{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r01-natural-number-0001","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1238350a71a47741b6985545a4325825fc14ffd690849e362dbfd46dd3f40b7","family_id":"qf-density11-natural-successor-window","seed":202610111001,"semantic_template":"qf-density11-natural-successor-window/[0,0]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=0, k=0. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[0]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [0]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0002","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15c585b5379f641c24d6eab193c437fb88193a67df189c6d54c6e993f974dfaf","family_id":"qf-density11-natural-successor-window","seed":202610111002,"semantic_template":"qf-density11-natural-successor-window/[0,8]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=0, k=8. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[0, 1, 2, 3, 4, 5, 6, 7, 8]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [0, 1, 2, 3, 4, 5, 6, 7, 8]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0003","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51318ec24875cb3e735d6d0ff72e8c71287c651d5f099c42232675efff4bba04","family_id":"qf-density11-natural-successor-window","seed":202610111003,"semantic_template":"qf-density11-natural-successor-window/[1,2]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=1, k=2. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[1, 2, 3]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [1, 2, 3]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0004","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf7e17df9e9bf42828441146a863a6d07bf6f023a145deff0936badbe75befdd","family_id":"qf-density11-natural-successor-window","seed":202610111004,"semantic_template":"qf-density11-natural-successor-window/[2,3]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=2, k=3. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[2, 3, 4, 5]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [2, 3, 4, 5]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0005","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4495ca7183de0a104530379a8b233db48d311d01b09a9219d902e38819137c8","family_id":"qf-density11-natural-successor-window","seed":202610111005,"semantic_template":"qf-density11-natural-successor-window/[3,4]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=3, k=4. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[3, 4, 5, 6, 7]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [3, 4, 5, 6, 7]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0006","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7463c7d3342733be56abd614e5c0c41ff2fdfe5b2e6cd0afe9ca2320c995b0ee","family_id":"qf-density11-natural-successor-window","seed":202610111006,"semantic_template":"qf-density11-natural-successor-window/[4,5]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=4, k=5. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[4, 5, 6, 7, 8, 9]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [4, 5, 6, 7, 8, 9]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0007","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07103d17f6b4a61361ab945f714c9b10483108c1eb8af088f50b5b004797ea4d","family_id":"qf-density11-natural-successor-window","seed":202610111007,"semantic_template":"qf-density11-natural-successor-window/[5,6]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=5, k=6. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[5, 6, 7, 8, 9, 10, 11]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [5, 6, 7, 8, 9, 10, 11]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0008","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fdddcb8c5d8e7ed8b43d718b2e4adb96e4f7923cefa974114cde52fcaaf0196","family_id":"qf-density11-natural-successor-window","seed":202610111008,"semantic_template":"qf-density11-natural-successor-window/[6,7]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=6, k=7. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[6, 7, 8, 9, 10, 11, 12, 13]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [6, 7, 8, 9, 10, 11, 12, 13]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0009","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88d8b041b427219d09bab8baf5071406e05c446d6fbde69fdce91e1acc4994c4","family_id":"qf-density11-natural-successor-window","seed":202610111009,"semantic_template":"qf-density11-natural-successor-window/[7,8]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=7, k=8. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[7, 8, 9, 10, 11, 12, 13, 14, 15]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [7, 8, 9, 10, 11, 12, 13, 14, 15]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0010","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f545ece788ec92f499a028d58f2a6e792278961aa827762d5c7c83fd251cb3f9","family_id":"qf-density11-natural-successor-window","seed":202610111010,"semantic_template":"qf-density11-natural-successor-window/[8,0]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=8, k=0. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[8]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [8]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0011","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcb56d52b2623ea7f13a461db33b3d0d3db2126cc45eea16726a04f73f189033","family_id":"qf-density11-natural-successor-window","seed":202610111011,"semantic_template":"qf-density11-natural-successor-window/[9,1]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=9, k=1. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[9, 10]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [9, 10]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0012","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e793a24c1e25292ba544353979ad6f050349ebbf6466840bc6a9c8e76d8a5bc1","family_id":"qf-density11-natural-successor-window","seed":202610111012,"semantic_template":"qf-density11-natural-successor-window/[10,2]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=10, k=2. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[10, 11, 12]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [10, 11, 12]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0013","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acd1ae5794d48b2e22eee4ba9a5eb3c7d725d212dd22ea9f7f64114cd7a48acd","family_id":"qf-density11-natural-successor-window","seed":202610111013,"semantic_template":"qf-density11-natural-successor-window/[11,3]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=11, k=3. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[11, 12, 13, 14]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [11, 12, 13, 14]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0014","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed2b699870e8187aab78877923714147a4dbcf6f87d617c6507cbab55b0fd06a","family_id":"qf-density11-natural-successor-window","seed":202610111014,"semantic_template":"qf-density11-natural-successor-window/[12,4]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=12, k=4. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[12, 13, 14, 15, 16]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [12, 13, 14, 15, 16]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0015","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85eaeb2c0be7b3f733f433ef4eb2367e6d6e347ba4384b177852134a218a7eea","family_id":"qf-density11-natural-successor-window","seed":202610111015,"semantic_template":"qf-density11-natural-successor-window/[13,5]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=13, k=5. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[13, 14, 15, 16, 17, 18]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [13, 14, 15, 16, 17, 18]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0016","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fbbfe353cd1448180111d691e34dd1343d27ed557f130e168db8da55e21f280","family_id":"qf-density11-natural-successor-window","seed":202610111016,"semantic_template":"qf-density11-natural-successor-window/[14,6]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=14, k=6. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[14, 15, 16, 17, 18, 19, 20]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [14, 15, 16, 17, 18, 19, 20]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0017","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec05f0f6474d6fa286734ef3c2c61c19041796bab7fa03734204361496d3c4cf","family_id":"qf-density11-natural-successor-window","seed":202610111017,"semantic_template":"qf-density11-natural-successor-window/[15,7]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=15, k=7. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[15, 16, 17, 18, 19, 20, 21, 22]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [15, 16, 17, 18, 19, 20, 21, 22]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0018","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cea8c019e429254308a5c1fd8c62d9f6296cf2fd92f213af3c95fa14bc1f1d46","family_id":"qf-density11-natural-successor-window","seed":202610111018,"semantic_template":"qf-density11-natural-successor-window/[16,8]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=16, k=8. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[16, 17, 18, 19, 20, 21, 22, 23, 24]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [16, 17, 18, 19, 20, 21, 22, 23, 24]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0019","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ab6204f7bba1f4fa28f9ef79797e965cc4ab103f9367c47e1473b2864baec76","family_id":"qf-density11-natural-successor-window","seed":202610111019,"semantic_template":"qf-density11-natural-successor-window/[17,0]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=17, k=0. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[17]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [17]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0020","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aaae4cae8964b73feafba0263930ca072b342d2c0a4b3ca11584bb5288c9919","family_id":"qf-density11-natural-successor-window","seed":202610111020,"semantic_template":"qf-density11-natural-successor-window/[18,1]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=18, k=1. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[18, 19]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [18, 19]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0021","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43a903ff9450567aae3dd59c674b190588597fd9149937916e021416e996465c","family_id":"qf-density11-natural-successor-window","seed":202610111021,"semantic_template":"qf-density11-natural-successor-window/[19,2]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=19, k=2. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[19, 20, 21]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [19, 20, 21]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0022","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cd3c383fc98b7ff553bf18e54611faf4cb3196cfc9a024eb1e9ab3dbc5d7756","family_id":"qf-density11-natural-successor-window","seed":202610111022,"semantic_template":"qf-density11-natural-successor-window/[20,3]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=20, k=3. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[20, 21, 22, 23]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [20, 21, 22, 23]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0023","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eec0991d75bc6883928111456379de954048acd29a45abc42d5544622b17955f","family_id":"qf-density11-natural-successor-window","seed":202610111023,"semantic_template":"qf-density11-natural-successor-window/[21,4]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=21, k=4. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[21, 22, 23, 24, 25]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [21, 22, 23, 24, 25]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0024","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"934e4b15fdd11d2533a280aac0b87427f9a67ad46c0928cf00b9386a628fcae2","family_id":"qf-density11-natural-successor-window","seed":202610111024,"semantic_template":"qf-density11-natural-successor-window/[22,5]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=22, k=5. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[22, 23, 24, 25, 26, 27]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [22, 23, 24, 25, 26, 27]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0025","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad6d5e09ff6f7d48e4124e049bee5ad7ebde298239730e641b0bafac65263ddc","family_id":"qf-density11-natural-successor-window","seed":202610111025,"semantic_template":"qf-density11-natural-successor-window/[23,6]"},"fields":{"title":"Inclusive Natural Successor Window","stem":"N includes 0. n=23, k=6. List the complete ordered window beginning at n and applying the successor k times.","answer":{"kind":"expression","value":"[23, 24, 25, 26, 27, 28, 29]","explanation":"Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [23, 24, 25, 26, 27, 28, 29]."},"skills":["Inclusive Natural Successor Window"],"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-20261008-advance-20261011-r01-natural-number-0026","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ceaa8b371d7c2c80930900514cfe1dc2152c9a78f06ca490739b5916873b9cf1","family_id":"qf-density11-natural-predecessor-pair","seed":202610111026,"semantic_template":"qf-density11-natural-predecessor-pair/[0,0]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [0, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(none, none)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (none, none)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0027","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bab8cee963f098fa89f2a98f4033c01499e8c2847253898f407a288213358552","family_id":"qf-density11-natural-predecessor-pair","seed":202610111027,"semantic_template":"qf-density11-natural-predecessor-pair/[0,1]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [0, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(none, 0)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (none, 0)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0028","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdac27686bfc638e58625d2256da87edb906f65d4d9c5aac3e6df96904a9702d","family_id":"qf-density11-natural-predecessor-pair","seed":202610111028,"semantic_template":"qf-density11-natural-predecessor-pair/[2,2]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [2, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(1, 1)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (1, 1)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0029","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a68c9ef96e68d359a5b4120b749a4cfb60ba707f4eac72fc88193ae93ff31a57","family_id":"qf-density11-natural-predecessor-pair","seed":202610111029,"semantic_template":"qf-density11-natural-predecessor-pair/[3,3]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [3, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(2, 2)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (2, 2)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0030","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0ec384f201d000c829b67ed7cc3ef42f0e992cd7c77204658e2ff565fa849e4","family_id":"qf-density11-natural-predecessor-pair","seed":202610111030,"semantic_template":"qf-density11-natural-predecessor-pair/[4,4]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [4, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(3, 3)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (3, 3)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0031","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f0392311487b877c459c013375c9c0289a1feeb740b9258cf70a3340a62cd13","family_id":"qf-density11-natural-predecessor-pair","seed":202610111031,"semantic_template":"qf-density11-natural-predecessor-pair/[5,0]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [5, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(4, none)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (4, none)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0032","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5ed9fbc974868b5bc2a55b8755cfa668e8e5f306c1bb5c2174d868ec8faa7dc","family_id":"qf-density11-natural-predecessor-pair","seed":202610111032,"semantic_template":"qf-density11-natural-predecessor-pair/[6,1]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [6, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(5, 0)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (5, 0)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0033","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ca597d971385cfb6d620831d1e9d4afa15050ebb82bf2d6994cd443f35401c5","family_id":"qf-density11-natural-predecessor-pair","seed":202610111033,"semantic_template":"qf-density11-natural-predecessor-pair/[7,2]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [7, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(6, 1)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (6, 1)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0034","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"560f658d1774c5c32a9ad7c7272bea84467b2ae841e75235c75395529fb1673e","family_id":"qf-density11-natural-predecessor-pair","seed":202610111034,"semantic_template":"qf-density11-natural-predecessor-pair/[8,3]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [8, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(7, 2)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (7, 2)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0035","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"199599fcc616f4e64bc83d367c075b497c53597dac08ba5d3c2a7707c8753171","family_id":"qf-density11-natural-predecessor-pair","seed":202610111035,"semantic_template":"qf-density11-natural-predecessor-pair/[9,4]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [9, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(8, 3)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (8, 3)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0036","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7148ea1f7c5df31247df9ed89b60802222175cf235d7914991b13ae6b8da0fa9","family_id":"qf-density11-natural-predecessor-pair","seed":202610111036,"semantic_template":"qf-density11-natural-predecessor-pair/[10,0]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [10, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(9, none)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (9, none)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0037","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5084c670fa8084464452ee5f5de16a63f51f14824b2ba3fb32b83a79ea81b009","family_id":"qf-density11-natural-predecessor-pair","seed":202610111037,"semantic_template":"qf-density11-natural-predecessor-pair/[11,1]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [11, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(10, 0)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (10, 0)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0038","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71efd260e8eae6bc5b6edec2ef6298990ed39b23880551ecc2f311b709f11459","family_id":"qf-density11-natural-predecessor-pair","seed":202610111038,"semantic_template":"qf-density11-natural-predecessor-pair/[12,2]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [12, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(11, 1)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (11, 1)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0039","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1038f60e9995a05734b77e015d81a19c2e24dac1b7f45bed4b2855ab23e679c3","family_id":"qf-density11-natural-predecessor-pair","seed":202610111039,"semantic_template":"qf-density11-natural-predecessor-pair/[13,3]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [13, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(12, 2)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (12, 2)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0040","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"385804aa195f4cb0f2fcaaaf2cde40d39cc8d0a535dd48b6756330842742d172","family_id":"qf-density11-natural-predecessor-pair","seed":202610111040,"semantic_template":"qf-density11-natural-predecessor-pair/[14,4]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [14, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(13, 3)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (13, 3)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0041","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae2caff3e623cde06fb2b0317c595b128ee9b467c4d3c28d1654de9a87be487a","family_id":"qf-density11-natural-predecessor-pair","seed":202610111041,"semantic_template":"qf-density11-natural-predecessor-pair/[15,0]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [15, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(14, none)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (14, none)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0042","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e1adf143b636045795fa6f50043eab6721920ee6216b3da8e7069b32d57789a","family_id":"qf-density11-natural-predecessor-pair","seed":202610111042,"semantic_template":"qf-density11-natural-predecessor-pair/[16,1]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [16, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(15, 0)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (15, 0)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0043","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4271e508a6db766d36883fdfb62d1472badfa4d8a7c122223092cd66d4bd615c","family_id":"qf-density11-natural-predecessor-pair","seed":202610111043,"semantic_template":"qf-density11-natural-predecessor-pair/[17,2]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [17, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(16, 1)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (16, 1)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0044","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c31ec4fcce278e21d40cee601dd682bce3f3c6ed40ab9853f5da88f6cde5fd1","family_id":"qf-density11-natural-predecessor-pair","seed":202610111044,"semantic_template":"qf-density11-natural-predecessor-pair/[18,3]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [18, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(17, 2)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (17, 2)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0045","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"422508dd256e3bf0acf9feb4580624c04b17cb57ba516bb7ac14f833d26e7567","family_id":"qf-density11-natural-predecessor-pair","seed":202610111045,"semantic_template":"qf-density11-natural-predecessor-pair/[19,4]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [19, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(18, 3)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (18, 3)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0046","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba36f59dd85ef6794c475fc1ff2bd62bff2a06c7bc28461c90a1ebdf1c2a1338","family_id":"qf-density11-natural-predecessor-pair","seed":202610111046,"semantic_template":"qf-density11-natural-predecessor-pair/[20,0]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [20, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(19, none)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (19, none)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0047","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2c56fa797eb460b22f1c315d3cd0f8de7d39d62ed983af8733a03441ea92157","family_id":"qf-density11-natural-predecessor-pair","seed":202610111047,"semantic_template":"qf-density11-natural-predecessor-pair/[21,1]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [21, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(20, 0)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (20, 0)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0048","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0721f81969c7a462c127a0bf14c5041c78909f00378b68eb27fc63f027f13a87","family_id":"qf-density11-natural-predecessor-pair","seed":202610111048,"semantic_template":"qf-density11-natural-predecessor-pair/[22,2]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [22, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(21, 1)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (21, 1)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0049","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7bf0c1d3f581f5250f97747b509985e3611c8306d354b46b012a3ce08b62603","family_id":"qf-density11-natural-predecessor-pair","seed":202610111049,"semantic_template":"qf-density11-natural-predecessor-pair/[23,3]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [23, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(22, 2)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (22, 2)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0050","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3083e8c210bcb9140b1afe302569c409047f76c94e52341293b321e207adfc0f","family_id":"qf-density11-natural-predecessor-pair","seed":202610111050,"semantic_template":"qf-density11-natural-predecessor-pair/[24,4]"},"fields":{"title":"Partial Natural Predecessors","stem":"N includes 0. The ordered values are [24, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.","answer":{"kind":"expression","value":"(23, 3)","explanation":"Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (23, 3)."},"skills":["Partial Natural Predecessors"],"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-20261008-advance-20261011-r01-natural-number-0051","batch_id":"batch-20261008-advance-20261011-r01-natural-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01263cd915a4a4be8e3de977b70d4f9729b428eb1b65630189af8bc43d4cf757","family_id":"qf-density11-natural-interval-size","seed":202610111051,"semantic_template":"qf-density11-natural-interval-size/[\"empty_interval\"]"},"fields":{"title":"Natural Closed-Interval Cardinality","stem":"N includes 0. lo=5, hi=4. Find |{n∈N: lo≤n≤hi}|, including endpoints.","answer":{"kind":"expression","value":"0","explanation":"Count both endpoints of a valid interval; hi