21, No], [5, 5, 11, a^16b^21, No], [5, 6, 10, a^15b^21, No], [5, 7, 9, a^14b^21, No], [5, 8, 8, a^13b^21, No], [5, 9, 7, a^12b^21, No], [5, 10, 6, a^11b^21, No], [5, 11, 5, a^10b^21, No], [5, 12, 4, a^9b^21, No], [5, 13, 3, a^8b^21, No], [5, 14, 2, a^7b^21, No], [5, 15, 1, a^6b^21, No], [5, 16, 0, a^5b^21, No], [6, 1, 14, a^20b^21, No], [6, 2, 13, a^19b^21, No], [6, 3, 12, a^18b^21, No], [6, 4, 11, a^17b^21, No], [6, 5, 10, a^16b^21, No], [6, 6, 9, a^15b^21, No], [6, 7, 8, a^14b^21, No], [6, 8, 7, a^13b^21, No], [6, 9, 6, a^12b^21, No], [6, 10, 5, a^11b^21, No], [6, 11, 4, a^10b^21, No], [6, 12, 3, a^9b^21, No], [6, 13, 2, a^8b^21, No], [6, 14, 1, a^7b^21, No], [6, 15, 0, a^6b^21, No], [7, 1, 13, a^20b^21, No], [7, 2, 12, a^19b^21, No], [7, 3, 11, a^18b^21, No], [7, 4, 10, a^17b^21, No], [7, 5, 9, a^16b^21, No], [7, 6, 8, a^15b^21, No], [7, 7, 7, a^14b^21, No], [7, 8, 6, a^13b^21, No], [7, 9, 5, a^12b^21, No], [7, 10, 4, a^11b^21, No], [7, 11, 3, a^10b^21, No], [7, 12, 2, a^9b^21, No], [7, 13, 1, a^8b^21, No], [7, 14, 0, a^7b^21, No], [8, 1, 12, a^20b^21, No], [8, 2, 11, a^19b^21, No], [8, 3, 10, a^18b^21, No], [8, 4, 9, a^17b^21, No], [8, 5, 8, a^16b^21, No], [8, 6, 7, a^15b^21, No], [8, 7, 6, a^14b^21, No], [8, 8, 5, a^13b^21, No], [8, 9, 4, a^12b^21, No], [8, 10, 3, a^11b^21, No], [8, 11, 2, a^10b^21, No], [8, 12, 1, a^9b^21, No], [8, 13, 0, a^8b^21, No], [9, 1, 11, a^20b^21, No], [9, 2, 10, a^19b^21, No], [9, 3, 9, a^18b^21, No], [9, 4, 8, a^17b^21, No], [9, 5, 7, a^16b^21, No], [9, 6, 6, a^15b^21, No], [9, 7, 5, a^14b^21, No], [9, 8, 4, a^13b^21, No], [9, 9, 3, a^12b^21, No], [9, 10, 2, a^11b^21, No], [9, 11, 1, a^10b^21, No], [9, 12, 0, a^9b^21, No], [10, 1, 10, a^20b^21, No], [10, 2, 9, a^19b^21, No], [10, 3, 8, a^18b^21, No], [10, 4, 7, a^17b^21, No], [10, 5, 6, a^16b^21, No], [10, 6, 5, a^15b^21, No], [10, 7, 4, a^14b^21, No], [10, 8, 3, a^13b^21, No], [10, 9, 2, a^12b^21, No], [10, 10, 1, a^11b^21, No], [10, 11, 0, a^10b^21, No], [11, 1, 9, a^20b^21, No], [11, 2, 8, a^19b^21, No], [11, 3, 7, a^18b^21, No], [11, 4, 6, a^17b^21, No], [11, 5, 5, a^16b^21, No], [11, 6, 4, a^15b^21, No], [11, 7, 3, a^14b^21, No], [11, 8, 2, a^13b^21, No], [11, 9, 1, a^12b^21, No], [11, 10, 0, a^11b^21, No], [12, 1, 8, a^20b^21, No], [12, 2, 7, a^19b^21, No], [12, 3, 6, a^18b^21, No], [12, 4, 5, a^17b^21, No], [12, 5, 4, a^16b^21, No], [12, 6, 3, a^15b^21, No], [12, 7, 2, a^14b^21, No], [12, 8, 1, a^13b^21, No], [12, 9, 0, a^12b^21, No], [13, 1, 7, a^20b^21, No], [13, 2, 6, a^19b^21, No], [13, 3, 5, a^18b^21, No], [13, 4, 4, a^17b^21, No], [13, 5, 3, a^16b^21, No], [13, 6, 2, a^15b^21, No], [13, 7, 1, a^14b^21, No], [13, 8, 0, a^13b^21, No], [14, 1, 6, a^20b^21, No], [14, 2, 5, a^19b^21, No], [14, 3, 4, a^18b^21, No], [14, 4, 3, a^17b^21, No], [14, 5, 2, a^16b^21, No], [14, 6, 1, a^15b^21, No], [14, 7, 0, a^14b^21, No], [15, 1, 5, a^20b^21, No], [15, 2, 4, a^19b^21, No], [15, 3, 3, a^18b^21, No], [15, 4, 2, a^17b^21, No], [15, 5, 1, a^16b^21, No], [15, 6, 0, a^15b^21, No], [16, 1, 4, a^20b^21, No], [16, 2, 3, a^19b^21, No], [16, 3, 2, a^18b^21, No], [16, 4, 1, a^17b^21, No], [16, 5, 0, a^16b^21, No], [17, 1, 3, a^20b^21, No], [17, 2, 2, a^19b^21, No], [17, 3, 1, a^18b^21, No], [17, 4, 0, a^17b^21, No], [18, 1, 2, a^20b^21, No], [18, 2, 1, a^19b^21, No], [18, 3, 0, a^18b^21, No], [19, 1, 1, a^20b^21, No], [19, 2, 0, a^19b^21, No], [20, 1, 0, a^20b^21, No]], 231, 231, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0072","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12ed5cbd245fb7442d39a0af1bd08791ed4125a1f526edfeb30e6d9cb653320e","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187072,"semantic_template":"pump-anbn-all-splits|{p:22}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=22 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(22, a^22b^22, [[0, 1, 21, a^21b^22, No], [0, 2, 20, a^20b^22, No], [0, 3, 19, a^19b^22, No], [0, 4, 18, a^18b^22, No], [0, 5, 17, a^17b^22, No], [0, 6, 16, a^16b^22, No], [0, 7, 15, a^15b^22, No], [0, 8, 14, a^14b^22, No], [0, 9, 13, a^13b^22, No], [0, 10, 12, a^12b^22, No], [0, 11, 11, a^11b^22, No], [0, 12, 10, a^10b^22, No], [0, 13, 9, a^9b^22, No], [0, 14, 8, a^8b^22, No], [0, 15, 7, a^7b^22, No], [0, 16, 6, a^6b^22, No], [0, 17, 5, a^5b^22, No], [0, 18, 4, a^4b^22, No], [0, 19, 3, a^3b^22, No], [0, 20, 2, a^2b^22, No], [0, 21, 1, a^1b^22, No], [0, 22, 0, a^0b^22, No], [1, 1, 20, a^21b^22, No], [1, 2, 19, a^20b^22, No], [1, 3, 18, a^19b^22, No], [1, 4, 17, a^18b^22, No], [1, 5, 16, a^17b^22, No], [1, 6, 15, a^16b^22, No], [1, 7, 14, a^15b^22, No], [1, 8, 13, a^14b^22, No], [1, 9, 12, a^13b^22, No], [1, 10, 11, a^12b^22, No], [1, 11, 10, a^11b^22, No], [1, 12, 9, a^10b^22, No], [1, 13, 8, a^9b^22, No], [1, 14, 7, a^8b^22, No], [1, 15, 6, a^7b^22, No], [1, 16, 5, a^6b^22, No], [1, 17, 4, a^5b^22, No], [1, 18, 3, a^4b^22, No], [1, 19, 2, a^3b^22, No], [1, 20, 1, a^2b^22, No], [1, 21, 0, a^1b^22, No], [2, 1, 19, a^21b^22, No], [2, 2, 18, a^20b^22, No], [2, 3, 17, a^19b^22, No], [2, 4, 16, a^18b^22, No], [2, 5, 15, a^17b^22, No], [2, 6, 14, a^16b^22, No], [2, 7, 13, a^15b^22, No], [2, 8, 12, a^14b^22, No], [2, 9, 11, a^13b^22, No], [2, 10, 10, a^12b^22, No], [2, 11, 9, a^11b^22, No], [2, 12, 8, a^10b^22, No], [2, 13, 7, a^9b^22, No], [2, 14, 6, a^8b^22, No], [2, 15, 5, a^7b^22, No], [2, 16, 4, a^6b^22, No], [2, 17, 3, a^5b^22, No], [2, 18, 2, a^4b^22, No], [2, 19, 1, a^3b^22, No], [2, 20, 0, a^2b^22, No], [3, 1, 18, a^21b^22, No], [3, 2, 17, a^20b^22, No], [3, 3, 16, a^19b^22, No], [3, 4, 15, a^18b^22, No], [3, 5, 14, a^17b^22, No], [3, 6, 13, a^16b^22, No], [3, 7, 12, a^15b^22, No], [3, 8, 11, a^14b^22, No], [3, 9, 10, a^13b^22, No], [3, 10, 9, a^12b^22, No], [3, 11, 8, a^11b^22, No], [3, 12, 7, a^10b^22, No], [3, 13, 6, a^9b^22, No], [3, 14, 5, a^8b^22, No], [3, 15, 4, a^7b^22, No], [3, 16, 3, a^6b^22, No], [3, 17, 2, a^5b^22, No], [3, 18, 1, a^4b^22, No], [3, 19, 0, a^3b^22, No], [4, 1, 17, a^21b^22, No], [4, 2, 16, a^20b^22, No], [4, 3, 15, a^19b^22, No], [4, 4, 14, a^18b^22, No], [4, 5, 13, a^17b^22, No], [4, 6, 12, a^16b^22, No], [4, 7, 11, a^15b^22, No], [4, 8, 10, a^14b^22, No], [4, 9, 9, a^13b^22, No], [4, 10, 8, a^12b^22, No], [4, 11, 7, a^11b^22, No], [4, 12, 6, a^10b^22, No], [4, 13, 5, a^9b^22, No], [4, 14, 4, a^8b^22, No], [4, 15, 3, a^7b^22, No], [4, 16, 2, a^6b^22, No], [4, 17, 1, a^5b^22, No], [4, 18, 0, a^4b^22, No], [5, 1, 16, a^21b^22, No], [5, 2, 15, a^20b^22, No], [5, 3, 14, a^19b^22, No], [5, 4, 13, a^18b^22, No], [5, 5, 12, a^17b^22, No], [5, 6, 11, a^16b^22, No], [5, 7, 10, a^15b^22, No], [5, 8, 9, a^14b^22, No], [5, 9, 8, a^13b^22, No], [5, 10, 7, a^12b^22, No], [5, 11, 6, a^11b^22, No], [5, 12, 5, a^10b^22, No], [5, 13, 4, a^9b^22, No], [5, 14, 3, a^8b^22, No], [5, 15, 2, a^7b^22, No], [5, 16, 1, a^6b^22, No], [5, 17, 0, a^5b^22, No], [6, 1, 15, a^21b^22, No], [6, 2, 14, a^20b^22, No], [6, 3, 13, a^19b^22, No], [6, 4, 12, a^18b^22, No], [6, 5, 11, a^17b^22, No], [6, 6, 10, a^16b^22, No], [6, 7, 9, a^15b^22, No], [6, 8, 8, a^14b^22, No], [6, 9, 7, a^13b^22, No], [6, 10, 6, a^12b^22, No], [6, 11, 5, a^11b^22, No], [6, 12, 4, a^10b^22, No], [6, 13, 3, a^9b^22, No], [6, 14, 2, a^8b^22, No], [6, 15, 1, a^7b^22, No], [6, 16, 0, a^6b^22, No], [7, 1, 14, a^21b^22, No], [7, 2, 13, a^20b^22, No], [7, 3, 12, a^19b^22, No], [7, 4, 11, a^18b^22, No], [7, 5, 10, a^17b^22, No], [7, 6, 9, a^16b^22, No], [7, 7, 8, a^15b^22, No], [7, 8, 7, a^14b^22, No], [7, 9, 6, a^13b^22, No], [7, 10, 5, a^12b^22, No], [7, 11, 4, a^11b^22, No], [7, 12, 3, a^10b^22, No], [7, 13, 2, a^9b^22, No], [7, 14, 1, a^8b^22, No], [7, 15, 0, a^7b^22, No], [8, 1, 13, a^21b^22, No], [8, 2, 12, a^20b^22, No], [8, 3, 11, a^19b^22, No], [8, 4, 10, a^18b^22, No], [8, 5, 9, a^17b^22, No], [8, 6, 8, a^16b^22, No], [8, 7, 7, a^15b^22, No], [8, 8, 6, a^14b^22, No], [8, 9, 5, a^13b^22, No], [8, 10, 4, a^12b^22, No], [8, 11, 3, a^11b^22, No], [8, 12, 2, a^10b^22, No], [8, 13, 1, a^9b^22, No], [8, 14, 0, a^8b^22, No], [9, 1, 12, a^21b^22, No], [9, 2, 11, a^20b^22, No], [9, 3, 10, a^19b^22, No], [9, 4, 9, a^18b^22, No], [9, 5, 8, a^17b^22, No], [9, 6, 7, a^16b^22, No], [9, 7, 6, a^15b^22, No], [9, 8, 5, a^14b^22, No], [9, 9, 4, a^13b^22, No], [9, 10, 3, a^12b^22, No], [9, 11, 2, a^11b^22, No], [9, 12, 1, a^10b^22, No], [9, 13, 0, a^9b^22, No], [10, 1, 11, a^21b^22, No], [10, 2, 10, a^20b^22, No], [10, 3, 9, a^19b^22, No], [10, 4, 8, a^18b^22, No], [10, 5, 7, a^17b^22, No], [10, 6, 6, a^16b^22, No], [10, 7, 5, a^15b^22, No], [10, 8, 4, a^14b^22, No], [10, 9, 3, a^13b^22, No], [10, 10, 2, a^12b^22, No], [10, 11, 1, a^11b^22, No], [10, 12, 0, a^10b^22, No], [11, 1, 10, a^21b^22, No], [11, 2, 9, a^20b^22, No], [11, 3, 8, a^19b^22, No], [11, 4, 7, a^18b^22, No], [11, 5, 6, a^17b^22, No], [11, 6, 5, a^16b^22, No], [11, 7, 4, a^15b^22, No], [11, 8, 3, a^14b^22, No], [11, 9, 2, a^13b^22, No], [11, 10, 1, a^12b^22, No], [11, 11, 0, a^11b^22, No], [12, 1, 9, a^21b^22, No], [12, 2, 8, a^20b^22, No], [12, 3, 7, a^19b^22, No], [12, 4, 6, a^18b^22, No], [12, 5, 5, a^17b^22, No], [12, 6, 4, a^16b^22, No], [12, 7, 3, a^15b^22, No], [12, 8, 2, a^14b^22, No], [12, 9, 1, a^13b^22, No], [12, 10, 0, a^12b^22, No], [13, 1, 8, a^21b^22, No], [13, 2, 7, a^20b^22, No], [13, 3, 6, a^19b^22, No], [13, 4, 5, a^18b^22, No], [13, 5, 4, a^17b^22, No], [13, 6, 3, a^16b^22, No], [13, 7, 2, a^15b^22, No], [13, 8, 1, a^14b^22, No], [13, 9, 0, a^13b^22, No], [14, 1, 7, a^21b^22, No], [14, 2, 6, a^20b^22, No], [14, 3, 5, a^19b^22, No], [14, 4, 4, a^18b^22, No], [14, 5, 3, a^17b^22, No], [14, 6, 2, a^16b^22, No], [14, 7, 1, a^15b^22, No], [14, 8, 0, a^14b^22, No], [15, 1, 6, a^21b^22, No], [15, 2, 5, a^20b^22, No], [15, 3, 4, a^19b^22, No], [15, 4, 3, a^18b^22, No], [15, 5, 2, a^17b^22, No], [15, 6, 1, a^16b^22, No], [15, 7, 0, a^15b^22, No], [16, 1, 5, a^21b^22, No], [16, 2, 4, a^20b^22, No], [16, 3, 3, a^19b^22, No], [16, 4, 2, a^18b^22, No], [16, 5, 1, a^17b^22, No], [16, 6, 0, a^16b^22, No], [17, 1, 4, a^21b^22, No], [17, 2, 3, a^20b^22, No], [17, 3, 2, a^19b^22, No], [17, 4, 1, a^18b^22, No], [17, 5, 0, a^17b^22, No], [18, 1, 3, a^21b^22, No], [18, 2, 2, a^20b^22, No], [18, 3, 1, a^19b^22, No], [18, 4, 0, a^18b^22, No], [19, 1, 2, a^21b^22, No], [19, 2, 1, a^20b^22, No], [19, 3, 0, a^19b^22, No], [20, 1, 1, a^21b^22, No], [20, 2, 0, a^20b^22, No], [21, 1, 0, a^21b^22, No]], 253, 253, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (22, a^22b^22, [[0, 1, 21, a^21b^22, No], [0, 2, 20, a^20b^22, No], [0, 3, 19, a^19b^22, No], [0, 4, 18, a^18b^22, No], [0, 5, 17, a^17b^22, No], [0, 6, 16, a^16b^22, No], [0, 7, 15, a^15b^22, No], [0, 8, 14, a^14b^22, No], [0, 9, 13, a^13b^22, No], [0, 10, 12, a^12b^22, No], [0, 11, 11, a^11b^22, No], [0, 12, 10, a^10b^22, No], [0, 13, 9, a^9b^22, No], [0, 14, 8, a^8b^22, No], [0, 15, 7, a^7b^22, No], [0, 16, 6, a^6b^22, No], [0, 17, 5, a^5b^22, No], [0, 18, 4, a^4b^22, No], [0, 19, 3, a^3b^22, No], [0, 20, 2, a^2b^22, No], [0, 21, 1, a^1b^22, No], [0, 22, 0, a^0b^22, No], [1, 1, 20, a^21b^22, No], [1, 2, 19, a^20b^22, No], [1, 3, 18, a^19b^22, No], [1, 4, 17, a^18b^22, No], [1, 5, 16, a^17b^22, No], [1, 6, 15, a^16b^22, No], [1, 7, 14, a^15b^22, No], [1, 8, 13, a^14b^22, No], [1, 9, 12, a^13b^22, No], [1, 10, 11, a^12b^22, No], [1, 11, 10, a^11b^22, No], [1, 12, 9, a^10b^22, No], [1, 13, 8, a^9b^22, No], [1, 14, 7, a^8b^22, No], [1, 15, 6, a^7b^22, No], [1, 16, 5, a^6b^22, No], [1, 17, 4, a^5b^22, No], [1, 18, 3, a^4b^22, No], [1, 19, 2, a^3b^22, No], [1, 20, 1, a^2b^22, No], [1, 21, 0, a^1b^22, No], [2, 1, 19, a^21b^22, No], [2, 2, 18, a^20b^22, No], [2, 3, 17, a^19b^22, No], [2, 4, 16, a^18b^22, No], [2, 5, 15, a^17b^22, No], [2, 6, 14, a^16b^22, No], [2, 7, 13, a^15b^22, No], [2, 8, 12, a^14b^22, No], [2, 9, 11, a^13b^22, No], [2, 10, 10, a^12b^22, No], [2, 11, 9, a^11b^22, No], [2, 12, 8, a^10b^22, No], [2, 13, 7, a^9b^22, No], [2, 14, 6, a^8b^22, No], [2, 15, 5, a^7b^22, No], [2, 16, 4, a^6b^22, No], [2, 17, 3, a^5b^22, No], [2, 18, 2, a^4b^22, No], [2, 19, 1, a^3b^22, No], [2, 20, 0, a^2b^22, No], [3, 1, 18, a^21b^22, No], [3, 2, 17, a^20b^22, No], [3, 3, 16, a^19b^22, No], [3, 4, 15, a^18b^22, No], [3, 5, 14, a^17b^22, No], [3, 6, 13, a^16b^22, No], [3, 7, 12, a^15b^22, No], [3, 8, 11, a^14b^22, No], [3, 9, 10, a^13b^22, No], [3, 10, 9, a^12b^22, No], [3, 11, 8, a^11b^22, No], [3, 12, 7, a^10b^22, No], [3, 13, 6, a^9b^22, No], [3, 14, 5, a^8b^22, No], [3, 15, 4, a^7b^22, No], [3, 16, 3, a^6b^22, No], [3, 17, 2, a^5b^22, No], [3, 18, 1, a^4b^22, No], [3, 19, 0, a^3b^22, No], [4, 1, 17, a^21b^22, No], [4, 2, 16, a^20b^22, No], [4, 3, 15, a^19b^22, No], [4, 4, 14, a^18b^22, No], [4, 5, 13, a^17b^22, No], [4, 6, 12, a^16b^22, No], [4, 7, 11, a^15b^22, No], [4, 8, 10, a^14b^22, No], [4, 9, 9, a^13b^22, No], [4, 10, 8, a^12b^22, No], [4, 11, 7, a^11b^22, No], [4, 12, 6, a^10b^22, No], [4, 13, 5, a^9b^22, No], [4, 14, 4, a^8b^22, No], [4, 15, 3, a^7b^22, No], [4, 16, 2, a^6b^22, No], [4, 17, 1, a^5b^22, No], [4, 18, 0, a^4b^22, No], [5, 1, 16, a^21b^22, No], [5, 2, 15, a^20b^22, No], [5, 3, 14, a^19b^22, No], [5, 4, 13, a^18b^22, No], [5, 5, 12, a^17b^22, No], [5, 6, 11, a^16b^22, No], [5, 7, 10, a^15b^22, No], [5, 8, 9, a^14b^22, No], [5, 9, 8, a^13b^22, No], [5, 10, 7, a^12b^22, No], [5, 11, 6, a^11b^22, No], [5, 12, 5, a^10b^22, No], [5, 13, 4, a^9b^22, No], [5, 14, 3, a^8b^22, No], [5, 15, 2, a^7b^22, No], [5, 16, 1, a^6b^22, No], [5, 17, 0, a^5b^22, No], [6, 1, 15, a^21b^22, No], [6, 2, 14, a^20b^22, No], [6, 3, 13, a^19b^22, No], [6, 4, 12, a^18b^22, No], [6, 5, 11, a^17b^22, No], [6, 6, 10, a^16b^22, No], [6, 7, 9, a^15b^22, No], [6, 8, 8, a^14b^22, No], [6, 9, 7, a^13b^22, No], [6, 10, 6, a^12b^22, No], [6, 11, 5, a^11b^22, No], [6, 12, 4, a^10b^22, No], [6, 13, 3, a^9b^22, No], [6, 14, 2, a^8b^22, No], [6, 15, 1, a^7b^22, No], [6, 16, 0, a^6b^22, No], [7, 1, 14, a^21b^22, No], [7, 2, 13, a^20b^22, No], [7, 3, 12, a^19b^22, No], [7, 4, 11, a^18b^22, No], [7, 5, 10, a^17b^22, No], [7, 6, 9, a^16b^22, No], [7, 7, 8, a^15b^22, No], [7, 8, 7, a^14b^22, No], [7, 9, 6, a^13b^22, No], [7, 10, 5, a^12b^22, No], [7, 11, 4, a^11b^22, No], [7, 12, 3, a^10b^22, No], [7, 13, 2, a^9b^22, No], [7, 14, 1, a^8b^22, No], [7, 15, 0, a^7b^22, No], [8, 1, 13, a^21b^22, No], [8, 2, 12, a^20b^22, No], [8, 3, 11, a^19b^22, No], [8, 4, 10, a^18b^22, No], [8, 5, 9, a^17b^22, No], [8, 6, 8, a^16b^22, No], [8, 7, 7, a^15b^22, No], [8, 8, 6, a^14b^22, No], [8, 9, 5, a^13b^22, No], [8, 10, 4, a^12b^22, No], [8, 11, 3, a^11b^22, No], [8, 12, 2, a^10b^22, No], [8, 13, 1, a^9b^22, No], [8, 14, 0, a^8b^22, No], [9, 1, 12, a^21b^22, No], [9, 2, 11, a^20b^22, No], [9, 3, 10, a^19b^22, No], [9, 4, 9, a^18b^22, No], [9, 5, 8, a^17b^22, No], [9, 6, 7, a^16b^22, No], [9, 7, 6, a^15b^22, No], [9, 8, 5, a^14b^22, No], [9, 9, 4, a^13b^22, No], [9, 10, 3, a^12b^22, No], [9, 11, 2, a^11b^22, No], [9, 12, 1, a^10b^22, No], [9, 13, 0, a^9b^22, No], [10, 1, 11, a^21b^22, No], [10, 2, 10, a^20b^22, No], [10, 3, 9, a^19b^22, No], [10, 4, 8, a^18b^22, No], [10, 5, 7, a^17b^22, No], [10, 6, 6, a^16b^22, No], [10, 7, 5, a^15b^22, No], [10, 8, 4, a^14b^22, No], [10, 9, 3, a^13b^22, No], [10, 10, 2, a^12b^22, No], [10, 11, 1, a^11b^22, No], [10, 12, 0, a^10b^22, No], [11, 1, 10, a^21b^22, No], [11, 2, 9, a^20b^22, No], [11, 3, 8, a^19b^22, No], [11, 4, 7, a^18b^22, No], [11, 5, 6, a^17b^22, No], [11, 6, 5, a^16b^22, No], [11, 7, 4, a^15b^22, No], [11, 8, 3, a^14b^22, No], [11, 9, 2, a^13b^22, No], [11, 10, 1, a^12b^22, No], [11, 11, 0, a^11b^22, No], [12, 1, 9, a^21b^22, No], [12, 2, 8, a^20b^22, No], [12, 3, 7, a^19b^22, No], [12, 4, 6, a^18b^22, No], [12, 5, 5, a^17b^22, No], [12, 6, 4, a^16b^22, No], [12, 7, 3, a^15b^22, No], [12, 8, 2, a^14b^22, No], [12, 9, 1, a^13b^22, No], [12, 10, 0, a^12b^22, No], [13, 1, 8, a^21b^22, No], [13, 2, 7, a^20b^22, No], [13, 3, 6, a^19b^22, No], [13, 4, 5, a^18b^22, No], [13, 5, 4, a^17b^22, No], [13, 6, 3, a^16b^22, No], [13, 7, 2, a^15b^22, No], [13, 8, 1, a^14b^22, No], [13, 9, 0, a^13b^22, No], [14, 1, 7, a^21b^22, No], [14, 2, 6, a^20b^22, No], [14, 3, 5, a^19b^22, No], [14, 4, 4, a^18b^22, No], [14, 5, 3, a^17b^22, No], [14, 6, 2, a^16b^22, No], [14, 7, 1, a^15b^22, No], [14, 8, 0, a^14b^22, No], [15, 1, 6, a^21b^22, No], [15, 2, 5, a^20b^22, No], [15, 3, 4, a^19b^22, No], [15, 4, 3, a^18b^22, No], [15, 5, 2, a^17b^22, No], [15, 6, 1, a^16b^22, No], [15, 7, 0, a^15b^22, No], [16, 1, 5, a^21b^22, No], [16, 2, 4, a^20b^22, No], [16, 3, 3, a^19b^22, No], [16, 4, 2, a^18b^22, No], [16, 5, 1, a^17b^22, No], [16, 6, 0, a^16b^22, No], [17, 1, 4, a^21b^22, No], [17, 2, 3, a^20b^22, No], [17, 3, 2, a^19b^22, No], [17, 4, 1, a^18b^22, No], [17, 5, 0, a^17b^22, No], [18, 1, 3, a^21b^22, No], [18, 2, 2, a^20b^22, No], [18, 3, 1, a^19b^22, No], [18, 4, 0, a^18b^22, No], [19, 1, 2, a^21b^22, No], [19, 2, 1, a^20b^22, No], [19, 3, 0, a^19b^22, No], [20, 1, 1, a^21b^22, No], [20, 2, 0, a^20b^22, No], [21, 1, 0, a^21b^22, No]], 253, 253, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0073","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a9ac69baf2b7025516bab5cfdce30020f1627ab2432baa765dbd8d88bd92d5b","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187073,"semantic_template":"pump-anbn-all-splits|{p:23}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=23 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(23, a^23b^23, [[0, 1, 22, a^22b^23, No], [0, 2, 21, a^21b^23, No], [0, 3, 20, a^20b^23, No], [0, 4, 19, a^19b^23, No], [0, 5, 18, a^18b^23, No], [0, 6, 17, a^17b^23, No], [0, 7, 16, a^16b^23, No], [0, 8, 15, a^15b^23, No], [0, 9, 14, a^14b^23, No], [0, 10, 13, a^13b^23, No], [0, 11, 12, a^12b^23, No], [0, 12, 11, a^11b^23, No], [0, 13, 10, a^10b^23, No], [0, 14, 9, a^9b^23, No], [0, 15, 8, a^8b^23, No], [0, 16, 7, a^7b^23, No], [0, 17, 6, a^6b^23, No], [0, 18, 5, a^5b^23, No], [0, 19, 4, a^4b^23, No], [0, 20, 3, a^3b^23, No], [0, 21, 2, a^2b^23, No], [0, 22, 1, a^1b^23, No], [0, 23, 0, a^0b^23, No], [1, 1, 21, a^22b^23, No], [1, 2, 20, a^21b^23, No], [1, 3, 19, a^20b^23, No], [1, 4, 18, a^19b^23, No], [1, 5, 17, a^18b^23, No], [1, 6, 16, a^17b^23, No], [1, 7, 15, a^16b^23, No], [1, 8, 14, a^15b^23, No], [1, 9, 13, a^14b^23, No], [1, 10, 12, a^13b^23, No], [1, 11, 11, a^12b^23, No], [1, 12, 10, a^11b^23, No], [1, 13, 9, a^10b^23, No], [1, 14, 8, a^9b^23, No], [1, 15, 7, a^8b^23, No], [1, 16, 6, a^7b^23, No], [1, 17, 5, a^6b^23, No], [1, 18, 4, a^5b^23, No], [1, 19, 3, a^4b^23, No], [1, 20, 2, a^3b^23, No], [1, 21, 1, a^2b^23, No], [1, 22, 0, a^1b^23, No], [2, 1, 20, a^22b^23, No], [2, 2, 19, a^21b^23, No], [2, 3, 18, a^20b^23, No], [2, 4, 17, a^19b^23, No], [2, 5, 16, a^18b^23, No], [2, 6, 15, a^17b^23, No], [2, 7, 14, a^16b^23, No], [2, 8, 13, a^15b^23, No], [2, 9, 12, a^14b^23, No], [2, 10, 11, a^13b^23, No], [2, 11, 10, a^12b^23, No], [2, 12, 9, a^11b^23, No], [2, 13, 8, a^10b^23, No], [2, 14, 7, a^9b^23, No], [2, 15, 6, a^8b^23, No], [2, 16, 5, a^7b^23, No], [2, 17, 4, a^6b^23, No], [2, 18, 3, a^5b^23, No], [2, 19, 2, a^4b^23, No], [2, 20, 1, a^3b^23, No], [2, 21, 0, a^2b^23, No], [3, 1, 19, a^22b^23, No], [3, 2, 18, a^21b^23, No], [3, 3, 17, a^20b^23, No], [3, 4, 16, a^19b^23, No], [3, 5, 15, a^18b^23, No], [3, 6, 14, a^17b^23, No], [3, 7, 13, a^16b^23, No], [3, 8, 12, a^15b^23, No], [3, 9, 11, a^14b^23, No], [3, 10, 10, a^13b^23, No], [3, 11, 9, a^12b^23, No], [3, 12, 8, a^11b^23, No], [3, 13, 7, a^10b^23, No], [3, 14, 6, a^9b^23, No], [3, 15, 5, a^8b^23, No], [3, 16, 4, a^7b^23, No], [3, 17, 3, a^6b^23, No], [3, 18, 2, a^5b^23, No], [3, 19, 1, a^4b^23, No], [3, 20, 0, a^3b^23, No], [4, 1, 18, a^22b^23, No], [4, 2, 17, a^21b^23, No], [4, 3, 16, a^20b^23, No], [4, 4, 15, a^19b^23, No], [4, 5, 14, a^18b^23, No], [4, 6, 13, a^17b^23, No], [4, 7, 12, a^16b^23, No], [4, 8, 11, a^15b^23, No], [4, 9, 10, a^14b^23, No], [4, 10, 9, a^13b^23, No], [4, 11, 8, a^12b^23, No], [4, 12, 7, a^11b^23, No], [4, 13, 6, a^10b^23, No], [4, 14, 5, a^9b^23, No], [4, 15, 4, a^8b^23, No], [4, 16, 3, a^7b^23, No], [4, 17, 2, a^6b^23, No], [4, 18, 1, a^5b^23, No], [4, 19, 0, a^4b^23, No], [5, 1, 17, a^22b^23, No], [5, 2, 16, a^21b^23, No], [5, 3, 15, a^20b^23, No], [5, 4, 14, a^19b^23, No], [5, 5, 13, a^18b^23, No], [5, 6, 12, a^17b^23, No], [5, 7, 11, a^16b^23, No], [5, 8, 10, a^15b^23, No], [5, 9, 9, a^14b^23, No], [5, 10, 8, a^13b^23, No], [5, 11, 7, a^12b^23, No], [5, 12, 6, a^11b^23, No], [5, 13, 5, a^10b^23, No], [5, 14, 4, a^9b^23, No], [5, 15, 3, a^8b^23, No], [5, 16, 2, a^7b^23, No], [5, 17, 1, a^6b^23, No], [5, 18, 0, a^5b^23, No], [6, 1, 16, a^22b^23, No], [6, 2, 15, a^21b^23, No], [6, 3, 14, a^20b^23, No], [6, 4, 13, a^19b^23, No], [6, 5, 12, a^18b^23, No], [6, 6, 11, a^17b^23, No], [6, 7, 10, a^16b^23, No], [6, 8, 9, a^15b^23, No], [6, 9, 8, a^14b^23, No], [6, 10, 7, a^13b^23, No], [6, 11, 6, a^12b^23, No], [6, 12, 5, a^11b^23, No], [6, 13, 4, a^10b^23, No], [6, 14, 3, a^9b^23, No], [6, 15, 2, a^8b^23, No], [6, 16, 1, a^7b^23, No], [6, 17, 0, a^6b^23, No], [7, 1, 15, a^22b^23, No], [7, 2, 14, a^21b^23, No], [7, 3, 13, a^20b^23, No], [7, 4, 12, a^19b^23, No], [7, 5, 11, a^18b^23, No], [7, 6, 10, a^17b^23, No], [7, 7, 9, a^16b^23, No], [7, 8, 8, a^15b^23, No], [7, 9, 7, a^14b^23, No], [7, 10, 6, a^13b^23, No], [7, 11, 5, a^12b^23, No], [7, 12, 4, a^11b^23, No], [7, 13, 3, a^10b^23, No], [7, 14, 2, a^9b^23, No], [7, 15, 1, a^8b^23, No], [7, 16, 0, a^7b^23, No], [8, 1, 14, a^22b^23, No], [8, 2, 13, a^21b^23, No], [8, 3, 12, a^20b^23, No], [8, 4, 11, a^19b^23, No], [8, 5, 10, a^18b^23, No], [8, 6, 9, a^17b^23, No], [8, 7, 8, a^16b^23, No], [8, 8, 7, a^15b^23, No], [8, 9, 6, a^14b^23, No], [8, 10, 5, a^13b^23, No], [8, 11, 4, a^12b^23, No], [8, 12, 3, a^11b^23, No], [8, 13, 2, a^10b^23, No], [8, 14, 1, a^9b^23, No], [8, 15, 0, a^8b^23, No], [9, 1, 13, a^22b^23, No], [9, 2, 12, a^21b^23, No], [9, 3, 11, a^20b^23, No], [9, 4, 10, a^19b^23, No], [9, 5, 9, a^18b^23, No], [9, 6, 8, a^17b^23, No], [9, 7, 7, a^16b^23, No], [9, 8, 6, a^15b^23, No], [9, 9, 5, a^14b^23, No], [9, 10, 4, a^13b^23, No], [9, 11, 3, a^12b^23, No], [9, 12, 2, a^11b^23, No], [9, 13, 1, a^10b^23, No], [9, 14, 0, a^9b^23, No], [10, 1, 12, a^22b^23, No], [10, 2, 11, a^21b^23, No], [10, 3, 10, a^20b^23, No], [10, 4, 9, a^19b^23, No], [10, 5, 8, a^18b^23, No], [10, 6, 7, a^17b^23, No], [10, 7, 6, a^16b^23, No], [10, 8, 5, a^15b^23, No], [10, 9, 4, a^14b^23, No], [10, 10, 3, a^13b^23, No], [10, 11, 2, a^12b^23, No], [10, 12, 1, a^11b^23, No], [10, 13, 0, a^10b^23, No], [11, 1, 11, a^22b^23, No], [11, 2, 10, a^21b^23, No], [11, 3, 9, a^20b^23, No], [11, 4, 8, a^19b^23, No], [11, 5, 7, a^18b^23, No], [11, 6, 6, a^17b^23, No], [11, 7, 5, a^16b^23, No], [11, 8, 4, a^15b^23, No], [11, 9, 3, a^14b^23, No], [11, 10, 2, a^13b^23, No], [11, 11, 1, a^12b^23, No], [11, 12, 0, a^11b^23, No], [12, 1, 10, a^22b^23, No], [12, 2, 9, a^21b^23, No], [12, 3, 8, a^20b^23, No], [12, 4, 7, a^19b^23, No], [12, 5, 6, a^18b^23, No], [12, 6, 5, a^17b^23, No], [12, 7, 4, a^16b^23, No], [12, 8, 3, a^15b^23, No], [12, 9, 2, a^14b^23, No], [12, 10, 1, a^13b^23, No], [12, 11, 0, a^12b^23, No], [13, 1, 9, a^22b^23, No], [13, 2, 8, a^21b^23, No], [13, 3, 7, a^20b^23, No], [13, 4, 6, a^19b^23, No], [13, 5, 5, a^18b^23, No], [13, 6, 4, a^17b^23, No], [13, 7, 3, a^16b^23, No], [13, 8, 2, a^15b^23, No], [13, 9, 1, a^14b^23, No], [13, 10, 0, a^13b^23, No], [14, 1, 8, a^22b^23, No], [14, 2, 7, a^21b^23, No], [14, 3, 6, a^20b^23, No], [14, 4, 5, a^19b^23, No], [14, 5, 4, a^18b^23, No], [14, 6, 3, a^17b^23, No], [14, 7, 2, a^16b^23, No], [14, 8, 1, a^15b^23, No], [14, 9, 0, a^14b^23, No], [15, 1, 7, a^22b^23, No], [15, 2, 6, a^21b^23, No], [15, 3, 5, a^20b^23, No], [15, 4, 4, a^19b^23, No], [15, 5, 3, a^18b^23, No], [15, 6, 2, a^17b^23, No], [15, 7, 1, a^16b^23, No], [15, 8, 0, a^15b^23, No], [16, 1, 6, a^22b^23, No], [16, 2, 5, a^21b^23, No], [16, 3, 4, a^20b^23, No], [16, 4, 3, a^19b^23, No], [16, 5, 2, a^18b^23, No], [16, 6, 1, a^17b^23, No], [16, 7, 0, a^16b^23, No], [17, 1, 5, a^22b^23, No], [17, 2, 4, a^21b^23, No], [17, 3, 3, a^20b^23, No], [17, 4, 2, a^19b^23, No], [17, 5, 1, a^18b^23, No], [17, 6, 0, a^17b^23, No], [18, 1, 4, a^22b^23, No], [18, 2, 3, a^21b^23, No], [18, 3, 2, a^20b^23, No], [18, 4, 1, a^19b^23, No], [18, 5, 0, a^18b^23, No], [19, 1, 3, a^22b^23, No], [19, 2, 2, a^21b^23, No], [19, 3, 1, a^20b^23, No], [19, 4, 0, a^19b^23, No], [20, 1, 2, a^22b^23, No], [20, 2, 1, a^21b^23, No], [20, 3, 0, a^20b^23, No], [21, 1, 1, a^22b^23, No], [21, 2, 0, a^21b^23, No], [22, 1, 0, a^22b^23, No]], 276, 276, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (23, a^23b^23, [[0, 1, 22, a^22b^23, No], [0, 2, 21, a^21b^23, No], [0, 3, 20, a^20b^23, No], [0, 4, 19, a^19b^23, No], [0, 5, 18, a^18b^23, No], [0, 6, 17, a^17b^23, No], [0, 7, 16, a^16b^23, No], [0, 8, 15, a^15b^23, No], [0, 9, 14, a^14b^23, No], [0, 10, 13, a^13b^23, No], [0, 11, 12, a^12b^23, No], [0, 12, 11, a^11b^23, No], [0, 13, 10, a^10b^23, No], [0, 14, 9, a^9b^23, No], [0, 15, 8, a^8b^23, No], [0, 16, 7, a^7b^23, No], [0, 17, 6, a^6b^23, No], [0, 18, 5, a^5b^23, No], [0, 19, 4, a^4b^23, No], [0, 20, 3, a^3b^23, No], [0, 21, 2, a^2b^23, No], [0, 22, 1, a^1b^23, No], [0, 23, 0, a^0b^23, No], [1, 1, 21, a^22b^23, No], [1, 2, 20, a^21b^23, No], [1, 3, 19, a^20b^23, No], [1, 4, 18, a^19b^23, No], [1, 5, 17, a^18b^23, No], [1, 6, 16, a^17b^23, No], [1, 7, 15, a^16b^23, No], [1, 8, 14, a^15b^23, No], [1, 9, 13, a^14b^23, No], [1, 10, 12, a^13b^23, No], [1, 11, 11, a^12b^23, No], [1, 12, 10, a^11b^23, No], [1, 13, 9, a^10b^23, No], [1, 14, 8, a^9b^23, No], [1, 15, 7, a^8b^23, No], [1, 16, 6, a^7b^23, No], [1, 17, 5, a^6b^23, No], [1, 18, 4, a^5b^23, No], [1, 19, 3, a^4b^23, No], [1, 20, 2, a^3b^23, No], [1, 21, 1, a^2b^23, No], [1, 22, 0, a^1b^23, No], [2, 1, 20, a^22b^23, No], [2, 2, 19, a^21b^23, No], [2, 3, 18, a^20b^23, No], [2, 4, 17, a^19b^23, No], [2, 5, 16, a^18b^23, No], [2, 6, 15, a^17b^23, No], [2, 7, 14, a^16b^23, No], [2, 8, 13, a^15b^23, No], [2, 9, 12, a^14b^23, No], [2, 10, 11, a^13b^23, No], [2, 11, 10, a^12b^23, No], [2, 12, 9, a^11b^23, No], [2, 13, 8, a^10b^23, No], [2, 14, 7, a^9b^23, No], [2, 15, 6, a^8b^23, No], [2, 16, 5, a^7b^23, No], [2, 17, 4, a^6b^23, No], [2, 18, 3, a^5b^23, No], [2, 19, 2, a^4b^23, No], [2, 20, 1, a^3b^23, No], [2, 21, 0, a^2b^23, No], [3, 1, 19, a^22b^23, No], [3, 2, 18, a^21b^23, No], [3, 3, 17, a^20b^23, No], [3, 4, 16, a^19b^23, No], [3, 5, 15, a^18b^23, No], [3, 6, 14, a^17b^23, No], [3, 7, 13, a^16b^23, No], [3, 8, 12, a^15b^23, No], [3, 9, 11, a^14b^23, No], [3, 10, 10, a^13b^23, No], [3, 11, 9, a^12b^23, No], [3, 12, 8, a^11b^23, No], [3, 13, 7, a^10b^23, No], [3, 14, 6, a^9b^23, No], [3, 15, 5, a^8b^23, No], [3, 16, 4, a^7b^23, No], [3, 17, 3, a^6b^23, No], [3, 18, 2, a^5b^23, No], [3, 19, 1, a^4b^23, No], [3, 20, 0, a^3b^23, No], [4, 1, 18, a^22b^23, No], [4, 2, 17, a^21b^23, No], [4, 3, 16, a^20b^23, No], [4, 4, 15, a^19b^23, No], [4, 5, 14, a^18b^23, No], [4, 6, 13, a^17b^23, No], [4, 7, 12, a^16b^23, No], [4, 8, 11, a^15b^23, No], [4, 9, 10, a^14b^23, No], [4, 10, 9, a^13b^23, No], [4, 11, 8, a^12b^23, No], [4, 12, 7, a^11b^23, No], [4, 13, 6, a^10b^23, No], [4, 14, 5, a^9b^23, No], [4, 15, 4, a^8b^23, No], [4, 16, 3, a^7b^23, No], [4, 17, 2, a^6b^23, No], [4, 18, 1, a^5b^23, No], [4, 19, 0, a^4b^23, No], [5, 1, 17, a^22b^23, No], [5, 2, 16, a^21b^23, No], [5, 3, 15, a^20b^23, No], [5, 4, 14, a^19b^23, No], [5, 5, 13, a^18b^23, No], [5, 6, 12, a^17b^23, No], [5, 7, 11, a^16b^23, No], [5, 8, 10, a^15b^23, No], [5, 9, 9, a^14b^23, No], [5, 10, 8, a^13b^23, No], [5, 11, 7, a^12b^23, No], [5, 12, 6, a^11b^23, No], [5, 13, 5, a^10b^23, No], [5, 14, 4, a^9b^23, No], [5, 15, 3, a^8b^23, No], [5, 16, 2, a^7b^23, No], [5, 17, 1, a^6b^23, No], [5, 18, 0, a^5b^23, No], [6, 1, 16, a^22b^23, No], [6, 2, 15, a^21b^23, No], [6, 3, 14, a^20b^23, No], [6, 4, 13, a^19b^23, No], [6, 5, 12, a^18b^23, No], [6, 6, 11, a^17b^23, No], [6, 7, 10, a^16b^23, No], [6, 8, 9, a^15b^23, No], [6, 9, 8, a^14b^23, No], [6, 10, 7, a^13b^23, No], [6, 11, 6, a^12b^23, No], [6, 12, 5, a^11b^23, No], [6, 13, 4, a^10b^23, No], [6, 14, 3, a^9b^23, No], [6, 15, 2, a^8b^23, No], [6, 16, 1, a^7b^23, No], [6, 17, 0, a^6b^23, No], [7, 1, 15, a^22b^23, No], [7, 2, 14, a^21b^23, No], [7, 3, 13, a^20b^23, No], [7, 4, 12, a^19b^23, No], [7, 5, 11, a^18b^23, No], [7, 6, 10, a^17b^23, No], [7, 7, 9, a^16b^23, No], [7, 8, 8, a^15b^23, No], [7, 9, 7, a^14b^23, No], [7, 10, 6, a^13b^23, No], [7, 11, 5, a^12b^23, No], [7, 12, 4, a^11b^23, No], [7, 13, 3, a^10b^23, No], [7, 14, 2, a^9b^23, No], [7, 15, 1, a^8b^23, No], [7, 16, 0, a^7b^23, No], [8, 1, 14, a^22b^23, No], [8, 2, 13, a^21b^23, No], [8, 3, 12, a^20b^23, No], [8, 4, 11, a^19b^23, No], [8, 5, 10, a^18b^23, No], [8, 6, 9, a^17b^23, No], [8, 7, 8, a^16b^23, No], [8, 8, 7, a^15b^23, No], [8, 9, 6, a^14b^23, No], [8, 10, 5, a^13b^23, No], [8, 11, 4, a^12b^23, No], [8, 12, 3, a^11b^23, No], [8, 13, 2, a^10b^23, No], [8, 14, 1, a^9b^23, No], [8, 15, 0, a^8b^23, No], [9, 1, 13, a^22b^23, No], [9, 2, 12, a^21b^23, No], [9, 3, 11, a^20b^23, No], [9, 4, 10, a^19b^23, No], [9, 5, 9, a^18b^23, No], [9, 6, 8, a^17b^23, No], [9, 7, 7, a^16b^23, No], [9, 8, 6, a^15b^23, No], [9, 9, 5, a^14b^23, No], [9, 10, 4, a^13b^23, No], [9, 11, 3, a^12b^23, No], [9, 12, 2, a^11b^23, No], [9, 13, 1, a^10b^23, No], [9, 14, 0, a^9b^23, No], [10, 1, 12, a^22b^23, No], [10, 2, 11, a^21b^23, No], [10, 3, 10, a^20b^23, No], [10, 4, 9, a^19b^23, No], [10, 5, 8, a^18b^23, No], [10, 6, 7, a^17b^23, No], [10, 7, 6, a^16b^23, No], [10, 8, 5, a^15b^23, No], [10, 9, 4, a^14b^23, No], [10, 10, 3, a^13b^23, No], [10, 11, 2, a^12b^23, No], [10, 12, 1, a^11b^23, No], [10, 13, 0, a^10b^23, No], [11, 1, 11, a^22b^23, No], [11, 2, 10, a^21b^23, No], [11, 3, 9, a^20b^23, No], [11, 4, 8, a^19b^23, No], [11, 5, 7, a^18b^23, No], [11, 6, 6, a^17b^23, No], [11, 7, 5, a^16b^23, No], [11, 8, 4, a^15b^23, No], [11, 9, 3, a^14b^23, No], [11, 10, 2, a^13b^23, No], [11, 11, 1, a^12b^23, No], [11, 12, 0, a^11b^23, No], [12, 1, 10, a^22b^23, No], [12, 2, 9, a^21b^23, No], [12, 3, 8, a^20b^23, No], [12, 4, 7, a^19b^23, No], [12, 5, 6, a^18b^23, No], [12, 6, 5, a^17b^23, No], [12, 7, 4, a^16b^23, No], [12, 8, 3, a^15b^23, No], [12, 9, 2, a^14b^23, No], [12, 10, 1, a^13b^23, No], [12, 11, 0, a^12b^23, No], [13, 1, 9, a^22b^23, No], [13, 2, 8, a^21b^23, No], [13, 3, 7, a^20b^23, No], [13, 4, 6, a^19b^23, No], [13, 5, 5, a^18b^23, No], [13, 6, 4, a^17b^23, No], [13, 7, 3, a^16b^23, No], [13, 8, 2, a^15b^23, No], [13, 9, 1, a^14b^23, No], [13, 10, 0, a^13b^23, No], [14, 1, 8, a^22b^23, No], [14, 2, 7, a^21b^23, No], [14, 3, 6, a^20b^23, No], [14, 4, 5, a^19b^23, No], [14, 5, 4, a^18b^23, No], [14, 6, 3, a^17b^23, No], [14, 7, 2, a^16b^23, No], [14, 8, 1, a^15b^23, No], [14, 9, 0, a^14b^23, No], [15, 1, 7, a^22b^23, No], [15, 2, 6, a^21b^23, No], [15, 3, 5, a^20b^23, No], [15, 4, 4, a^19b^23, No], [15, 5, 3, a^18b^23, No], [15, 6, 2, a^17b^23, No], [15, 7, 1, a^16b^23, No], [15, 8, 0, a^15b^23, No], [16, 1, 6, a^22b^23, No], [16, 2, 5, a^21b^23, No], [16, 3, 4, a^20b^23, No], [16, 4, 3, a^19b^23, No], [16, 5, 2, a^18b^23, No], [16, 6, 1, a^17b^23, No], [16, 7, 0, a^16b^23, No], [17, 1, 5, a^22b^23, No], [17, 2, 4, a^21b^23, No], [17, 3, 3, a^20b^23, No], [17, 4, 2, a^19b^23, No], [17, 5, 1, a^18b^23, No], [17, 6, 0, a^17b^23, No], [18, 1, 4, a^22b^23, No], [18, 2, 3, a^21b^23, No], [18, 3, 2, a^20b^23, No], [18, 4, 1, a^19b^23, No], [18, 5, 0, a^18b^23, No], [19, 1, 3, a^22b^23, No], [19, 2, 2, a^21b^23, No], [19, 3, 1, a^20b^23, No], [19, 4, 0, a^19b^23, No], [20, 1, 2, a^22b^23, No], [20, 2, 1, a^21b^23, No], [20, 3, 0, a^20b^23, No], [21, 1, 1, a^22b^23, No], [21, 2, 0, a^21b^23, No], [22, 1, 0, a^22b^23, No]], 276, 276, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0074","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7476cde028ea5879d246c42477c5e2f28a05aa42b85c24a283f49930947979e","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187074,"semantic_template":"pump-anbn-all-splits|{p:24}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=24 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(24, a^24b^24, [[0, 1, 23, a^23b^24, No], [0, 2, 22, a^22b^24, No], [0, 3, 21, a^21b^24, No], [0, 4, 20, a^20b^24, No], [0, 5, 19, a^19b^24, No], [0, 6, 18, a^18b^24, No], [0, 7, 17, a^17b^24, No], [0, 8, 16, a^16b^24, No], [0, 9, 15, a^15b^24, No], [0, 10, 14, a^14b^24, No], [0, 11, 13, a^13b^24, No], [0, 12, 12, a^12b^24, No], [0, 13, 11, a^11b^24, No], [0, 14, 10, a^10b^24, No], [0, 15, 9, a^9b^24, No], [0, 16, 8, a^8b^24, No], [0, 17, 7, a^7b^24, No], [0, 18, 6, a^6b^24, No], [0, 19, 5, a^5b^24, No], [0, 20, 4, a^4b^24, No], [0, 21, 3, a^3b^24, No], [0, 22, 2, a^2b^24, No], [0, 23, 1, a^1b^24, No], [0, 24, 0, a^0b^24, No], [1, 1, 22, a^23b^24, No], [1, 2, 21, a^22b^24, No], [1, 3, 20, a^21b^24, No], [1, 4, 19, a^20b^24, No], [1, 5, 18, a^19b^24, No], [1, 6, 17, a^18b^24, No], [1, 7, 16, a^17b^24, No], [1, 8, 15, a^16b^24, No], [1, 9, 14, a^15b^24, No], [1, 10, 13, a^14b^24, No], [1, 11, 12, a^13b^24, No], [1, 12, 11, a^12b^24, No], [1, 13, 10, a^11b^24, No], [1, 14, 9, a^10b^24, No], [1, 15, 8, a^9b^24, No], [1, 16, 7, a^8b^24, No], [1, 17, 6, a^7b^24, No], [1, 18, 5, a^6b^24, No], [1, 19, 4, a^5b^24, No], [1, 20, 3, a^4b^24, No], [1, 21, 2, a^3b^24, No], [1, 22, 1, a^2b^24, No], [1, 23, 0, a^1b^24, No], [2, 1, 21, a^23b^24, No], [2, 2, 20, a^22b^24, No], [2, 3, 19, a^21b^24, No], [2, 4, 18, a^20b^24, No], [2, 5, 17, a^19b^24, No], [2, 6, 16, a^18b^24, No], [2, 7, 15, a^17b^24, No], [2, 8, 14, a^16b^24, No], [2, 9, 13, a^15b^24, No], [2, 10, 12, a^14b^24, No], [2, 11, 11, a^13b^24, No], [2, 12, 10, a^12b^24, No], [2, 13, 9, a^11b^24, No], [2, 14, 8, a^10b^24, No], [2, 15, 7, a^9b^24, No], [2, 16, 6, a^8b^24, No], [2, 17, 5, a^7b^24, No], [2, 18, 4, a^6b^24, No], [2, 19, 3, a^5b^24, No], [2, 20, 2, a^4b^24, No], [2, 21, 1, a^3b^24, No], [2, 22, 0, a^2b^24, No], [3, 1, 20, a^23b^24, No], [3, 2, 19, a^22b^24, No], [3, 3, 18, a^21b^24, No], [3, 4, 17, a^20b^24, No], [3, 5, 16, a^19b^24, No], [3, 6, 15, a^18b^24, No], [3, 7, 14, a^17b^24, No], [3, 8, 13, a^16b^24, No], [3, 9, 12, a^15b^24, No], [3, 10, 11, a^14b^24, No], [3, 11, 10, a^13b^24, No], [3, 12, 9, a^12b^24, No], [3, 13, 8, a^11b^24, No], [3, 14, 7, a^10b^24, No], [3, 15, 6, a^9b^24, No], [3, 16, 5, a^8b^24, No], [3, 17, 4, a^7b^24, No], [3, 18, 3, a^6b^24, No], [3, 19, 2, a^5b^24, No], [3, 20, 1, a^4b^24, No], [3, 21, 0, a^3b^24, No], [4, 1, 19, a^23b^24, No], [4, 2, 18, a^22b^24, No], [4, 3, 17, a^21b^24, No], [4, 4, 16, a^20b^24, No], [4, 5, 15, a^19b^24, No], [4, 6, 14, a^18b^24, No], [4, 7, 13, a^17b^24, No], [4, 8, 12, a^16b^24, No], [4, 9, 11, a^15b^24, No], [4, 10, 10, a^14b^24, No], [4, 11, 9, a^13b^24, No], [4, 12, 8, a^12b^24, No], [4, 13, 7, a^11b^24, No], [4, 14, 6, a^10b^24, No], [4, 15, 5, a^9b^24, No], [4, 16, 4, a^8b^24, No], [4, 17, 3, a^7b^24, No], [4, 18, 2, a^6b^24, No], [4, 19, 1, a^5b^24, No], [4, 20, 0, a^4b^24, No], [5, 1, 18, a^23b^24, No], [5, 2, 17, a^22b^24, No], [5, 3, 16, a^21b^24, No], [5, 4, 15, a^20b^24, No], [5, 5, 14, a^19b^24, No], [5, 6, 13, a^18b^24, No], [5, 7, 12, a^17b^24, No], [5, 8, 11, a^16b^24, No], [5, 9, 10, a^15b^24, No], [5, 10, 9, a^14b^24, No], [5, 11, 8, a^13b^24, No], [5, 12, 7, a^12b^24, No], [5, 13, 6, a^11b^24, No], [5, 14, 5, a^10b^24, No], [5, 15, 4, a^9b^24, No], [5, 16, 3, a^8b^24, No], [5, 17, 2, a^7b^24, No], [5, 18, 1, a^6b^24, No], [5, 19, 0, a^5b^24, No], [6, 1, 17, a^23b^24, No], [6, 2, 16, a^22b^24, No], [6, 3, 15, a^21b^24, No], [6, 4, 14, a^20b^24, No], [6, 5, 13, a^19b^24, No], [6, 6, 12, a^18b^24, No], [6, 7, 11, a^17b^24, No], [6, 8, 10, a^16b^24, No], [6, 9, 9, a^15b^24, No], [6, 10, 8, a^14b^24, No], [6, 11, 7, a^13b^24, No], [6, 12, 6, a^12b^24, No], [6, 13, 5, a^11b^24, No], [6, 14, 4, a^10b^24, No], [6, 15, 3, a^9b^24, No], [6, 16, 2, a^8b^24, No], [6, 17, 1, a^7b^24, No], [6, 18, 0, a^6b^24, No], [7, 1, 16, a^23b^24, No], [7, 2, 15, a^22b^24, No], [7, 3, 14, a^21b^24, No], [7, 4, 13, a^20b^24, No], [7, 5, 12, a^19b^24, No], [7, 6, 11, a^18b^24, No], [7, 7, 10, a^17b^24, No], [7, 8, 9, a^16b^24, No], [7, 9, 8, a^15b^24, No], [7, 10, 7, a^14b^24, No], [7, 11, 6, a^13b^24, No], [7, 12, 5, a^12b^24, No], [7, 13, 4, a^11b^24, No], [7, 14, 3, a^10b^24, No], [7, 15, 2, a^9b^24, No], [7, 16, 1, a^8b^24, No], [7, 17, 0, a^7b^24, No], [8, 1, 15, a^23b^24, No], [8, 2, 14, a^22b^24, No], [8, 3, 13, a^21b^24, No], [8, 4, 12, a^20b^24, No], [8, 5, 11, a^19b^24, No], [8, 6, 10, a^18b^24, No], [8, 7, 9, a^17b^24, No], [8, 8, 8, a^16b^24, No], [8, 9, 7, a^15b^24, No], [8, 10, 6, a^14b^24, No], [8, 11, 5, a^13b^24, No], [8, 12, 4, a^12b^24, No], [8, 13, 3, a^11b^24, No], [8, 14, 2, a^10b^24, No], [8, 15, 1, a^9b^24, No], [8, 16, 0, a^8b^24, No], [9, 1, 14, a^23b^24, No], [9, 2, 13, a^22b^24, No], [9, 3, 12, a^21b^24, No], [9, 4, 11, a^20b^24, No], [9, 5, 10, a^19b^24, No], [9, 6, 9, a^18b^24, No], [9, 7, 8, a^17b^24, No], [9, 8, 7, a^16b^24, No], [9, 9, 6, a^15b^24, No], [9, 10, 5, a^14b^24, No], [9, 11, 4, a^13b^24, No], [9, 12, 3, a^12b^24, No], [9, 13, 2, a^11b^24, No], [9, 14, 1, a^10b^24, No], [9, 15, 0, a^9b^24, No], [10, 1, 13, a^23b^24, No], [10, 2, 12, a^22b^24, No], [10, 3, 11, a^21b^24, No], [10, 4, 10, a^20b^24, No], [10, 5, 9, a^19b^24, No], [10, 6, 8, a^18b^24, No], [10, 7, 7, a^17b^24, No], [10, 8, 6, a^16b^24, No], [10, 9, 5, a^15b^24, No], [10, 10, 4, a^14b^24, No], [10, 11, 3, a^13b^24, No], [10, 12, 2, a^12b^24, No], [10, 13, 1, a^11b^24, No], [10, 14, 0, a^10b^24, No], [11, 1, 12, a^23b^24, No], [11, 2, 11, a^22b^24, No], [11, 3, 10, a^21b^24, No], [11, 4, 9, a^20b^24, No], [11, 5, 8, a^19b^24, No], [11, 6, 7, a^18b^24, No], [11, 7, 6, a^17b^24, No], [11, 8, 5, a^16b^24, No], [11, 9, 4, a^15b^24, No], [11, 10, 3, a^14b^24, No], [11, 11, 2, a^13b^24, No], [11, 12, 1, a^12b^24, No], [11, 13, 0, a^11b^24, No], [12, 1, 11, a^23b^24, No], [12, 2, 10, a^22b^24, No], [12, 3, 9, a^21b^24, No], [12, 4, 8, a^20b^24, No], [12, 5, 7, a^19b^24, No], [12, 6, 6, a^18b^24, No], [12, 7, 5, a^17b^24, No], [12, 8, 4, a^16b^24, No], [12, 9, 3, a^15b^24, No], [12, 10, 2, a^14b^24, No], [12, 11, 1, a^13b^24, No], [12, 12, 0, a^12b^24, No], [13, 1, 10, a^23b^24, No], [13, 2, 9, a^22b^24, No], [13, 3, 8, a^21b^24, No], [13, 4, 7, a^20b^24, No], [13, 5, 6, a^19b^24, No], [13, 6, 5, a^18b^24, No], [13, 7, 4, a^17b^24, No], [13, 8, 3, a^16b^24, No], [13, 9, 2, a^15b^24, No], [13, 10, 1, a^14b^24, No], [13, 11, 0, a^13b^24, No], [14, 1, 9, a^23b^24, No], [14, 2, 8, a^22b^24, No], [14, 3, 7, a^21b^24, No], [14, 4, 6, a^20b^24, No], [14, 5, 5, a^19b^24, No], [14, 6, 4, a^18b^24, No], [14, 7, 3, a^17b^24, No], [14, 8, 2, a^16b^24, No], [14, 9, 1, a^15b^24, No], [14, 10, 0, a^14b^24, No], [15, 1, 8, a^23b^24, No], [15, 2, 7, a^22b^24, No], [15, 3, 6, a^21b^24, No], [15, 4, 5, a^20b^24, No], [15, 5, 4, a^19b^24, No], [15, 6, 3, a^18b^24, No], [15, 7, 2, a^17b^24, No], [15, 8, 1, a^16b^24, No], [15, 9, 0, a^15b^24, No], [16, 1, 7, a^23b^24, No], [16, 2, 6, a^22b^24, No], [16, 3, 5, a^21b^24, No], [16, 4, 4, a^20b^24, No], [16, 5, 3, a^19b^24, No], [16, 6, 2, a^18b^24, No], [16, 7, 1, a^17b^24, No], [16, 8, 0, a^16b^24, No], [17, 1, 6, a^23b^24, No], [17, 2, 5, a^22b^24, No], [17, 3, 4, a^21b^24, No], [17, 4, 3, a^20b^24, No], [17, 5, 2, a^19b^24, No], [17, 6, 1, a^18b^24, No], [17, 7, 0, a^17b^24, No], [18, 1, 5, a^23b^24, No], [18, 2, 4, a^22b^24, No], [18, 3, 3, a^21b^24, No], [18, 4, 2, a^20b^24, No], [18, 5, 1, a^19b^24, No], [18, 6, 0, a^18b^24, No], [19, 1, 4, a^23b^24, No], [19, 2, 3, a^22b^24, No], [19, 3, 2, a^21b^24, No], [19, 4, 1, a^20b^24, No], [19, 5, 0, a^19b^24, No], [20, 1, 3, a^23b^24, No], [20, 2, 2, a^22b^24, No], [20, 3, 1, a^21b^24, No], [20, 4, 0, a^20b^24, No], [21, 1, 2, a^23b^24, No], [21, 2, 1, a^22b^24, No], [21, 3, 0, a^21b^24, No], [22, 1, 1, a^23b^24, No], [22, 2, 0, a^22b^24, No], [23, 1, 0, a^23b^24, No]], 300, 300, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (24, a^24b^24, [[0, 1, 23, a^23b^24, No], [0, 2, 22, a^22b^24, No], [0, 3, 21, a^21b^24, No], [0, 4, 20, a^20b^24, No], [0, 5, 19, a^19b^24, No], [0, 6, 18, a^18b^24, No], [0, 7, 17, a^17b^24, No], [0, 8, 16, a^16b^24, No], [0, 9, 15, a^15b^24, No], [0, 10, 14, a^14b^24, No], [0, 11, 13, a^13b^24, No], [0, 12, 12, a^12b^24, No], [0, 13, 11, a^11b^24, No], [0, 14, 10, a^10b^24, No], [0, 15, 9, a^9b^24, No], [0, 16, 8, a^8b^24, No], [0, 17, 7, a^7b^24, No], [0, 18, 6, a^6b^24, No], [0, 19, 5, a^5b^24, No], [0, 20, 4, a^4b^24, No], [0, 21, 3, a^3b^24, No], [0, 22, 2, a^2b^24, No], [0, 23, 1, a^1b^24, No], [0, 24, 0, a^0b^24, No], [1, 1, 22, a^23b^24, No], [1, 2, 21, a^22b^24, No], [1, 3, 20, a^21b^24, No], [1, 4, 19, a^20b^24, No], [1, 5, 18, a^19b^24, No], [1, 6, 17, a^18b^24, No], [1, 7, 16, a^17b^24, No], [1, 8, 15, a^16b^24, No], [1, 9, 14, a^15b^24, No], [1, 10, 13, a^14b^24, No], [1, 11, 12, a^13b^24, No], [1, 12, 11, a^12b^24, No], [1, 13, 10, a^11b^24, No], [1, 14, 9, a^10b^24, No], [1, 15, 8, a^9b^24, No], [1, 16, 7, a^8b^24, No], [1, 17, 6, a^7b^24, No], [1, 18, 5, a^6b^24, No], [1, 19, 4, a^5b^24, No], [1, 20, 3, a^4b^24, No], [1, 21, 2, a^3b^24, No], [1, 22, 1, a^2b^24, No], [1, 23, 0, a^1b^24, No], [2, 1, 21, a^23b^24, No], [2, 2, 20, a^22b^24, No], [2, 3, 19, a^21b^24, No], [2, 4, 18, a^20b^24, No], [2, 5, 17, a^19b^24, No], [2, 6, 16, a^18b^24, No], [2, 7, 15, a^17b^24, No], [2, 8, 14, a^16b^24, No], [2, 9, 13, a^15b^24, No], [2, 10, 12, a^14b^24, No], [2, 11, 11, a^13b^24, No], [2, 12, 10, a^12b^24, No], [2, 13, 9, a^11b^24, No], [2, 14, 8, a^10b^24, No], [2, 15, 7, a^9b^24, No], [2, 16, 6, a^8b^24, No], [2, 17, 5, a^7b^24, No], [2, 18, 4, a^6b^24, No], [2, 19, 3, a^5b^24, No], [2, 20, 2, a^4b^24, No], [2, 21, 1, a^3b^24, No], [2, 22, 0, a^2b^24, No], [3, 1, 20, a^23b^24, No], [3, 2, 19, a^22b^24, No], [3, 3, 18, a^21b^24, No], [3, 4, 17, a^20b^24, No], [3, 5, 16, a^19b^24, No], [3, 6, 15, a^18b^24, No], [3, 7, 14, a^17b^24, No], [3, 8, 13, a^16b^24, No], [3, 9, 12, a^15b^24, No], [3, 10, 11, a^14b^24, No], [3, 11, 10, a^13b^24, No], [3, 12, 9, a^12b^24, No], [3, 13, 8, a^11b^24, No], [3, 14, 7, a^10b^24, No], [3, 15, 6, a^9b^24, No], [3, 16, 5, a^8b^24, No], [3, 17, 4, a^7b^24, No], [3, 18, 3, a^6b^24, No], [3, 19, 2, a^5b^24, No], [3, 20, 1, a^4b^24, No], [3, 21, 0, a^3b^24, No], [4, 1, 19, a^23b^24, No], [4, 2, 18, a^22b^24, No], [4, 3, 17, a^21b^24, No], [4, 4, 16, a^20b^24, No], [4, 5, 15, a^19b^24, No], [4, 6, 14, a^18b^24, No], [4, 7, 13, a^17b^24, No], [4, 8, 12, a^16b^24, No], [4, 9, 11, a^15b^24, No], [4, 10, 10, a^14b^24, No], [4, 11, 9, a^13b^24, No], [4, 12, 8, a^12b^24, No], [4, 13, 7, a^11b^24, No], [4, 14, 6, a^10b^24, No], [4, 15, 5, a^9b^24, No], [4, 16, 4, a^8b^24, No], [4, 17, 3, a^7b^24, No], [4, 18, 2, a^6b^24, No], [4, 19, 1, a^5b^24, No], [4, 20, 0, a^4b^24, No], [5, 1, 18, a^23b^24, No], [5, 2, 17, a^22b^24, No], [5, 3, 16, a^21b^24, No], [5, 4, 15, a^20b^24, No], [5, 5, 14, a^19b^24, No], [5, 6, 13, a^18b^24, No], [5, 7, 12, a^17b^24, No], [5, 8, 11, a^16b^24, No], [5, 9, 10, a^15b^24, No], [5, 10, 9, a^14b^24, No], [5, 11, 8, a^13b^24, No], [5, 12, 7, a^12b^24, No], [5, 13, 6, a^11b^24, No], [5, 14, 5, a^10b^24, No], [5, 15, 4, a^9b^24, No], [5, 16, 3, a^8b^24, No], [5, 17, 2, a^7b^24, No], [5, 18, 1, a^6b^24, No], [5, 19, 0, a^5b^24, No], [6, 1, 17, a^23b^24, No], [6, 2, 16, a^22b^24, No], [6, 3, 15, a^21b^24, No], [6, 4, 14, a^20b^24, No], [6, 5, 13, a^19b^24, No], [6, 6, 12, a^18b^24, No], [6, 7, 11, a^17b^24, No], [6, 8, 10, a^16b^24, No], [6, 9, 9, a^15b^24, No], [6, 10, 8, a^14b^24, No], [6, 11, 7, a^13b^24, No], [6, 12, 6, a^12b^24, No], [6, 13, 5, a^11b^24, No], [6, 14, 4, a^10b^24, No], [6, 15, 3, a^9b^24, No], [6, 16, 2, a^8b^24, No], [6, 17, 1, a^7b^24, No], [6, 18, 0, a^6b^24, No], [7, 1, 16, a^23b^24, No], [7, 2, 15, a^22b^24, No], [7, 3, 14, a^21b^24, No], [7, 4, 13, a^20b^24, No], [7, 5, 12, a^19b^24, No], [7, 6, 11, a^18b^24, No], [7, 7, 10, a^17b^24, No], [7, 8, 9, a^16b^24, No], [7, 9, 8, a^15b^24, No], [7, 10, 7, a^14b^24, No], [7, 11, 6, a^13b^24, No], [7, 12, 5, a^12b^24, No], [7, 13, 4, a^11b^24, No], [7, 14, 3, a^10b^24, No], [7, 15, 2, a^9b^24, No], [7, 16, 1, a^8b^24, No], [7, 17, 0, a^7b^24, No], [8, 1, 15, a^23b^24, No], [8, 2, 14, a^22b^24, No], [8, 3, 13, a^21b^24, No], [8, 4, 12, a^20b^24, No], [8, 5, 11, a^19b^24, No], [8, 6, 10, a^18b^24, No], [8, 7, 9, a^17b^24, No], [8, 8, 8, a^16b^24, No], [8, 9, 7, a^15b^24, No], [8, 10, 6, a^14b^24, No], [8, 11, 5, a^13b^24, No], [8, 12, 4, a^12b^24, No], [8, 13, 3, a^11b^24, No], [8, 14, 2, a^10b^24, No], [8, 15, 1, a^9b^24, No], [8, 16, 0, a^8b^24, No], [9, 1, 14, a^23b^24, No], [9, 2, 13, a^22b^24, No], [9, 3, 12, a^21b^24, No], [9, 4, 11, a^20b^24, No], [9, 5, 10, a^19b^24, No], [9, 6, 9, a^18b^24, No], [9, 7, 8, a^17b^24, No], [9, 8, 7, a^16b^24, No], [9, 9, 6, a^15b^24, No], [9, 10, 5, a^14b^24, No], [9, 11, 4, a^13b^24, No], [9, 12, 3, a^12b^24, No], [9, 13, 2, a^11b^24, No], [9, 14, 1, a^10b^24, No], [9, 15, 0, a^9b^24, No], [10, 1, 13, a^23b^24, No], [10, 2, 12, a^22b^24, No], [10, 3, 11, a^21b^24, No], [10, 4, 10, a^20b^24, No], [10, 5, 9, a^19b^24, No], [10, 6, 8, a^18b^24, No], [10, 7, 7, a^17b^24, No], [10, 8, 6, a^16b^24, No], [10, 9, 5, a^15b^24, No], [10, 10, 4, a^14b^24, No], [10, 11, 3, a^13b^24, No], [10, 12, 2, a^12b^24, No], [10, 13, 1, a^11b^24, No], [10, 14, 0, a^10b^24, No], [11, 1, 12, a^23b^24, No], [11, 2, 11, a^22b^24, No], [11, 3, 10, a^21b^24, No], [11, 4, 9, a^20b^24, No], [11, 5, 8, a^19b^24, No], [11, 6, 7, a^18b^24, No], [11, 7, 6, a^17b^24, No], [11, 8, 5, a^16b^24, No], [11, 9, 4, a^15b^24, No], [11, 10, 3, a^14b^24, No], [11, 11, 2, a^13b^24, No], [11, 12, 1, a^12b^24, No], [11, 13, 0, a^11b^24, No], [12, 1, 11, a^23b^24, No], [12, 2, 10, a^22b^24, No], [12, 3, 9, a^21b^24, No], [12, 4, 8, a^20b^24, No], [12, 5, 7, a^19b^24, No], [12, 6, 6, a^18b^24, No], [12, 7, 5, a^17b^24, No], [12, 8, 4, a^16b^24, No], [12, 9, 3, a^15b^24, No], [12, 10, 2, a^14b^24, No], [12, 11, 1, a^13b^24, No], [12, 12, 0, a^12b^24, No], [13, 1, 10, a^23b^24, No], [13, 2, 9, a^22b^24, No], [13, 3, 8, a^21b^24, No], [13, 4, 7, a^20b^24, No], [13, 5, 6, a^19b^24, No], [13, 6, 5, a^18b^24, No], [13, 7, 4, a^17b^24, No], [13, 8, 3, a^16b^24, No], [13, 9, 2, a^15b^24, No], [13, 10, 1, a^14b^24, No], [13, 11, 0, a^13b^24, No], [14, 1, 9, a^23b^24, No], [14, 2, 8, a^22b^24, No], [14, 3, 7, a^21b^24, No], [14, 4, 6, a^20b^24, No], [14, 5, 5, a^19b^24, No], [14, 6, 4, a^18b^24, No], [14, 7, 3, a^17b^24, No], [14, 8, 2, a^16b^24, No], [14, 9, 1, a^15b^24, No], [14, 10, 0, a^14b^24, No], [15, 1, 8, a^23b^24, No], [15, 2, 7, a^22b^24, No], [15, 3, 6, a^21b^24, No], [15, 4, 5, a^20b^24, No], [15, 5, 4, a^19b^24, No], [15, 6, 3, a^18b^24, No], [15, 7, 2, a^17b^24, No], [15, 8, 1, a^16b^24, No], [15, 9, 0, a^15b^24, No], [16, 1, 7, a^23b^24, No], [16, 2, 6, a^22b^24, No], [16, 3, 5, a^21b^24, No], [16, 4, 4, a^20b^24, No], [16, 5, 3, a^19b^24, No], [16, 6, 2, a^18b^24, No], [16, 7, 1, a^17b^24, No], [16, 8, 0, a^16b^24, No], [17, 1, 6, a^23b^24, No], [17, 2, 5, a^22b^24, No], [17, 3, 4, a^21b^24, No], [17, 4, 3, a^20b^24, No], [17, 5, 2, a^19b^24, No], [17, 6, 1, a^18b^24, No], [17, 7, 0, a^17b^24, No], [18, 1, 5, a^23b^24, No], [18, 2, 4, a^22b^24, No], [18, 3, 3, a^21b^24, No], [18, 4, 2, a^20b^24, No], [18, 5, 1, a^19b^24, No], [18, 6, 0, a^18b^24, No], [19, 1, 4, a^23b^24, No], [19, 2, 3, a^22b^24, No], [19, 3, 2, a^21b^24, No], [19, 4, 1, a^20b^24, No], [19, 5, 0, a^19b^24, No], [20, 1, 3, a^23b^24, No], [20, 2, 2, a^22b^24, No], [20, 3, 1, a^21b^24, No], [20, 4, 0, a^20b^24, No], [21, 1, 2, a^23b^24, No], [21, 2, 1, a^22b^24, No], [21, 3, 0, a^21b^24, No], [22, 1, 1, a^23b^24, No], [22, 2, 0, a^22b^24, No], [23, 1, 0, a^23b^24, No]], 300, 300, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0075","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f934cbc17f85ed7fd9646586d0df89790a82c1f2dc0ef093650869417ae69c36","family_id":"qf-mod49-pump-anbn-all-splits","seed":202611187075,"semantic_template":"pump-anbn-all-splits|{p:25}"},"fields":{"title":"Every Legal Split of a^p b^p Is Defeated by Pumping Down","stem":"Use positive pumping length p=25 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).","answer":{"kind":"expression","value":"(25, a^25b^25, [[0, 1, 24, a^24b^25, No], [0, 2, 23, a^23b^25, No], [0, 3, 22, a^22b^25, No], [0, 4, 21, a^21b^25, No], [0, 5, 20, a^20b^25, No], [0, 6, 19, a^19b^25, No], [0, 7, 18, a^18b^25, No], [0, 8, 17, a^17b^25, No], [0, 9, 16, a^16b^25, No], [0, 10, 15, a^15b^25, No], [0, 11, 14, a^14b^25, No], [0, 12, 13, a^13b^25, No], [0, 13, 12, a^12b^25, No], [0, 14, 11, a^11b^25, No], [0, 15, 10, a^10b^25, No], [0, 16, 9, a^9b^25, No], [0, 17, 8, a^8b^25, No], [0, 18, 7, a^7b^25, No], [0, 19, 6, a^6b^25, No], [0, 20, 5, a^5b^25, No], [0, 21, 4, a^4b^25, No], [0, 22, 3, a^3b^25, No], [0, 23, 2, a^2b^25, No], [0, 24, 1, a^1b^25, No], [0, 25, 0, a^0b^25, No], [1, 1, 23, a^24b^25, No], [1, 2, 22, a^23b^25, No], [1, 3, 21, a^22b^25, No], [1, 4, 20, a^21b^25, No], [1, 5, 19, a^20b^25, No], [1, 6, 18, a^19b^25, No], [1, 7, 17, a^18b^25, No], [1, 8, 16, a^17b^25, No], [1, 9, 15, a^16b^25, No], [1, 10, 14, a^15b^25, No], [1, 11, 13, a^14b^25, No], [1, 12, 12, a^13b^25, No], [1, 13, 11, a^12b^25, No], [1, 14, 10, a^11b^25, No], [1, 15, 9, a^10b^25, No], [1, 16, 8, a^9b^25, No], [1, 17, 7, a^8b^25, No], [1, 18, 6, a^7b^25, No], [1, 19, 5, a^6b^25, No], [1, 20, 4, a^5b^25, No], [1, 21, 3, a^4b^25, No], [1, 22, 2, a^3b^25, No], [1, 23, 1, a^2b^25, No], [1, 24, 0, a^1b^25, No], [2, 1, 22, a^24b^25, No], [2, 2, 21, a^23b^25, No], [2, 3, 20, a^22b^25, No], [2, 4, 19, a^21b^25, No], [2, 5, 18, a^20b^25, No], [2, 6, 17, a^19b^25, No], [2, 7, 16, a^18b^25, No], [2, 8, 15, a^17b^25, No], [2, 9, 14, a^16b^25, No], [2, 10, 13, a^15b^25, No], [2, 11, 12, a^14b^25, No], [2, 12, 11, a^13b^25, No], [2, 13, 10, a^12b^25, No], [2, 14, 9, a^11b^25, No], [2, 15, 8, a^10b^25, No], [2, 16, 7, a^9b^25, No], [2, 17, 6, a^8b^25, No], [2, 18, 5, a^7b^25, No], [2, 19, 4, a^6b^25, No], [2, 20, 3, a^5b^25, No], [2, 21, 2, a^4b^25, No], [2, 22, 1, a^3b^25, No], [2, 23, 0, a^2b^25, No], [3, 1, 21, a^24b^25, No], [3, 2, 20, a^23b^25, No], [3, 3, 19, a^22b^25, No], [3, 4, 18, a^21b^25, No], [3, 5, 17, a^20b^25, No], [3, 6, 16, a^19b^25, No], [3, 7, 15, a^18b^25, No], [3, 8, 14, a^17b^25, No], [3, 9, 13, a^16b^25, No], [3, 10, 12, a^15b^25, No], [3, 11, 11, a^14b^25, No], [3, 12, 10, a^13b^25, No], [3, 13, 9, a^12b^25, No], [3, 14, 8, a^11b^25, No], [3, 15, 7, a^10b^25, No], [3, 16, 6, a^9b^25, No], [3, 17, 5, a^8b^25, No], [3, 18, 4, a^7b^25, No], [3, 19, 3, a^6b^25, No], [3, 20, 2, a^5b^25, No], [3, 21, 1, a^4b^25, No], [3, 22, 0, a^3b^25, No], [4, 1, 20, a^24b^25, No], [4, 2, 19, a^23b^25, No], [4, 3, 18, a^22b^25, No], [4, 4, 17, a^21b^25, No], [4, 5, 16, a^20b^25, No], [4, 6, 15, a^19b^25, No], [4, 7, 14, a^18b^25, No], [4, 8, 13, a^17b^25, No], [4, 9, 12, a^16b^25, No], [4, 10, 11, a^15b^25, No], [4, 11, 10, a^14b^25, No], [4, 12, 9, a^13b^25, No], [4, 13, 8, a^12b^25, No], [4, 14, 7, a^11b^25, No], [4, 15, 6, a^10b^25, No], [4, 16, 5, a^9b^25, No], [4, 17, 4, a^8b^25, No], [4, 18, 3, a^7b^25, No], [4, 19, 2, a^6b^25, No], [4, 20, 1, a^5b^25, No], [4, 21, 0, a^4b^25, No], [5, 1, 19, a^24b^25, No], [5, 2, 18, a^23b^25, No], [5, 3, 17, a^22b^25, No], [5, 4, 16, a^21b^25, No], [5, 5, 15, a^20b^25, No], [5, 6, 14, a^19b^25, No], [5, 7, 13, a^18b^25, No], [5, 8, 12, a^17b^25, No], [5, 9, 11, a^16b^25, No], [5, 10, 10, a^15b^25, No], [5, 11, 9, a^14b^25, No], [5, 12, 8, a^13b^25, No], [5, 13, 7, a^12b^25, No], [5, 14, 6, a^11b^25, No], [5, 15, 5, a^10b^25, No], [5, 16, 4, a^9b^25, No], [5, 17, 3, a^8b^25, No], [5, 18, 2, a^7b^25, No], [5, 19, 1, a^6b^25, No], [5, 20, 0, a^5b^25, No], [6, 1, 18, a^24b^25, No], [6, 2, 17, a^23b^25, No], [6, 3, 16, a^22b^25, No], [6, 4, 15, a^21b^25, No], [6, 5, 14, a^20b^25, No], [6, 6, 13, a^19b^25, No], [6, 7, 12, a^18b^25, No], [6, 8, 11, a^17b^25, No], [6, 9, 10, a^16b^25, No], [6, 10, 9, a^15b^25, No], [6, 11, 8, a^14b^25, No], [6, 12, 7, a^13b^25, No], [6, 13, 6, a^12b^25, No], [6, 14, 5, a^11b^25, No], [6, 15, 4, a^10b^25, No], [6, 16, 3, a^9b^25, No], [6, 17, 2, a^8b^25, No], [6, 18, 1, a^7b^25, No], [6, 19, 0, a^6b^25, No], [7, 1, 17, a^24b^25, No], [7, 2, 16, a^23b^25, No], [7, 3, 15, a^22b^25, No], [7, 4, 14, a^21b^25, No], [7, 5, 13, a^20b^25, No], [7, 6, 12, a^19b^25, No], [7, 7, 11, a^18b^25, No], [7, 8, 10, a^17b^25, No], [7, 9, 9, a^16b^25, No], [7, 10, 8, a^15b^25, No], [7, 11, 7, a^14b^25, No], [7, 12, 6, a^13b^25, No], [7, 13, 5, a^12b^25, No], [7, 14, 4, a^11b^25, No], [7, 15, 3, a^10b^25, No], [7, 16, 2, a^9b^25, No], [7, 17, 1, a^8b^25, No], [7, 18, 0, a^7b^25, No], [8, 1, 16, a^24b^25, No], [8, 2, 15, a^23b^25, No], [8, 3, 14, a^22b^25, No], [8, 4, 13, a^21b^25, No], [8, 5, 12, a^20b^25, No], [8, 6, 11, a^19b^25, No], [8, 7, 10, a^18b^25, No], [8, 8, 9, a^17b^25, No], [8, 9, 8, a^16b^25, No], [8, 10, 7, a^15b^25, No], [8, 11, 6, a^14b^25, No], [8, 12, 5, a^13b^25, No], [8, 13, 4, a^12b^25, No], [8, 14, 3, a^11b^25, No], [8, 15, 2, a^10b^25, No], [8, 16, 1, a^9b^25, No], [8, 17, 0, a^8b^25, No], [9, 1, 15, a^24b^25, No], [9, 2, 14, a^23b^25, No], [9, 3, 13, a^22b^25, No], [9, 4, 12, a^21b^25, No], [9, 5, 11, a^20b^25, No], [9, 6, 10, a^19b^25, No], [9, 7, 9, a^18b^25, No], [9, 8, 8, a^17b^25, No], [9, 9, 7, a^16b^25, No], [9, 10, 6, a^15b^25, No], [9, 11, 5, a^14b^25, No], [9, 12, 4, a^13b^25, No], [9, 13, 3, a^12b^25, No], [9, 14, 2, a^11b^25, No], [9, 15, 1, a^10b^25, No], [9, 16, 0, a^9b^25, No], [10, 1, 14, a^24b^25, No], [10, 2, 13, a^23b^25, No], [10, 3, 12, a^22b^25, No], [10, 4, 11, a^21b^25, No], [10, 5, 10, a^20b^25, No], [10, 6, 9, a^19b^25, No], [10, 7, 8, a^18b^25, No], [10, 8, 7, a^17b^25, No], [10, 9, 6, a^16b^25, No], [10, 10, 5, a^15b^25, No], [10, 11, 4, a^14b^25, No], [10, 12, 3, a^13b^25, No], [10, 13, 2, a^12b^25, No], [10, 14, 1, a^11b^25, No], [10, 15, 0, a^10b^25, No], [11, 1, 13, a^24b^25, No], [11, 2, 12, a^23b^25, No], [11, 3, 11, a^22b^25, No], [11, 4, 10, a^21b^25, No], [11, 5, 9, a^20b^25, No], [11, 6, 8, a^19b^25, No], [11, 7, 7, a^18b^25, No], [11, 8, 6, a^17b^25, No], [11, 9, 5, a^16b^25, No], [11, 10, 4, a^15b^25, No], [11, 11, 3, a^14b^25, No], [11, 12, 2, a^13b^25, No], [11, 13, 1, a^12b^25, No], [11, 14, 0, a^11b^25, No], [12, 1, 12, a^24b^25, No], [12, 2, 11, a^23b^25, No], [12, 3, 10, a^22b^25, No], [12, 4, 9, a^21b^25, No], [12, 5, 8, a^20b^25, No], [12, 6, 7, a^19b^25, No], [12, 7, 6, a^18b^25, No], [12, 8, 5, a^17b^25, No], [12, 9, 4, a^16b^25, No], [12, 10, 3, a^15b^25, No], [12, 11, 2, a^14b^25, No], [12, 12, 1, a^13b^25, No], [12, 13, 0, a^12b^25, No], [13, 1, 11, a^24b^25, No], [13, 2, 10, a^23b^25, No], [13, 3, 9, a^22b^25, No], [13, 4, 8, a^21b^25, No], [13, 5, 7, a^20b^25, No], [13, 6, 6, a^19b^25, No], [13, 7, 5, a^18b^25, No], [13, 8, 4, a^17b^25, No], [13, 9, 3, a^16b^25, No], [13, 10, 2, a^15b^25, No], [13, 11, 1, a^14b^25, No], [13, 12, 0, a^13b^25, No], [14, 1, 10, a^24b^25, No], [14, 2, 9, a^23b^25, No], [14, 3, 8, a^22b^25, No], [14, 4, 7, a^21b^25, No], [14, 5, 6, a^20b^25, No], [14, 6, 5, a^19b^25, No], [14, 7, 4, a^18b^25, No], [14, 8, 3, a^17b^25, No], [14, 9, 2, a^16b^25, No], [14, 10, 1, a^15b^25, No], [14, 11, 0, a^14b^25, No], [15, 1, 9, a^24b^25, No], [15, 2, 8, a^23b^25, No], [15, 3, 7, a^22b^25, No], [15, 4, 6, a^21b^25, No], [15, 5, 5, a^20b^25, No], [15, 6, 4, a^19b^25, No], [15, 7, 3, a^18b^25, No], [15, 8, 2, a^17b^25, No], [15, 9, 1, a^16b^25, No], [15, 10, 0, a^15b^25, No], [16, 1, 8, a^24b^25, No], [16, 2, 7, a^23b^25, No], [16, 3, 6, a^22b^25, No], [16, 4, 5, a^21b^25, No], [16, 5, 4, a^20b^25, No], [16, 6, 3, a^19b^25, No], [16, 7, 2, a^18b^25, No], [16, 8, 1, a^17b^25, No], [16, 9, 0, a^16b^25, No], [17, 1, 7, a^24b^25, No], [17, 2, 6, a^23b^25, No], [17, 3, 5, a^22b^25, No], [17, 4, 4, a^21b^25, No], [17, 5, 3, a^20b^25, No], [17, 6, 2, a^19b^25, No], [17, 7, 1, a^18b^25, No], [17, 8, 0, a^17b^25, No], [18, 1, 6, a^24b^25, No], [18, 2, 5, a^23b^25, No], [18, 3, 4, a^22b^25, No], [18, 4, 3, a^21b^25, No], [18, 5, 2, a^20b^25, No], [18, 6, 1, a^19b^25, No], [18, 7, 0, a^18b^25, No], [19, 1, 5, a^24b^25, No], [19, 2, 4, a^23b^25, No], [19, 3, 3, a^22b^25, No], [19, 4, 2, a^21b^25, No], [19, 5, 1, a^20b^25, No], [19, 6, 0, a^19b^25, No], [20, 1, 4, a^24b^25, No], [20, 2, 3, a^23b^25, No], [20, 3, 2, a^22b^25, No], [20, 4, 1, a^21b^25, No], [20, 5, 0, a^20b^25, No], [21, 1, 3, a^24b^25, No], [21, 2, 2, a^23b^25, No], [21, 3, 1, a^22b^25, No], [21, 4, 0, a^21b^25, No], [22, 1, 2, a^24b^25, No], [22, 2, 1, a^23b^25, No], [22, 3, 0, a^22b^25, No], [23, 1, 1, a^24b^25, No], [23, 2, 0, a^23b^25, No], [24, 1, 0, a^24b^25, No]], 325, 325, Yes, every_legal_split_is_defeated_by_i=0)","explanation":"Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (25, a^25b^25, [[0, 1, 24, a^24b^25, No], [0, 2, 23, a^23b^25, No], [0, 3, 22, a^22b^25, No], [0, 4, 21, a^21b^25, No], [0, 5, 20, a^20b^25, No], [0, 6, 19, a^19b^25, No], [0, 7, 18, a^18b^25, No], [0, 8, 17, a^17b^25, No], [0, 9, 16, a^16b^25, No], [0, 10, 15, a^15b^25, No], [0, 11, 14, a^14b^25, No], [0, 12, 13, a^13b^25, No], [0, 13, 12, a^12b^25, No], [0, 14, 11, a^11b^25, No], [0, 15, 10, a^10b^25, No], [0, 16, 9, a^9b^25, No], [0, 17, 8, a^8b^25, No], [0, 18, 7, a^7b^25, No], [0, 19, 6, a^6b^25, No], [0, 20, 5, a^5b^25, No], [0, 21, 4, a^4b^25, No], [0, 22, 3, a^3b^25, No], [0, 23, 2, a^2b^25, No], [0, 24, 1, a^1b^25, No], [0, 25, 0, a^0b^25, No], [1, 1, 23, a^24b^25, No], [1, 2, 22, a^23b^25, No], [1, 3, 21, a^22b^25, No], [1, 4, 20, a^21b^25, No], [1, 5, 19, a^20b^25, No], [1, 6, 18, a^19b^25, No], [1, 7, 17, a^18b^25, No], [1, 8, 16, a^17b^25, No], [1, 9, 15, a^16b^25, No], [1, 10, 14, a^15b^25, No], [1, 11, 13, a^14b^25, No], [1, 12, 12, a^13b^25, No], [1, 13, 11, a^12b^25, No], [1, 14, 10, a^11b^25, No], [1, 15, 9, a^10b^25, No], [1, 16, 8, a^9b^25, No], [1, 17, 7, a^8b^25, No], [1, 18, 6, a^7b^25, No], [1, 19, 5, a^6b^25, No], [1, 20, 4, a^5b^25, No], [1, 21, 3, a^4b^25, No], [1, 22, 2, a^3b^25, No], [1, 23, 1, a^2b^25, No], [1, 24, 0, a^1b^25, No], [2, 1, 22, a^24b^25, No], [2, 2, 21, a^23b^25, No], [2, 3, 20, a^22b^25, No], [2, 4, 19, a^21b^25, No], [2, 5, 18, a^20b^25, No], [2, 6, 17, a^19b^25, No], [2, 7, 16, a^18b^25, No], [2, 8, 15, a^17b^25, No], [2, 9, 14, a^16b^25, No], [2, 10, 13, a^15b^25, No], [2, 11, 12, a^14b^25, No], [2, 12, 11, a^13b^25, No], [2, 13, 10, a^12b^25, No], [2, 14, 9, a^11b^25, No], [2, 15, 8, a^10b^25, No], [2, 16, 7, a^9b^25, No], [2, 17, 6, a^8b^25, No], [2, 18, 5, a^7b^25, No], [2, 19, 4, a^6b^25, No], [2, 20, 3, a^5b^25, No], [2, 21, 2, a^4b^25, No], [2, 22, 1, a^3b^25, No], [2, 23, 0, a^2b^25, No], [3, 1, 21, a^24b^25, No], [3, 2, 20, a^23b^25, No], [3, 3, 19, a^22b^25, No], [3, 4, 18, a^21b^25, No], [3, 5, 17, a^20b^25, No], [3, 6, 16, a^19b^25, No], [3, 7, 15, a^18b^25, No], [3, 8, 14, a^17b^25, No], [3, 9, 13, a^16b^25, No], [3, 10, 12, a^15b^25, No], [3, 11, 11, a^14b^25, No], [3, 12, 10, a^13b^25, No], [3, 13, 9, a^12b^25, No], [3, 14, 8, a^11b^25, No], [3, 15, 7, a^10b^25, No], [3, 16, 6, a^9b^25, No], [3, 17, 5, a^8b^25, No], [3, 18, 4, a^7b^25, No], [3, 19, 3, a^6b^25, No], [3, 20, 2, a^5b^25, No], [3, 21, 1, a^4b^25, No], [3, 22, 0, a^3b^25, No], [4, 1, 20, a^24b^25, No], [4, 2, 19, a^23b^25, No], [4, 3, 18, a^22b^25, No], [4, 4, 17, a^21b^25, No], [4, 5, 16, a^20b^25, No], [4, 6, 15, a^19b^25, No], [4, 7, 14, a^18b^25, No], [4, 8, 13, a^17b^25, No], [4, 9, 12, a^16b^25, No], [4, 10, 11, a^15b^25, No], [4, 11, 10, a^14b^25, No], [4, 12, 9, a^13b^25, No], [4, 13, 8, a^12b^25, No], [4, 14, 7, a^11b^25, No], [4, 15, 6, a^10b^25, No], [4, 16, 5, a^9b^25, No], [4, 17, 4, a^8b^25, No], [4, 18, 3, a^7b^25, No], [4, 19, 2, a^6b^25, No], [4, 20, 1, a^5b^25, No], [4, 21, 0, a^4b^25, No], [5, 1, 19, a^24b^25, No], [5, 2, 18, a^23b^25, No], [5, 3, 17, a^22b^25, No], [5, 4, 16, a^21b^25, No], [5, 5, 15, a^20b^25, No], [5, 6, 14, a^19b^25, No], [5, 7, 13, a^18b^25, No], [5, 8, 12, a^17b^25, No], [5, 9, 11, a^16b^25, No], [5, 10, 10, a^15b^25, No], [5, 11, 9, a^14b^25, No], [5, 12, 8, a^13b^25, No], [5, 13, 7, a^12b^25, No], [5, 14, 6, a^11b^25, No], [5, 15, 5, a^10b^25, No], [5, 16, 4, a^9b^25, No], [5, 17, 3, a^8b^25, No], [5, 18, 2, a^7b^25, No], [5, 19, 1, a^6b^25, No], [5, 20, 0, a^5b^25, No], [6, 1, 18, a^24b^25, No], [6, 2, 17, a^23b^25, No], [6, 3, 16, a^22b^25, No], [6, 4, 15, a^21b^25, No], [6, 5, 14, a^20b^25, No], [6, 6, 13, a^19b^25, No], [6, 7, 12, a^18b^25, No], [6, 8, 11, a^17b^25, No], [6, 9, 10, a^16b^25, No], [6, 10, 9, a^15b^25, No], [6, 11, 8, a^14b^25, No], [6, 12, 7, a^13b^25, No], [6, 13, 6, a^12b^25, No], [6, 14, 5, a^11b^25, No], [6, 15, 4, a^10b^25, No], [6, 16, 3, a^9b^25, No], [6, 17, 2, a^8b^25, No], [6, 18, 1, a^7b^25, No], [6, 19, 0, a^6b^25, No], [7, 1, 17, a^24b^25, No], [7, 2, 16, a^23b^25, No], [7, 3, 15, a^22b^25, No], [7, 4, 14, a^21b^25, No], [7, 5, 13, a^20b^25, No], [7, 6, 12, a^19b^25, No], [7, 7, 11, a^18b^25, No], [7, 8, 10, a^17b^25, No], [7, 9, 9, a^16b^25, No], [7, 10, 8, a^15b^25, No], [7, 11, 7, a^14b^25, No], [7, 12, 6, a^13b^25, No], [7, 13, 5, a^12b^25, No], [7, 14, 4, a^11b^25, No], [7, 15, 3, a^10b^25, No], [7, 16, 2, a^9b^25, No], [7, 17, 1, a^8b^25, No], [7, 18, 0, a^7b^25, No], [8, 1, 16, a^24b^25, No], [8, 2, 15, a^23b^25, No], [8, 3, 14, a^22b^25, No], [8, 4, 13, a^21b^25, No], [8, 5, 12, a^20b^25, No], [8, 6, 11, a^19b^25, No], [8, 7, 10, a^18b^25, No], [8, 8, 9, a^17b^25, No], [8, 9, 8, a^16b^25, No], [8, 10, 7, a^15b^25, No], [8, 11, 6, a^14b^25, No], [8, 12, 5, a^13b^25, No], [8, 13, 4, a^12b^25, No], [8, 14, 3, a^11b^25, No], [8, 15, 2, a^10b^25, No], [8, 16, 1, a^9b^25, No], [8, 17, 0, a^8b^25, No], [9, 1, 15, a^24b^25, No], [9, 2, 14, a^23b^25, No], [9, 3, 13, a^22b^25, No], [9, 4, 12, a^21b^25, No], [9, 5, 11, a^20b^25, No], [9, 6, 10, a^19b^25, No], [9, 7, 9, a^18b^25, No], [9, 8, 8, a^17b^25, No], [9, 9, 7, a^16b^25, No], [9, 10, 6, a^15b^25, No], [9, 11, 5, a^14b^25, No], [9, 12, 4, a^13b^25, No], [9, 13, 3, a^12b^25, No], [9, 14, 2, a^11b^25, No], [9, 15, 1, a^10b^25, No], [9, 16, 0, a^9b^25, No], [10, 1, 14, a^24b^25, No], [10, 2, 13, a^23b^25, No], [10, 3, 12, a^22b^25, No], [10, 4, 11, a^21b^25, No], [10, 5, 10, a^20b^25, No], [10, 6, 9, a^19b^25, No], [10, 7, 8, a^18b^25, No], [10, 8, 7, a^17b^25, No], [10, 9, 6, a^16b^25, No], [10, 10, 5, a^15b^25, No], [10, 11, 4, a^14b^25, No], [10, 12, 3, a^13b^25, No], [10, 13, 2, a^12b^25, No], [10, 14, 1, a^11b^25, No], [10, 15, 0, a^10b^25, No], [11, 1, 13, a^24b^25, No], [11, 2, 12, a^23b^25, No], [11, 3, 11, a^22b^25, No], [11, 4, 10, a^21b^25, No], [11, 5, 9, a^20b^25, No], [11, 6, 8, a^19b^25, No], [11, 7, 7, a^18b^25, No], [11, 8, 6, a^17b^25, No], [11, 9, 5, a^16b^25, No], [11, 10, 4, a^15b^25, No], [11, 11, 3, a^14b^25, No], [11, 12, 2, a^13b^25, No], [11, 13, 1, a^12b^25, No], [11, 14, 0, a^11b^25, No], [12, 1, 12, a^24b^25, No], [12, 2, 11, a^23b^25, No], [12, 3, 10, a^22b^25, No], [12, 4, 9, a^21b^25, No], [12, 5, 8, a^20b^25, No], [12, 6, 7, a^19b^25, No], [12, 7, 6, a^18b^25, No], [12, 8, 5, a^17b^25, No], [12, 9, 4, a^16b^25, No], [12, 10, 3, a^15b^25, No], [12, 11, 2, a^14b^25, No], [12, 12, 1, a^13b^25, No], [12, 13, 0, a^12b^25, No], [13, 1, 11, a^24b^25, No], [13, 2, 10, a^23b^25, No], [13, 3, 9, a^22b^25, No], [13, 4, 8, a^21b^25, No], [13, 5, 7, a^20b^25, No], [13, 6, 6, a^19b^25, No], [13, 7, 5, a^18b^25, No], [13, 8, 4, a^17b^25, No], [13, 9, 3, a^16b^25, No], [13, 10, 2, a^15b^25, No], [13, 11, 1, a^14b^25, No], [13, 12, 0, a^13b^25, No], [14, 1, 10, a^24b^25, No], [14, 2, 9, a^23b^25, No], [14, 3, 8, a^22b^25, No], [14, 4, 7, a^21b^25, No], [14, 5, 6, a^20b^25, No], [14, 6, 5, a^19b^25, No], [14, 7, 4, a^18b^25, No], [14, 8, 3, a^17b^25, No], [14, 9, 2, a^16b^25, No], [14, 10, 1, a^15b^25, No], [14, 11, 0, a^14b^25, No], [15, 1, 9, a^24b^25, No], [15, 2, 8, a^23b^25, No], [15, 3, 7, a^22b^25, No], [15, 4, 6, a^21b^25, No], [15, 5, 5, a^20b^25, No], [15, 6, 4, a^19b^25, No], [15, 7, 3, a^18b^25, No], [15, 8, 2, a^17b^25, No], [15, 9, 1, a^16b^25, No], [15, 10, 0, a^15b^25, No], [16, 1, 8, a^24b^25, No], [16, 2, 7, a^23b^25, No], [16, 3, 6, a^22b^25, No], [16, 4, 5, a^21b^25, No], [16, 5, 4, a^20b^25, No], [16, 6, 3, a^19b^25, No], [16, 7, 2, a^18b^25, No], [16, 8, 1, a^17b^25, No], [16, 9, 0, a^16b^25, No], [17, 1, 7, a^24b^25, No], [17, 2, 6, a^23b^25, No], [17, 3, 5, a^22b^25, No], [17, 4, 4, a^21b^25, No], [17, 5, 3, a^20b^25, No], [17, 6, 2, a^19b^25, No], [17, 7, 1, a^18b^25, No], [17, 8, 0, a^17b^25, No], [18, 1, 6, a^24b^25, No], [18, 2, 5, a^23b^25, No], [18, 3, 4, a^22b^25, No], [18, 4, 3, a^21b^25, No], [18, 5, 2, a^20b^25, No], [18, 6, 1, a^19b^25, No], [18, 7, 0, a^18b^25, No], [19, 1, 5, a^24b^25, No], [19, 2, 4, a^23b^25, No], [19, 3, 3, a^22b^25, No], [19, 4, 2, a^21b^25, No], [19, 5, 1, a^20b^25, No], [19, 6, 0, a^19b^25, No], [20, 1, 4, a^24b^25, No], [20, 2, 3, a^23b^25, No], [20, 3, 2, a^22b^25, No], [20, 4, 1, a^21b^25, No], [20, 5, 0, a^20b^25, No], [21, 1, 3, a^24b^25, No], [21, 2, 2, a^23b^25, No], [21, 3, 1, a^22b^25, No], [21, 4, 0, a^21b^25, No], [22, 1, 2, a^24b^25, No], [22, 2, 1, a^23b^25, No], [22, 3, 0, a^22b^25, No], [23, 1, 1, a^24b^25, No], [23, 2, 0, a^23b^25, No], [24, 1, 0, a^24b^25, No]], 325, 325, Yes, every_legal_split_is_defeated_by_i=0)."},"skills":["Every Legal Split of a^p b^p Is Defeated by Pumping Down"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0076","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a40ff535875aedd47f257b2e6727e6a561f738f6000ec477d23219f9f6eae2c8","family_id":"qf-mod49-pump-boundary","seed":202611187076,"semantic_template":"pump-boundary|{p:1,word:[\"a\",\"b\"],x_length:0,y_length:1}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=single_split, complete model={\"p\":1,\"word\":[\"a\",\"b\"],\"x_length\":0,\"y_length\":1}. Give (mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary).","answer":{"kind":"expression","value":"(single_split, 1, [a, b], 0, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (single_split, 1, [a, b], 0, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0077","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55bff0b5aa91a5aeef07b0205b696af5d500cd6a1101e217ec5cf8b342303374","family_id":"qf-mod49-pump-boundary","seed":202611187077,"semantic_template":"pump-boundary|{p:2,word:[\"a\",\"a\",\"b\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=wrong_quantifiers, complete model={\"p\":2,\"word\":[\"a\",\"a\",\"b\"]}. Give (mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim).","answer":{"kind":"expression","value":"(wrong_quantifiers, 2, [a, a, b], 3, adversary_word_then_prover_split_then_adversary_i, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (wrong_quantifiers, 2, [a, a, b], 3, adversary_word_then_prover_split_then_adversary_i, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0078","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d29b6c21fbcc771b6f2c92ddcdfa9b25e31a34b5eda79a07c61e7c888e30fa87","family_id":"qf-mod49-pump-boundary","seed":202611187078,"semantic_template":"pump-boundary|{alphabet:[\"a\",\"b\",\"c\"],max_checked_length:3}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=bounded_words, complete model={\"alphabet\":[\"a\",\"b\",\"c\"],\"max_checked_length\":3}. Give (mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary).","answer":{"kind":"expression","value":"(bounded_words, [a, b, c], 3, 40, No, bounded_words_do_not_prove_regular)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (bounded_words, [a, b, c], 3, 40, No, bounded_words_do_not_prove_regular)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0079","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"776998308fe59d51adc99c38d5f3c2c6d77eba096b9a0bb6594b472d1c19ada8","family_id":"qf-mod49-pump-boundary","seed":202611187079,"semantic_template":"pump-boundary|{p:4,word:[\"a\",\"a\",\"a\",\"a\",\"b\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=decomposition_order, complete model={\"p\":4,\"word\":[\"a\",\"a\",\"a\",\"a\",\"b\"]}. Give (mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim).","answer":{"kind":"expression","value":"(decomposition_order, 4, [a, a, a, a, b], 10, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (decomposition_order, 4, [a, a, a, a, b], 10, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0080","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e94eed4da2e75ddd96c68a84966ba408c4c249b95c3b6447433a06cf81200572","family_id":"qf-mod49-pump-boundary","seed":202611187080,"semantic_template":"pump-boundary|{p:5,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:4,y_length:1}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=single_split, complete model={\"p\":5,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],\"x_length\":4,\"y_length\":1}. Give (mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary).","answer":{"kind":"expression","value":"(single_split, 5, [a, a, a, a, a, b], 4, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (single_split, 5, [a, a, a, a, a, b], 4, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0081","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5bd923ceb70fb68795b41109d630b505ba933939d94599ed7484c1f152c9d75","family_id":"qf-mod49-pump-boundary","seed":202611187081,"semantic_template":"pump-boundary|{p:6,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=wrong_quantifiers, complete model={\"p\":6,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}. Give (mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim).","answer":{"kind":"expression","value":"(wrong_quantifiers, 6, [a, a, a, a, a, a, b], 21, adversary_word_then_prover_split_then_adversary_i, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (wrong_quantifiers, 6, [a, a, a, a, a, a, b], 21, adversary_word_then_prover_split_then_adversary_i, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0082","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9d4f663c102b53c0d4d1019b148fdc37ef1e4d85143695f94b7d019c41756b5","family_id":"qf-mod49-pump-boundary","seed":202611187082,"semantic_template":"pump-boundary|{alphabet:[\"a\"],max_checked_length:2}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=bounded_words, complete model={\"alphabet\":[\"a\"],\"max_checked_length\":2}. Give (mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary).","answer":{"kind":"expression","value":"(bounded_words, [a], 2, 3, No, bounded_words_do_not_prove_regular)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (bounded_words, [a], 2, 3, No, bounded_words_do_not_prove_regular)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0083","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34794ebff8a185f02e26a2f3a1a1ad9d94d1f13cacc2025d3c2d1e81cdc17154","family_id":"qf-mod49-pump-boundary","seed":202611187083,"semantic_template":"pump-boundary|{p:8,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=decomposition_order, complete model={\"p\":8,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}. Give (mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim).","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, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (decomposition_order, 8, [a, a, a, a, a, a, a, a, b], 36, all_legal_splits_must_be_addressed_not_a_preselected_decomposition, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0084","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b1576112191271f9e8b02c087cdefc6255d119208066706f498920e3282449","family_id":"qf-mod49-pump-boundary","seed":202611187084,"semantic_template":"pump-boundary|{p:9,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:8,y_length:1}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=single_split, complete model={\"p\":9,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],\"x_length\":8,\"y_length\":1}. Give (mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary).","answer":{"kind":"expression","value":"(single_split, 9, [a, a, a, a, a, a, a, a, a, b], 8, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (single_split, 9, [a, a, a, a, a, a, a, a, a, b], 8, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0085","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"224bfb9487335f1a7c26690709385580bcc6b22dbf2bc2e77c5db7ed202498fb","family_id":"qf-mod49-pump-boundary","seed":202611187085,"semantic_template":"pump-boundary|{p:10,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=wrong_quantifiers, complete model={\"p\":10,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}. Give (mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim).","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, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (wrong_quantifiers, 10, [a, a, a, a, a, a, a, a, a, a, b], 55, adversary_word_then_prover_split_then_adversary_i, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0086","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4514667d33649a9396692c2e4853d7fa115e74784c59fcc22a8dcfd56e910c4","family_id":"qf-mod49-pump-boundary","seed":202611187086,"semantic_template":"pump-boundary|{alphabet:[\"a\",\"b\"],max_checked_length:1}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=bounded_words, complete model={\"alphabet\":[\"a\",\"b\"],\"max_checked_length\":1}. Give (mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary).","answer":{"kind":"expression","value":"(bounded_words, [a, b], 1, 3, No, bounded_words_do_not_prove_regular)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (bounded_words, [a, b], 1, 3, No, bounded_words_do_not_prove_regular)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0087","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0369f27dd5a2ae60114fa4735ec3b52fcb14cb2ecc532a15fcb91e624948e202","family_id":"qf-mod49-pump-boundary","seed":202611187087,"semantic_template":"pump-boundary|{p:12,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=decomposition_order, complete model={\"p\":12,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}. Give (mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim).","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, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (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, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0088","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2cdc341d93534faee4b54e132d8041c3ca6ff1aba0c57da92420291cf0d6c96","family_id":"qf-mod49-pump-boundary","seed":202611187088,"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}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=single_split, complete model={\"p\":13,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],\"x_length\":12,\"y_length\":1}. Give (mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary).","answer":{"kind":"expression","value":"(single_split, 13, [a, a, a, a, a, a, a, a, a, a, a, a, a, b], 12, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (single_split, 13, [a, a, a, a, a, a, a, a, a, a, a, a, a, b], 12, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0089","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7293fb1d100cce4e9b44f7b7786fdb979e1b758a164a14b4aea3e71540b3c909","family_id":"qf-mod49-pump-boundary","seed":202611187089,"semantic_template":"pump-boundary|{p:14,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=wrong_quantifiers, complete model={\"p\":14,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}. Give (mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim).","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, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (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, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0090","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b19c10a3a6552842026edc3043d275380dc851ebf0a1652d53c4730ec53264f","family_id":"qf-mod49-pump-boundary","seed":202611187090,"semantic_template":"pump-boundary|{alphabet:[\"a\",\"b\",\"c\"],max_checked_length:5}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=bounded_words, complete model={\"alphabet\":[\"a\",\"b\",\"c\"],\"max_checked_length\":5}. Give (mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary).","answer":{"kind":"expression","value":"(bounded_words, [a, b, c], 5, 364, No, bounded_words_do_not_prove_regular)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (bounded_words, [a, b, c], 5, 364, No, bounded_words_do_not_prove_regular)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0091","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69d4c6c212ac2c41e3da6bd3bb93cc9638b9e8eaf96fb366df0f4dca5dbcd380","family_id":"qf-mod49-pump-boundary","seed":202611187091,"semantic_template":"pump-boundary|{p:16,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=decomposition_order, complete model={\"p\":16,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}. Give (mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim).","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, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (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, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0092","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b18374bd51b7692c813da8c7df82128769e49dcb5a72e4a5ce52a30342a941e","family_id":"qf-mod49-pump-boundary","seed":202611187092,"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}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=single_split, complete 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}. Give (mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary).","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, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (single_split, 17, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 16, 1, 1, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0093","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30608ad0daf85083510e9e555533bf19f8674106538fe3a9e5f357a21c99305d","family_id":"qf-mod49-pump-boundary","seed":202611187093,"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\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=wrong_quantifiers, complete model={\"p\":18,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"]}. Give (mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim).","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, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (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, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0094","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41f8e33a2c2d8f93e97e463548e80a7f900bf1dbfa634e7c2ba972d6b5a9fe16","family_id":"qf-mod49-pump-boundary","seed":202611187094,"semantic_template":"pump-boundary|{alphabet:[\"a\"],max_checked_length:4}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=bounded_words, complete model={\"alphabet\":[\"a\"],\"max_checked_length\":4}. Give (mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary).","answer":{"kind":"expression","value":"(bounded_words, [a], 4, 5, No, bounded_words_do_not_prove_regular)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (bounded_words, [a], 4, 5, No, bounded_words_do_not_prove_regular)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0095","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e0fd95c80d69cc37ee3c0f9c4df159fd7a63b04fef36587a72bb11f65c18f0f","family_id":"qf-mod49-pump-boundary","seed":202611187095,"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\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=decomposition_order, complete 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\"]}. Give (mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim).","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, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (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, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0096","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb5b0870b566149027e1d4f2ed076e668aa7d945fe74a100094c25c7262f7047","family_id":"qf-mod49-pump-boundary","seed":202611187096,"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}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=single_split, complete 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}. Give (mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary).","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, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (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, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0097","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a84555e585c8805f5d1295c1c0e91f3b0f03138813de72dcb19fe2a37023e372","family_id":"qf-mod49-pump-boundary","seed":202611187097,"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\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=wrong_quantifiers, complete 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\"]}. Give (mode, p, word, legal_split_count, adversarial_choice_order, reversed_quantifier_claim).","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, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (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, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0098","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c01c9a06b4e25a4082e044356473aa7dbddda7ce14a36125857eac402a200b2","family_id":"qf-mod49-pump-boundary","seed":202611187098,"semantic_template":"pump-boundary|{alphabet:[\"a\",\"b\"],max_checked_length:3}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=bounded_words, complete model={\"alphabet\":[\"a\",\"b\"],\"max_checked_length\":3}. Give (mode, alphabet, max_checked_length, checked_word_count, regularity_proof_claim, boundary).","answer":{"kind":"expression","value":"(bounded_words, [a, b], 3, 15, No, bounded_words_do_not_prove_regular)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (bounded_words, [a, b], 3, 15, No, bounded_words_do_not_prove_regular)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0099","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b9d071d66dd48cfce4a4bc1e8443d34ac7bc93a845f94f3bcb420505a73a7af","family_id":"qf-mod49-pump-boundary","seed":202611187099,"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\"]}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=decomposition_order, complete 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\"]}. Give (mode, p, word, legal_split_count, all_splits_obligation, preselected_split_suffices_claim).","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, No)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (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, No)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0100","batch_id":"batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7fa8735b0c74cbdadd535fe18d1b24bdb626fd0ac696aad2db883564b780650","family_id":"qf-mod49-pump-boundary","seed":202611187100,"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}"},"fields":{"title":"One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity","stem":"Use mode=single_split, complete 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}. Give (mode, p, word, x_length, y_length, z_length, split_legal, pumping_quantifiers_resolved_claim, boundary).","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, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)","explanation":"A single split may be legal without resolving exists split forall i. The adversary chooses the word, the prover then chooses a split, and the adversary then chooses i. A nonregularity argument must address all legal splits, not a preselected one. Bounded word checks do not prove regularity, and the pumping property is necessary but not sufficient. The exact result, retaining the requested order, is (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, Yes, No, one_decomposition_does_not_resolve_exists_split_forall_i)."},"skills":["One Split or Bounded Words Cannot Resolve the Pumping Quantifiers or Prove Regularity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}