{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r02-supremum-and-infimum-0001","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab07bbe2310c6cdabc7d578658055a2b053284557ef99f42c8e745c8a6a6c8f7","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072001,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[1,2],[-2,3],[3,4]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[1/2, -2/3, 3/4] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([1/2, -2/3, 3/4], 3/4, -2/3, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([1/2, -2/3, 3/4], 3/4, -2/3, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0002","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ebdaeda2ec229b39b4a5bb8453374216d3419507e4b6a2e66969fe8eac380c5","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072002,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[2,3],[-3,4],[5,5]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[2/3, -3/4, 5/5] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([2/3, -3/4, 1], 1, -3/4, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([2/3, -3/4, 1], 1, -3/4, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0003","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9d7fd812ee191c536557e8c3a24858b8bd12bbd173ff3dd67741113d24301ad","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072003,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[3,4],[-4,5],[7,6]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[3/4, -4/5, 7/6] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([3/4, -4/5, 7/6], 7/6, -4/5, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([3/4, -4/5, 7/6], 7/6, -4/5, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0004","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90b22a6d4afbb7cd03157aadf021bff837c02e3c3a3d32bec199cce04fd6f897","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072004,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[4,5],[-5,6],[9,7]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[4/5, -5/6, 9/7] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([4/5, -5/6, 9/7], 9/7, -5/6, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([4/5, -5/6, 9/7], 9/7, -5/6, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0005","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f89d8d732063ddcecd2b57a78c2535abec48ae7d894afca88be97a9e697fd3a3","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072005,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[5,6],[-6,7],[11,8]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[5/6, -6/7, 11/8] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([5/6, -6/7, 11/8], 11/8, -6/7, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([5/6, -6/7, 11/8], 11/8, -6/7, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0006","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78f18980e654e6deef26713cfd84f05149e0361834daf6b38082e6b746c36e2d","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072006,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[6,7],[-7,8],[13,9]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[6/7, -7/8, 13/9] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([6/7, -7/8, 13/9], 13/9, -7/8, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([6/7, -7/8, 13/9], 13/9, -7/8, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0007","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55a615262d97367b181227da29f89d9db4fb7fed46252ec4fa1dd949b96b6af9","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072007,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[7,8],[-8,9],[15,10]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[7/8, -8/9, 15/10] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([7/8, -8/9, 3/2], 3/2, -8/9, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([7/8, -8/9, 3/2], 3/2, -8/9, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0008","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"816c61ad14cef120f8926eb7762d32182c3031ac63ff582c262e164b5378c19e","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072008,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[8,9],[-9,10],[17,11]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[8/9, -9/10, 17/11] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([8/9, -9/10, 17/11], 17/11, -9/10, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([8/9, -9/10, 17/11], 17/11, -9/10, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0009","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11ac571407a42f8e93b2e0dc3809a231bab3d4a0655a8d2920d5c53a70324083","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072009,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[9,10],[-10,11],[19,12]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[9/10, -10/11, 19/12] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([9/10, -10/11, 19/12], 19/12, -10/11, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([9/10, -10/11, 19/12], 19/12, -10/11, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0010","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52b8f42dd4f8d1917143db9712667ca7a410814ecc0e100559fb714fd6123d64","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072010,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[10,11],[-11,12],[21,13]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[10/11, -11/12, 21/13] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([10/11, -11/12, 21/13], 21/13, -11/12, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([10/11, -11/12, 21/13], 21/13, -11/12, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0011","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16729c7cf63d0210a42d8fc542e599fa950a4e327cf3f36cb1144a625df528dd","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072011,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[11,12],[-12,13],[23,14]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[11/12, -12/13, 23/14] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([11/12, -12/13, 23/14], 23/14, -12/13, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([11/12, -12/13, 23/14], 23/14, -12/13, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0012","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8082ea7ba5d46be897f4065e60484e8469307a49d903a3def61955232d35900c","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072012,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[12,13],[-13,14],[25,15]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[12/13, -13/14, 25/15] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([12/13, -13/14, 5/3], 5/3, -13/14, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([12/13, -13/14, 5/3], 5/3, -13/14, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0013","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dac13fafd5149499ebf4a59c1126aa6ffcc435da7742fc5b54b5f99e2815b46","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072013,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[13,14],[-14,15],[27,16]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[13/14, -14/15, 27/16] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([13/14, -14/15, 27/16], 27/16, -14/15, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([13/14, -14/15, 27/16], 27/16, -14/15, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0014","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e15f731d27b9e91f862fe7ed7e6271ec609804de0b9eecff334d700f863f473","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072014,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[14,15],[-15,16],[29,17]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[14/15, -15/16, 29/17] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([14/15, -15/16, 29/17], 29/17, -15/16, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([14/15, -15/16, 29/17], 29/17, -15/16, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0015","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffd101b6ef09220a36e9b2df054549f91ce3f27e27a05a9ae853fe4345cc27ce","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072015,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[15,16],[-16,17],[31,18]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[15/16, -16/17, 31/18] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([15/16, -16/17, 31/18], 31/18, -16/17, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([15/16, -16/17, 31/18], 31/18, -16/17, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0016","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60451d8fa94e093a08082678eb9a50adfd13b13deb773a45514f8d8ebc0dfa32","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072016,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[16,17],[-17,18],[33,19]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[16/17, -17/18, 33/19] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([16/17, -17/18, 33/19], 33/19, -17/18, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([16/17, -17/18, 33/19], 33/19, -17/18, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0017","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cea93a2965a7377e1ccc0a52e5d6c9a19930f6da7013edd1272aa0828970e228","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072017,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[17,18],[-18,19],[35,20]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[17/18, -18/19, 35/20] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([17/18, -18/19, 7/4], 7/4, -18/19, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([17/18, -18/19, 7/4], 7/4, -18/19, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0018","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7916a23c0e5fac910d008e6df052c18653e51a5e57511816aac954bac2032f9","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072018,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[18,19],[-19,20],[37,21]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[18/19, -19/20, 37/21] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([18/19, -19/20, 37/21], 37/21, -19/20, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([18/19, -19/20, 37/21], 37/21, -19/20, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0019","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"095c30c973f8f223c08219dac77b6fd137d1cf60b58fc951640a874e6e76b2eb","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072019,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[19,20],[-20,21],[39,22]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[19/20, -20/21, 39/22] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([19/20, -20/21, 39/22], 39/22, -20/21, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([19/20, -20/21, 39/22], 39/22, -20/21, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0020","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2e568a43ab1ad00d77fa45c18305eba6699eb0282aa5af7bf4a14fff3700a86","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072020,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[20,21],[-21,22],[41,23]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[20/21, -21/22, 41/23] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([20/21, -21/22, 41/23], 41/23, -21/22, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([20/21, -21/22, 41/23], 41/23, -21/22, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0021","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4ff11f9831f9c1fa1bab90fc89352ed912e34ad7010bdd506c0265864c3fffd","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072021,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[21,22],[-22,23],[43,24]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[21/22, -22/23, 43/24] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([21/22, -22/23, 43/24], 43/24, -22/23, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([21/22, -22/23, 43/24], 43/24, -22/23, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0022","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69df99aea1654916d30bba12f28a3ce2bfc2833f27054aa3081d78330afbb2e9","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072022,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[22,23],[-23,24],[45,25]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[22/23, -23/24, 45/25] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([22/23, -23/24, 9/5], 9/5, -23/24, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([22/23, -23/24, 9/5], 9/5, -23/24, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0023","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe84c2269177813a2cabbb64ab75bcb164901d2885cf8e91c9533ff354e3a327","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072023,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[23,24],[-24,25],[47,26]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[23/24, -24/25, 47/26] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([23/24, -24/25, 47/26], 47/26, -24/25, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([23/24, -24/25, 47/26], 47/26, -24/25, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0024","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d64bb4cf4bd953d139db5068e8a7a5c836b2754565bbe5190e4fc9838b3b359d","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072024,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[24,25],[-25,26],[49,27]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[24/25, -25/26, 49/27] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([24/25, -25/26, 49/27], 49/27, -25/26, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([24/25, -25/26, 49/27], 49/27, -25/26, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0025","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5c4f089bfaed588c235cacf9760d82aa650840feda01b35c4edd0bf5ef655b2","family_id":"qf-mod38-supinf-finite-extrema","seed":202611072025,"semantic_template":"qf-mod38-supinf-finite-extrema/{\"values\":[[25,26],[-26,27],[51,28]]}"},"fields":{"title":"Finite Nonempty Real Sets Attain Both Their Bounds","stem":"The ordered rational values=[25/26, -26/27, 51/28] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).","answer":{"kind":"expression","value":"([25/26, -26/27, 51/28], 51/28, -26/27, Yes, Yes, finite_nonempty_set_extrema)","explanation":"Compute the largest and smallest values exactly. For this complete finite nonempty set they are the supremum and infimum and are both attained; do not extend this attainment conclusion to arbitrary bounded real sets. The exact result, retaining the requested order, is ([25/26, -26/27, 51/28], 51/28, -26/27, Yes, Yes, finite_nonempty_set_extrema)."},"skills":["Finite Nonempty Real Sets Attain Both Their Bounds"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0026","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ec6b3d57fb2801d66b14d7c1367a0ea5679881a219aa78bd642d522b78b6e68","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072026,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[3,4],\"values\":[[1,2],[-2,3],[3,4]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[1/2, -2/3, 3/4] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=3/4. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(3/4, Yes, 3/4, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (3/4, Yes, 3/4, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0027","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b40cd7e7205b7247694dd9a51271a3407e35b6176f0b1581024cabd957263233","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072027,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[1,1],\"values\":[[2,3],[-3,4],[5,5]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[2/3, -3/4, 5/5] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=1/1. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(1, Yes, 1, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (1, Yes, 1, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0028","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c87698261d9a5b0d9550b655268e1d05adf99f1500cfff72c2a2000734a13510","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072028,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[7,6],\"values\":[[3,4],[-4,5],[7,6]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[3/4, -4/5, 7/6] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=7/6. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(7/6, Yes, 7/6, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (7/6, Yes, 7/6, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0029","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fccd962be41aa01de4f9fb6dc84fb333df25c99bcfd484006474ae92a47de357","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072029,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[9,7],\"values\":[[4,5],[-5,6],[9,7]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[4/5, -5/6, 9/7] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=9/7. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(9/7, Yes, 9/7, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (9/7, Yes, 9/7, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0030","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"043319e05ecaf447531d24e0ad82e29b1a67ebe1a9c028f2aa2a8fce5afdc4ed","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072030,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[11,8],\"values\":[[5,6],[-6,7],[11,8]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[5/6, -6/7, 11/8] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=11/8. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(11/8, Yes, 11/8, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (11/8, Yes, 11/8, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0031","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b96b1645a76309663a444a13bb39658d92b95025d4de8eb7b478e51fab9eba19","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072031,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[13,9],\"values\":[[6,7],[-7,8],[13,9]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[6/7, -7/8, 13/9] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=13/9. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(13/9, Yes, 13/9, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (13/9, Yes, 13/9, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0032","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"913c45c4faef094e501c3ef6ec9bc249997dd48bed74f64fce2b004e5891cf5c","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072032,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[3,2],\"values\":[[7,8],[-8,9],[15,10]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[7/8, -8/9, 15/10] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=3/2. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(3/2, Yes, 3/2, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (3/2, Yes, 3/2, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0033","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f23bebd393f2c170797406bf84fc5131fc9967b776b00d33879db2190371f0e6","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072033,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[17,11],\"values\":[[8,9],[-9,10],[17,11]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[8/9, -9/10, 17/11] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=17/11. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(17/11, Yes, 17/11, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (17/11, Yes, 17/11, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0034","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c371a515f67c8600b8fb20ee36af0531200b3d8b0b35ead13f8de6dd0f1572e","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072034,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[19,12],\"values\":[[9,10],[-10,11],[19,12]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[9/10, -10/11, 19/12] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=19/12. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(19/12, Yes, 19/12, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (19/12, Yes, 19/12, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0035","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9919a64a35eb092d9d267dfce77099e44a38b8d3466c181331709a43ae38dad","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072035,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[21,13],\"values\":[[10,11],[-11,12],[21,13]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[10/11, -11/12, 21/13] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=21/13. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(21/13, Yes, 21/13, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (21/13, Yes, 21/13, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0036","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d56d6884553d322524f5e4686cea7c5df7880948511964ab46d35099b6275c5","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072036,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[23,14],\"values\":[[11,12],[-12,13],[23,14]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[11/12, -12/13, 23/14] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=23/14. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(23/14, Yes, 23/14, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (23/14, Yes, 23/14, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0037","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd452327e8e89bbc8420f26affcdc20efa7125ac0f17ea919129b53cf4d53256","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072037,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[5,3],\"values\":[[12,13],[-13,14],[25,15]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[12/13, -13/14, 25/15] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=5/3. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(5/3, Yes, 5/3, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (5/3, Yes, 5/3, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0038","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4131febe6bc2ab4a002a47fd63f4fec4bbda9ec594d0d293e6ea837f9c4a84e5","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072038,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[27,16],\"values\":[[13,14],[-14,15],[27,16]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[13/14, -14/15, 27/16] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=27/16. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(27/16, Yes, 27/16, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (27/16, Yes, 27/16, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0039","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65475699a11712df1ae66610b8fdf9f9028e248a05424d03381ba63ed6da953f","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072039,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[29,17],\"values\":[[14,15],[-15,16],[29,17]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[14/15, -15/16, 29/17] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=29/17. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(29/17, Yes, 29/17, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (29/17, Yes, 29/17, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0040","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7999a77030a2c916cf486b511759ea159e4499fe4550c7e71cefcde8ed8c7c78","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072040,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[31,18],\"values\":[[15,16],[-16,17],[31,18]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[15/16, -16/17, 31/18] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=31/18. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(31/18, Yes, 31/18, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (31/18, Yes, 31/18, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0041","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65ca6555d265e9ad84929ca3d9bc5f5aee39c065aaaa4cf551dffad63deb5649","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072041,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[33,19],\"values\":[[16,17],[-17,18],[33,19]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[16/17, -17/18, 33/19] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=33/19. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(33/19, Yes, 33/19, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (33/19, Yes, 33/19, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0042","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7912d5e5d08a0179769bdfbee55820cf8aec0eee743f7bb21fb0fd906b96604","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072042,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[7,4],\"values\":[[17,18],[-18,19],[35,20]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[17/18, -18/19, 35/20] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=7/4. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(7/4, Yes, 7/4, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (7/4, Yes, 7/4, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0043","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"042e355eb678a07e0d81f33ede0f0fc8ecaa0c2af78f8e410fe878b3d69c3801","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072043,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[37,21],\"values\":[[18,19],[-19,20],[37,21]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[18/19, -19/20, 37/21] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=37/21. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(37/21, Yes, 37/21, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (37/21, Yes, 37/21, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0044","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b304fae0d7b3b7e1dfc1333f301cf6dd6f0de8cb658a1b410951496d698b1a7","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072044,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[39,22],\"values\":[[19,20],[-20,21],[39,22]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[19/20, -20/21, 39/22] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=39/22. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(39/22, Yes, 39/22, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (39/22, Yes, 39/22, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0045","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"168e4375eea47b8c37ccbb385382e4ab6519b4c36ab818671027df1a0f1546b1","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072045,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[41,23],\"values\":[[20,21],[-21,22],[41,23]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[20/21, -21/22, 41/23] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=41/23. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(41/23, Yes, 41/23, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (41/23, Yes, 41/23, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0046","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb9045682beeb44d1c1ce36a11eff341a075f05ae66f531103151be9002c7374","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072046,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[43,24],\"values\":[[21,22],[-22,23],[43,24]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[21/22, -22/23, 43/24] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=43/24. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(43/24, Yes, 43/24, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (43/24, Yes, 43/24, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0047","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cd6edf5e873ff92ad16e995c4dc2d5a59c323b6f80400573179303f5bde5665","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072047,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[9,5],\"values\":[[22,23],[-23,24],[45,25]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[22/23, -23/24, 45/25] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=9/5. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(9/5, Yes, 9/5, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (9/5, Yes, 9/5, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0048","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df59b47046da6109b15e1acc2ec2d119d448767e940c45a43eab34fddd50aa64","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072048,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[47,26],\"values\":[[23,24],[-24,25],[47,26]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[23/24, -24/25, 47/26] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=47/26. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(47/26, Yes, 47/26, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (47/26, Yes, 47/26, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0049","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6a96a43e852606a42016a89a4e4aa7ed324264685edbcd1a77998819f3deddd","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072049,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[49,27],\"values\":[[24,25],[-25,26],[49,27]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[24/25, -25/26, 49/27] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=49/27. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(49/27, Yes, 49/27, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (49/27, Yes, 49/27, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0050","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1fc6e7b0a232cfa38064d354546e29e051b028d4e11e3d5cd2f5e9648166c93","family_id":"qf-mod38-supinf-upper-bound-leastness","seed":202611072050,"semantic_template":"qf-mod38-supinf-upper-bound-leastness/{\"candidate\":[51,28],\"values\":[[25,26],[-26,27],[51,28]]}"},"fields":{"title":"An Upper Bound Still Needs Leastness to Be the Supremum","stem":"The ordered rational values=[25/26, -26/27, 51/28] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=51/28. Give (candidate, is_upper_bound, supremum, equals_supremum, boundary).","answer":{"kind":"expression","value":"(51/28, Yes, 51/28, Yes, upper_bound_and_leastness_are_separate)","explanation":"Check every value is at most the candidate, then separately compare the candidate with the exact supremum. A loose upper bound passes the first check but is not the least upper bound. The exact result, retaining the requested order, is (51/28, Yes, 51/28, Yes, upper_bound_and_leastness_are_separate)."},"skills":["An Upper Bound Still Needs Leastness to Be the Supremum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0051","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c615c0152d517e807d1b7b00d19ef83e112c864ec63c867a4dcf590c35e3686","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072051,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-2,3],\"values\":[[1,2],[-2,3],[3,4]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[1/2, -2/3, 3/4] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-2/3. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-2/3, Yes, -2/3, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-2/3, Yes, -2/3, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0052","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bffa7969a4de8b49773f906e07438b541f506bfefa33df97733a3dd0d9936b6b","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072052,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-3,4],\"values\":[[2,3],[-3,4],[5,5]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[2/3, -3/4, 5/5] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-3/4. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-3/4, Yes, -3/4, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-3/4, Yes, -3/4, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0053","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71ede18a1adc61db02d94b651d142ba20e859d0332775ec77a81226ef33b353e","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072053,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-4,5],\"values\":[[3,4],[-4,5],[7,6]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[3/4, -4/5, 7/6] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-4/5. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-4/5, Yes, -4/5, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-4/5, Yes, -4/5, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0054","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9930bd8ff86df6434ad2872b7ad0d5154855becd765e69d8b76fa3f2495fba2c","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072054,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-5,6],\"values\":[[4,5],[-5,6],[9,7]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[4/5, -5/6, 9/7] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-5/6. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-5/6, Yes, -5/6, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-5/6, Yes, -5/6, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0055","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c15b795c30860e600f446f58c0efd19d1c59c68041d487f5f80a4c8d383b9ef3","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072055,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-6,7],\"values\":[[5,6],[-6,7],[11,8]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[5/6, -6/7, 11/8] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-6/7. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-6/7, Yes, -6/7, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-6/7, Yes, -6/7, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0056","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a130ad1b71825a352efc2d78e10b8bfcd43103b56219c34b73a7b6ded208a77","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072056,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-7,8],\"values\":[[6,7],[-7,8],[13,9]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[6/7, -7/8, 13/9] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-7/8. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-7/8, Yes, -7/8, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-7/8, Yes, -7/8, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0057","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92dcefd808473806d04d4bf907c9275aeb50186f331460ccb68fed6140fd10d5","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072057,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-8,9],\"values\":[[7,8],[-8,9],[15,10]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[7/8, -8/9, 15/10] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-8/9. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-8/9, Yes, -8/9, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-8/9, Yes, -8/9, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0058","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea0efe9aa7b22cd0313de0dd5bf93ed5dbd10f18a55613ffe0d2010645331f77","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072058,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-9,10],\"values\":[[8,9],[-9,10],[17,11]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[8/9, -9/10, 17/11] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-9/10. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-9/10, Yes, -9/10, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-9/10, Yes, -9/10, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0059","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b68843ee81381fa82aa5164397678c025ab787be651d279b3a8fa12836c5a62b","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072059,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-10,11],\"values\":[[9,10],[-10,11],[19,12]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[9/10, -10/11, 19/12] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-10/11. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-10/11, Yes, -10/11, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-10/11, Yes, -10/11, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0060","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b68c8c24aa6a2d4d65f10457881acfdc72e9263233ac2fb32a4b2d8a7f370ee","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072060,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-11,12],\"values\":[[10,11],[-11,12],[21,13]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[10/11, -11/12, 21/13] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-11/12. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-11/12, Yes, -11/12, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-11/12, Yes, -11/12, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0061","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4706ab2897d072eb03fa466fe22df386c1a8f42a7e275b8bbc52c97bf9659224","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072061,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-12,13],\"values\":[[11,12],[-12,13],[23,14]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[11/12, -12/13, 23/14] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-12/13. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-12/13, Yes, -12/13, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-12/13, Yes, -12/13, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0062","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9f680189e102bd92ef618c94044f094303b232ee1d26b1ea10ea729a8a28a4a","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072062,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-13,14],\"values\":[[12,13],[-13,14],[25,15]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[12/13, -13/14, 25/15] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-13/14. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-13/14, Yes, -13/14, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-13/14, Yes, -13/14, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0063","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c8d8ccbe424d9b978a265dbba9476c1f4e6991209a9159ec4a4460d127ee4f9","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072063,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-14,15],\"values\":[[13,14],[-14,15],[27,16]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[13/14, -14/15, 27/16] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-14/15. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-14/15, Yes, -14/15, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-14/15, Yes, -14/15, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0064","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01a28f9bc81d1eba81eab7a2bac4d1ed1f3e9b139bc9d283ff2a5f5faee99131","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072064,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-15,16],\"values\":[[14,15],[-15,16],[29,17]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[14/15, -15/16, 29/17] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-15/16. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-15/16, Yes, -15/16, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-15/16, Yes, -15/16, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0065","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9bdee257e29ee41eaff958988f4aa6e27fdb5cb59b69779738d72e55ec12ba1","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072065,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-16,17],\"values\":[[15,16],[-16,17],[31,18]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[15/16, -16/17, 31/18] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-16/17. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-16/17, Yes, -16/17, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-16/17, Yes, -16/17, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0066","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c819e4369388ba74209572b723e4196940431117e35cff9d43509ffddef59a7c","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072066,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-17,18],\"values\":[[16,17],[-17,18],[33,19]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[16/17, -17/18, 33/19] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-17/18. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-17/18, Yes, -17/18, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-17/18, Yes, -17/18, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0067","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6ce5c3e6390302a545134c3dba94348b08223fbc07b1d7e1354ba93cc711842","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072067,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-18,19],\"values\":[[17,18],[-18,19],[35,20]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[17/18, -18/19, 35/20] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-18/19. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-18/19, Yes, -18/19, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-18/19, Yes, -18/19, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0068","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d91f3f9618bbb08195b802e25d4b104dc8c5ea262dd1c81465e650adefc631d4","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072068,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-19,20],\"values\":[[18,19],[-19,20],[37,21]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[18/19, -19/20, 37/21] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-19/20. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-19/20, Yes, -19/20, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-19/20, Yes, -19/20, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0069","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1d3da29b412896aedffe1a3c6c7ee11a6754f8557dc27806c1daacbefaffa84","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072069,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-20,21],\"values\":[[19,20],[-20,21],[39,22]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[19/20, -20/21, 39/22] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-20/21. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-20/21, Yes, -20/21, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-20/21, Yes, -20/21, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0070","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"311fc4a80ebed541c1f4f1de56289b0bbc63eafb9cabcebed0dfa14d59e3d491","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072070,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-21,22],\"values\":[[20,21],[-21,22],[41,23]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[20/21, -21/22, 41/23] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-21/22. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-21/22, Yes, -21/22, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-21/22, Yes, -21/22, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0071","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a7fd1bb90121688e9f5a547b06207fde3b310db3354d06f5527886985f0c0e4","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072071,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-22,23],\"values\":[[21,22],[-22,23],[43,24]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[21/22, -22/23, 43/24] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-22/23. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-22/23, Yes, -22/23, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-22/23, Yes, -22/23, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0072","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2ee167eab7dac03263dd3d1800f22e8a695f376c506308dfa00b1bfc42f5c96","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072072,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-23,24],\"values\":[[22,23],[-23,24],[45,25]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[22/23, -23/24, 45/25] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-23/24. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-23/24, Yes, -23/24, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-23/24, Yes, -23/24, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0073","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caffae2ed01becc084a0862e0fa413b244a9604b1d8bd8e1b052f5f4e011a9bc","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072073,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-24,25],\"values\":[[23,24],[-24,25],[47,26]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[23/24, -24/25, 47/26] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-24/25. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-24/25, Yes, -24/25, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-24/25, Yes, -24/25, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0074","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f67c849a35fe61854cae0262ebd7310651df8c3c6dac2f2bc2c16edd1ad7cf4","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072074,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-25,26],\"values\":[[24,25],[-25,26],[49,27]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[24/25, -25/26, 49/27] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-25/26. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-25/26, Yes, -25/26, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-25/26, Yes, -25/26, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0075","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd2ebe1ed101861beea572b42b4586e5058ee185e98b77cc19aca7a75a74ae16","family_id":"qf-mod38-supinf-lower-bound-greatestness","seed":202611072075,"semantic_template":"qf-mod38-supinf-lower-bound-greatestness/{\"candidate\":[-26,27],\"values\":[[25,26],[-26,27],[51,28]]}"},"fields":{"title":"A Lower Bound Still Needs Greatestness to Be the Infimum","stem":"The ordered rational values=[25/26, -26/27, 51/28] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Candidate=-26/27. Give (candidate, is_lower_bound, infimum, equals_infimum, boundary).","answer":{"kind":"expression","value":"(-26/27, Yes, -26/27, Yes, lower_bound_and_greatestness_are_separate)","explanation":"Check every value is at least the candidate, then separately compare it with the exact infimum. A smaller lower bound is not the greatest lower bound. The exact result, retaining the requested order, is (-26/27, Yes, -26/27, Yes, lower_bound_and_greatestness_are_separate)."},"skills":["A Lower Bound Still Needs Greatestness to Be the Infimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0076","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05020e5994b275c0381f987b89620b4dc2f2fedadfbab346af4338e84490a18b","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072076,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[0,1],\"right\":[2,1]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (0/1, 2/1), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(0, 2, 2, 0, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (0, 2, 2, 0, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0077","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d98cc2a564f0006bbe4a3735c96d2170863168c38d6417383a25c97fa98191cd","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072077,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[1,2],\"right\":[3,2]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (1/2, 3/2), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(1/2, 3/2, 3/2, 1/2, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (1/2, 3/2, 3/2, 1/2, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0078","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48ca3f502949a204ae0649fc00df532e6414c5f5d2c2f59c7f5b2912d463760a","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072078,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[2,3],\"right\":[4,3]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (2/3, 4/3), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(2/3, 4/3, 4/3, 2/3, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (2/3, 4/3, 4/3, 2/3, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0079","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dc4ac1560b8927cc5c82cc2b3dbf551a659d6141032d026deb69879606d02f6","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072079,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[3,4],\"right\":[5,4]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (3/4, 5/4), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(3/4, 5/4, 5/4, 3/4, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (3/4, 5/4, 5/4, 3/4, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0080","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fc0d8b02b7a4a8dd41f5623ff57a60d5349296da007484b9ffeb97eb438c650","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072080,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[4,5],\"right\":[6,5]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (4/5, 6/5), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(4/5, 6/5, 6/5, 4/5, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (4/5, 6/5, 6/5, 4/5, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0081","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d3b0f6e30dbef964b07416f03952b48483bbfdccd89b68f3d520a6c939b0013","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072081,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[5,6],\"right\":[7,6]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (5/6, 7/6), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(5/6, 7/6, 7/6, 5/6, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (5/6, 7/6, 7/6, 5/6, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0082","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1da6613b4ba8d1611dd45fc1838ddd9b011a1d414c093aeb24f687c8c240cc1","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072082,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[6,7],\"right\":[8,7]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (6/7, 8/7), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(6/7, 8/7, 8/7, 6/7, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (6/7, 8/7, 8/7, 6/7, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0083","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bca42279101578d6ad23e251c2ebb75cd1d7d0584464740fa1cb7888b406984","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072083,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[7,8],\"right\":[9,8]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (7/8, 9/8), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(7/8, 9/8, 9/8, 7/8, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (7/8, 9/8, 9/8, 7/8, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0084","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2af6780f38ed51af864011c0343306430e0749f8295c48a497f02091f7344708","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072084,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[8,9],\"right\":[10,9]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (8/9, 10/9), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(8/9, 10/9, 10/9, 8/9, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (8/9, 10/9, 10/9, 8/9, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0085","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e3490bcaf1801c1ae0b64c9cd7216a3af07a7cc00b068b338cc9c8d78f30742","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072085,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[9,10],\"right\":[11,10]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (9/10, 11/10), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(9/10, 11/10, 11/10, 9/10, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (9/10, 11/10, 11/10, 9/10, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0086","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99b9db7c226251fcc03451241d218cd93b3769a1916bd04cbe86085f430852ce","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072086,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[10,11],\"right\":[12,11]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (10/11, 12/11), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(10/11, 12/11, 12/11, 10/11, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (10/11, 12/11, 12/11, 10/11, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0087","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"127cd6509cee714386cdb8a09baaba04a518650bd99a555d0206f98ec28b1c55","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072087,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[11,12],\"right\":[13,12]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (11/12, 13/12), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(11/12, 13/12, 13/12, 11/12, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (11/12, 13/12, 13/12, 11/12, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0088","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb97636c548975c49b0719603e39750584aa1149820b8fae8938bfa7d704a2f3","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072088,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[12,13],\"right\":[14,13]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (12/13, 14/13), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(12/13, 14/13, 14/13, 12/13, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (12/13, 14/13, 14/13, 12/13, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0089","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ba7e97cf022d2205c37b24d190c698ad1407f1f2c7230de84a9dd44bd07a87c","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072089,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[13,14],\"right\":[15,14]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (13/14, 15/14), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(13/14, 15/14, 15/14, 13/14, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (13/14, 15/14, 15/14, 13/14, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0090","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16be2187438cc0334627d65c1ba756c2f5cd5d33f7791566c0022527f7e693d4","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072090,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[14,15],\"right\":[16,15]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (14/15, 16/15), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(14/15, 16/15, 16/15, 14/15, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (14/15, 16/15, 16/15, 14/15, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0091","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b18baadee6d5a2f1ee4b3f2a40c59e3770a99e9d57b58199f6bbd0b2eb8f11a","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072091,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[15,16],\"right\":[17,16]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (15/16, 17/16), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(15/16, 17/16, 17/16, 15/16, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (15/16, 17/16, 17/16, 15/16, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0092","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3dcb78377791409a5f8027e344efc381e153893f86901cc36c1bd898a3310b8","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072092,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[16,17],\"right\":[18,17]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (16/17, 18/17), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(16/17, 18/17, 18/17, 16/17, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (16/17, 18/17, 18/17, 16/17, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0093","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a16b6c1fd93941a21acfad90fb97f9950dc232ab729444055ecd1eb94cc2dc81","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072093,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[17,18],\"right\":[19,18]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (17/18, 19/18), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(17/18, 19/18, 19/18, 17/18, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (17/18, 19/18, 19/18, 17/18, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0094","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5720deb56d706117c96e967e4fe7248c9b5d8b8efdc069d04b1f55c1bde4b83","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072094,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[18,19],\"right\":[20,19]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (18/19, 20/19), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(18/19, 20/19, 20/19, 18/19, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (18/19, 20/19, 20/19, 18/19, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0095","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"795d1f0603947e088ed61f009b1b37ac4de1f41f828c443b846390dd965225ca","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072095,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[19,20],\"right\":[21,20]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (19/20, 21/20), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(19/20, 21/20, 21/20, 19/20, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (19/20, 21/20, 21/20, 19/20, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0096","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87c4c714e7f65643a71ebca05002a9a1359d240abd22dee88c9295f7562945e6","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072096,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[20,21],\"right\":[22,21]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (20/21, 22/21), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(20/21, 22/21, 22/21, 20/21, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (20/21, 22/21, 22/21, 20/21, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0097","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9129b84586e157f91d524a22ec0e6d0ef8f5f081784d9919080d79985103f29a","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072097,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[21,22],\"right\":[23,22]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (21/22, 23/22), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(21/22, 23/22, 23/22, 21/22, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (21/22, 23/22, 23/22, 21/22, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0098","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2e3e48610bec7e63c8008763295b51829706bd18c0644ffadc86a722a982be7","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072098,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[22,23],\"right\":[24,23]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (22/23, 24/23), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(22/23, 24/23, 24/23, 22/23, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (22/23, 24/23, 24/23, 22/23, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0099","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21deba30937987a765e0da390c9d1b9fc9551a88546cea0531fbf18ae24c57ca","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072099,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[23,24],\"right\":[25,24]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (23/24, 25/24), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(23/24, 25/24, 25/24, 23/24, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (23/24, 25/24, 25/24, 23/24, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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-20261014-advance-20261107-r02-supremum-and-infimum-0100","batch_id":"batch-20261014-advance-20261107-r02-supremum-and-infimum-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f19e5fbfca70c4bb90d2ddd314490f9b5226e930987d3b655981138277bd6172","family_id":"qf-mod38-supinf-open-interval-nonattainment","seed":202611072100,"semantic_template":"qf-mod38-supinf-open-interval-nonattainment/{\"left\":[24,25],\"right\":[26,25]}"},"fields":{"title":"The Bounds of an Open Interval Are Not Its Maximum or Minimum","stem":"Use the entire nonempty bounded real open interval (24/25, 26/25), with left<right. The endpoints are exact rationals; this is not a finite sample of interval points. Give (left, right, supremum, infimum, supremum_attained, infimum_attained, boundary).","answer":{"kind":"expression","value":"(24/25, 26/25, 26/25, 24/25, No, No, supremum_and_infimum_not_attained)","explanation":"The right endpoint is the supremum and the left endpoint the infimum. Points arbitrarily close to each endpoint give leastness/greatestness, but neither endpoint is in the open interval, so neither bound is attained. The exact result, retaining the requested order, is (24/25, 26/25, 26/25, 24/25, No, No, supremum_and_infimum_not_attained)."},"skills":["The Bounds of an Open Interval Are Not Its Maximum or Minimum"],"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."}}