{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0001","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40d47cca205685d031e578d1fdd6bb9dc733e4046ce3ce3a70ab14da832ad479","family_id":"qf-density10-quantifier-truth-partition","seed":202610107001,"semantic_template":"qf-density10-quantifier-truth-partition/[[],[]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([], [])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([], [])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0002","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62b04534ce3d413ee12a2ae7a2d493feeee0a168a98fb169b57f0671b63370ed","family_id":"qf-density10-quantifier-truth-partition","seed":202610107002,"semantic_template":"qf-density10-quantifier-truth-partition/[[-2,0,2],[]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-2, 0, 2] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([-2, 0, 2], [])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([-2, 0, 2], [])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0003","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1561d24969081b08afefb4c5703f3802ca0e78751aca1b80257a9bebbd76b0e3","family_id":"qf-density10-quantifier-truth-partition","seed":202610107003,"semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-1,1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-1, 0, 1] and P(x) means x=0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0], [-1, 1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-1, 1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0004","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01f3cd271ea457753009f8bab9e38c865b967b5a309d263114ad4ea4639fdd13","family_id":"qf-density10-quantifier-truth-partition","seed":202610107004,"semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,3],[-3,-1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-3, -1, 0, 1, 3] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0, 1, 3], [-3, -1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 3], [-3, -1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0005","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ffc176ca992de577ca925b995b1b559ec020020df75c2c17219fc5fba878e80","family_id":"qf-density10-quantifier-truth-partition","seed":202610107005,"semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-4,-1,0,4]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-4, -1, 0, 1, 4] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([1], [-4, -1, 0, 4])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-4, -1, 0, 4])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0006","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"335dd47c16360a59a25b83bd33a6e376e67d91f54d88386f78ff911f16d27506","family_id":"qf-density10-quantifier-truth-partition","seed":202610107006,"semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-5,-1,1,5]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-5, -1, 0, 1, 5] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0], [-5, -1, 1, 5])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-5, -1, 1, 5])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0007","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"314ae53c7befcfb60968922cd323a49eeee2216d123735170c63e9165963fe4c","family_id":"qf-density10-quantifier-truth-partition","seed":202610107007,"semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,6],[-6,-1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-6, -1, 0, 1, 6] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0, 1, 6], [-6, -1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 6], [-6, -1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0008","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e98a52f17b846b77a9cb47c7f8025561adc4a0789886e49cac9723f4a88d4181","family_id":"qf-density10-quantifier-truth-partition","seed":202610107008,"semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-7,-1,0,7]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-7, -1, 0, 1, 7] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([1], [-7, -1, 0, 7])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-7, -1, 0, 7])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0009","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c39ba4e8208bf19c642c585db14184327c8d79b2a6050215ac8fcb43a9dfa0d4","family_id":"qf-density10-quantifier-truth-partition","seed":202610107009,"semantic_template":"qf-density10-quantifier-truth-partition/[[-8,0,8],[-1,1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-8, -1, 0, 1, 8] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([-8, 0, 8], [-1, 1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([-8, 0, 8], [-1, 1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0010","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c986e4aff6d4b252a03b1e402cdffb455aec6718864e4227d9718c726b0439af","family_id":"qf-density10-quantifier-truth-partition","seed":202610107010,"semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,9],[-9,-1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-9, -1, 0, 1, 9] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0, 1, 9], [-9, -1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 9], [-9, -1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0011","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c28f08c50b7f0fc687d82cddca77ed18b6fca2a1d75784ced542faf5f4e2766d","family_id":"qf-density10-quantifier-truth-partition","seed":202610107011,"semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-10,-1,0,10]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-10, -1, 0, 1, 10] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([1], [-10, -1, 0, 10])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-10, -1, 0, 10])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0012","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26b6df1444d73b2e2ee236d1af57aa4d8bdba0a733969cc36ec22d324c1ee5f7","family_id":"qf-density10-quantifier-truth-partition","seed":202610107012,"semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-11,-1,1,11]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-11, -1, 0, 1, 11] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0], [-11, -1, 1, 11])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-11, -1, 1, 11])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0013","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f9d4245bfb7210aaa3357f23d85e95ecf4d804a4647c93b01b531ed4a69642b","family_id":"qf-density10-quantifier-truth-partition","seed":202610107013,"semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,12],[-12,-1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-12, -1, 0, 1, 12] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0, 1, 12], [-12, -1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 12], [-12, -1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0014","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbdd0350f4b7929d6229d3e2bc1ffe768e1385dbbc77eb880d180c99506bc35c","family_id":"qf-density10-quantifier-truth-partition","seed":202610107014,"semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-13,-1,0,13]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-13, -1, 0, 1, 13] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([1], [-13, -1, 0, 13])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-13, -1, 0, 13])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0015","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e06cd984701cbcf9b49306f1aee7d14733def893aadd5a2b2f8cbe163624b66","family_id":"qf-density10-quantifier-truth-partition","seed":202610107015,"semantic_template":"qf-density10-quantifier-truth-partition/[[-14,0,14],[-1,1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-14, -1, 0, 1, 14] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([-14, 0, 14], [-1, 1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([-14, 0, 14], [-1, 1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0016","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f92c2380f85a9b40307b5857abd0ea461899c2f1dc4fb4f8e8e5e4f64076f6c1","family_id":"qf-density10-quantifier-truth-partition","seed":202610107016,"semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,15],[-15,-1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-15, -1, 0, 1, 15] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0, 1, 15], [-15, -1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 15], [-15, -1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0017","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7662216779e34b4f8dce9ebfaa8650cc1e64c9707e10e22ca36aa01bdbf1405f","family_id":"qf-density10-quantifier-truth-partition","seed":202610107017,"semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-16,-1,0,16]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-16, -1, 0, 1, 16] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([1], [-16, -1, 0, 16])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-16, -1, 0, 16])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0018","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"104470243025be254b0e6448f843d6ca0754bb99ddb071050d811aca1cfb91a3","family_id":"qf-density10-quantifier-truth-partition","seed":202610107018,"semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-17,-1,1,17]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-17, -1, 0, 1, 17] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0], [-17, -1, 1, 17])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-17, -1, 1, 17])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0019","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2c617df9ac79a9d0c563bcd6188c25de54e328e2979588f4a9af5043c74ea92","family_id":"qf-density10-quantifier-truth-partition","seed":202610107019,"semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,18],[-18,-1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-18, -1, 0, 1, 18] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0, 1, 18], [-18, -1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 18], [-18, -1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0020","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d51de49123617774dd5cdce3d3583cbdb10fa87ed02cce620ee398bfeefc5c7e","family_id":"qf-density10-quantifier-truth-partition","seed":202610107020,"semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-19,-1,0,19]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-19, -1, 0, 1, 19] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([1], [-19, -1, 0, 19])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-19, -1, 0, 19])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0021","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59febfda5bfe672af2575e0cd2fde40fa155cea54de91a23697548b912ad3435","family_id":"qf-density10-quantifier-truth-partition","seed":202610107021,"semantic_template":"qf-density10-quantifier-truth-partition/[[-20,0,20],[-1,1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-20, -1, 0, 1, 20] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([-20, 0, 20], [-1, 1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([-20, 0, 20], [-1, 1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0022","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7220a2a0b8690037ce492d6b144ee8b78d87907ead8937d14cd9bb013fc5a1a1","family_id":"qf-density10-quantifier-truth-partition","seed":202610107022,"semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,21],[-21,-1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-21, -1, 0, 1, 21] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0, 1, 21], [-21, -1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 21], [-21, -1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0023","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"deba8d94f41607cad3892e2cd0bb9b84e5c74fd9f0423a6ed36e800d0233f8db","family_id":"qf-density10-quantifier-truth-partition","seed":202610107023,"semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-22,-1,0,22]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-22, -1, 0, 1, 22] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([1], [-22, -1, 0, 22])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-22, -1, 0, 22])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0024","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a89e620a61c19fdba03ba3278128992fcb955b908c800532f937ea7674c09bc","family_id":"qf-density10-quantifier-truth-partition","seed":202610107024,"semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-23,-1,1,23]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-23, -1, 0, 1, 23] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0], [-23, -1, 1, 23])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-23, -1, 1, 23])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0025","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4a8ddf9b1eb58e70bfef24c5c0813c1654cbae06dca1806249d45c410852029","family_id":"qf-density10-quantifier-truth-partition","seed":202610107025,"semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,24],[-24,-1]]"},"fields":{"title":"Complete Finite Predicate Partition","stem":"The explicitly finite integer domain is U=[-24, -1, 0, 1, 24] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.","answer":{"kind":"expression","value":"([0, 1, 24], [-24, -1])","explanation":"Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 24], [-24, -1])."},"skills":["Complete Finite Predicate Partition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0026","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"511005560cfe9af070b8c4f9bb962640ab04cb27fc82a195f06dd9f865f4098d","family_id":"qf-density10-quantifier-negation-pair","seed":202610107026,"semantic_template":"qf-density10-quantifier-negation-pair/[0,0]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(False, True)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, True)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0027","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13d5c932ef659df6529acdaa3006043eda5a0354fb2f749e757f746207866596","family_id":"qf-density10-quantifier-negation-pair","seed":202610107027,"semantic_template":"qf-density10-quantifier-negation-pair/[0,1]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[-1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, True)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0028","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e06802eada1ed39124af605b80c872af2813f6564e00a7f122a4208b64e78b52","family_id":"qf-density10-quantifier-negation-pair","seed":202610107028,"semantic_template":"qf-density10-quantifier-negation-pair/[1,0]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(False, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0029","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82be50a240c042ef53e1edf47494d1fe55991522efade02b72d9f1adba1ba041","family_id":"qf-density10-quantifier-negation-pair","seed":202610107029,"semantic_template":"qf-density10-quantifier-negation-pair/[0,2]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[-2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, True)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0030","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78d2b38f8a48dd6ff931b8f20968ae2a135b52a3b44f3505531e752e32ed520d","family_id":"qf-density10-quantifier-negation-pair","seed":202610107030,"semantic_template":"qf-density10-quantifier-negation-pair/[1,1]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0031","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f54431394a4e85ffd57328b2d632e41b5fadb82bbfb6de09247c20ca67c638d","family_id":"qf-density10-quantifier-negation-pair","seed":202610107031,"semantic_template":"qf-density10-quantifier-negation-pair/[2,0]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(False, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0032","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"831dcffee96e3c43d7b474e154adc772ccf87cf4a0e74b99850bfae98f28f1e8","family_id":"qf-density10-quantifier-negation-pair","seed":202610107032,"semantic_template":"qf-density10-quantifier-negation-pair/[0,3]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[-3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, True)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0033","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e584714c11354fe23adc08e66605bae67e83cb1c012befb960fb2ca9f7ec3533","family_id":"qf-density10-quantifier-negation-pair","seed":202610107033,"semantic_template":"qf-density10-quantifier-negation-pair/[1,2]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0034","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb5552d7526d35801859fe53e2acd179d591746ded913ad49758416832edba6","family_id":"qf-density10-quantifier-negation-pair","seed":202610107034,"semantic_template":"qf-density10-quantifier-negation-pair/[2,1]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0035","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4308960cea24db7be002fd7b714cd631f138e4e38118484e6a9222da4c15e11","family_id":"qf-density10-quantifier-negation-pair","seed":202610107035,"semantic_template":"qf-density10-quantifier-negation-pair/[3,0]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, 3] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(False, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0036","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73366b6914daafd096acf50b947a4cbefee4e30c7fa59b207c5dc49218af22ee","family_id":"qf-density10-quantifier-negation-pair","seed":202610107036,"semantic_template":"qf-density10-quantifier-negation-pair/[0,4]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[-4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, True)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0037","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2731f0141feaffee13f6c2361e4222a1f3e0ef7bfef3f9c93407104d516104bb","family_id":"qf-density10-quantifier-negation-pair","seed":202610107037,"semantic_template":"qf-density10-quantifier-negation-pair/[1,3]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0038","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad3345340aa66fd66f39febc18d8d83e2694e13b1a38e59e692be85373836b4a","family_id":"qf-density10-quantifier-negation-pair","seed":202610107038,"semantic_template":"qf-density10-quantifier-negation-pair/[2,2]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0039","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9991e57034c83a812950b5ee7224bf4bb563f49a159ec4069dd832b8752488f9","family_id":"qf-density10-quantifier-negation-pair","seed":202610107039,"semantic_template":"qf-density10-quantifier-negation-pair/[3,1]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, 3, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0040","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ee829273024472c28c07a10c1ac01f393c72347481adf087b24f248588196ca","family_id":"qf-density10-quantifier-negation-pair","seed":202610107040,"semantic_template":"qf-density10-quantifier-negation-pair/[4,0]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, 3, 4] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(False, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0041","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c4d6f7710fc98ac9561308d064f389be37e2aa61b205006d245d30ed34d4487","family_id":"qf-density10-quantifier-negation-pair","seed":202610107041,"semantic_template":"qf-density10-quantifier-negation-pair/[0,5]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[-5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, True)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0042","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5386af65bdc152aa8de8f5f45724604f5c7b13016e33ef074330b8bfe6a4ad85","family_id":"qf-density10-quantifier-negation-pair","seed":202610107042,"semantic_template":"qf-density10-quantifier-negation-pair/[1,4]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0043","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3a73fba97b31878981ee7a2821dc5ac27d478a8f2ec123cd7416f44425f6d17","family_id":"qf-density10-quantifier-negation-pair","seed":202610107043,"semantic_template":"qf-density10-quantifier-negation-pair/[2,3]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0044","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ec91a0abb17d3c08862f78602aade9b358a4add54d00a95a2414e3d46562f70","family_id":"qf-density10-quantifier-negation-pair","seed":202610107044,"semantic_template":"qf-density10-quantifier-negation-pair/[3,2]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, 3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0045","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de95a593abdc8665f92828ff3281260ea9187728ae5821ec0e41274bacc1af47","family_id":"qf-density10-quantifier-negation-pair","seed":202610107045,"semantic_template":"qf-density10-quantifier-negation-pair/[4,1]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, 3, 4, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0046","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"721723927bc2a3d979c184c6120684ecae007424dcdfde4a2b242835da7af884","family_id":"qf-density10-quantifier-negation-pair","seed":202610107046,"semantic_template":"qf-density10-quantifier-negation-pair/[5,0]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, 3, 4, 5] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(False, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0047","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec19db9a7fa743c1540f2560f10d0835dcccb7d6d509ae30ed88d15a387fb49f","family_id":"qf-density10-quantifier-negation-pair","seed":202610107047,"semantic_template":"qf-density10-quantifier-negation-pair/[0,6]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[-6, -5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, True)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0048","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1636eb52884698d28c16c2ae0261b9c240ee948ee49d2115517dabd04571ac5","family_id":"qf-density10-quantifier-negation-pair","seed":202610107048,"semantic_template":"qf-density10-quantifier-negation-pair/[1,5]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, -5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0049","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35f84253212a8bbdfc1e9d531d0c9f7dc3fcaecc85d582990f45c7eb1b17219e","family_id":"qf-density10-quantifier-negation-pair","seed":202610107049,"semantic_template":"qf-density10-quantifier-negation-pair/[2,4]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0050","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4317f69304863833ea58f9fdb1b9580b53a68b0e4b6a137c53eaa0ad5b2a9392","family_id":"qf-density10-quantifier-negation-pair","seed":202610107050,"semantic_template":"qf-density10-quantifier-negation-pair/[3,3]"},"fields":{"title":"Negated Universal and Existential Pair","stem":"The explicitly finite integer domain is U=[1, 2, 3, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.","answer":{"kind":"expression","value":"(True, False)","explanation":"A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False)."},"skills":["Negated Universal and Existential Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0051","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"646e021f5dc3de36fdb76d811864070a4fb0f7f0bf549f992b036d79f2d4f244","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107051,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"outside\",\"outside\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[] and P(x) means x≥0. Ordered candidates=[100, 101, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0052","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9da62eab5ac842e0c49e2a3e9043270340bf13f16b2d9871b12f9246190f3ae9","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107052,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"outside\",\"outside\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 101, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[Yes, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0053","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6888ab22751e54241d8b79a2922f9f2df965199c300e5fab93b4183a0e2efbf5","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107053,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"satisfies\",\"outside\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0054","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf718ea5edd6001e9ad223790ec6308c84a7d3e0a6c46a1dba8ed34adaeb6511","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107054,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"satisfies\",\"outside\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[Yes, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0055","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef130c356ebff7f20625173689a89a2448af6328d6a58db117a34cf09819c399","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107055,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"violates\",\"outside\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, -2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, Yes, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0056","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f19146375ea7924f97699396f388487cc4c137f9d35f91306c5bd221a862c657","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107056,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"violates\",\"outside\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, -2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, Yes, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0057","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aab558dcccaef552991ad103e11a360a52b5c3cb0dced2fc392b2f634777d30a","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107057,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"violates\",\"outside\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, -2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[Yes, Yes, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, Yes, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0058","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7325b7af1f57e79e7b6eb71f1d3282b26b69d20cd18225ea40e9c0c601ff39ca","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107058,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"outside\",\"satisfies\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, 101, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0059","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0109aef89a149086d0b940ae50bd14687cd72b3eb4bf5e74d3a4e5c8e1dbe060","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107059,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"outside\",\"satisfies\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 101, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0060","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77edebd3a792c1b5132ed5b62613ee8dc86aee76bd29a8540488f0879ca90f19","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107060,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"outside\",\"satisfies\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 101, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[Yes, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0061","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"034290483d172f791f576b2708d821803097717a48bb6a53661f0b172b39e67f","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107061,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"satisfies\",\"satisfies\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, 2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0062","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1bc3c27eb10ab95c3d7245275940ff89f0cc51e8d157d1dbf044c69f3a4bf64","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107062,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"satisfies\",\"satisfies\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0063","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35b20d76fb42996c66596e766ef970a1fe65afcaf3b9bf1175508a8fc0ca6f3b","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107063,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"satisfies\",\"satisfies\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[Yes, No, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0064","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"272c2ef4e550e1e674bbbba68ec405e53565cb973c3df314499d388a44c9f845","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107064,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"violates\",\"satisfies\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, -2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, Yes, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0065","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6470fdfcd5852aeafc7c2a13050a627dec98477a0ede44a56a911c526096bb91","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107065,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"violates\",\"satisfies\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, -2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, Yes, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0066","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d129f218ee085fa5bbcf6ef3466be442766eb35e2a796d54abbae5bc4c3c644f","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107066,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"violates\",\"satisfies\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, -2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[Yes, Yes, No]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, Yes, No]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0067","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeec59e534fc8a434d83f9d1d61e4fcf1b4fdd8204b06f5c2b88327e83e23c02","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107067,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"outside\",\"violates\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, 101, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, Yes]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, Yes]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0068","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"993da7a27d19d8bdaea7c1658d736705e28f7ae08d95d121fe662b3aa8c80343","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107068,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"outside\",\"violates\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 101, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, Yes]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, Yes]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0069","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2e83bffcb81e236d1a5de1d6864096b6588d2b1a96f4d7160ef16de5e0178d0","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107069,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"outside\",\"violates\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 101, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[Yes, No, Yes]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, Yes]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0070","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98fac0ac5bf46a4c5e4b61093c43fc2d635b1a136202be96f36989814a7a54c8","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107070,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"satisfies\",\"violates\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, 2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, Yes]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, Yes]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0071","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10884c52a81a5b2c02bddc2917bb7525e799f911db5e11ddc3a8c1b1ffd087e4","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107071,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"satisfies\",\"violates\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, No, Yes]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, Yes]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0072","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8268fe391928787b27559c700a3abe6f0a71b10c2d40b685f5d6f2a5e9f7a36c","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107072,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"satisfies\",\"violates\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[Yes, No, Yes]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, Yes]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0073","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"764b6bbd15bfa1eaf55d1b1b2f4d50b839718e4304af7def167ac9bb8e668e71","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107073,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"violates\",\"violates\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, -2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, Yes, Yes]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, Yes]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0074","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c0e8151d0efd9e2de7a762feb34eb138468e89ab8b099353899fffcdf055d36","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107074,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"violates\",\"violates\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, -2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[No, Yes, Yes]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, Yes]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0075","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcc5353c54dbd4d9382dffbad736db480ac9c3cca404879b7bac629019cf6bdd","family_id":"qf-density10-quantifier-candidate-mask","seed":202610107075,"semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"violates\",\"violates\"]"},"fields":{"title":"In-Domain Counterexample Candidate Mask","stem":"The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, -2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.","answer":{"kind":"expression","value":"[Yes, Yes, Yes]","explanation":"Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, Yes, Yes]."},"skills":["In-Domain Counterexample Candidate Mask"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0076","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"810ce7c91906da37ca3ca2b3e03e279dd69d9dcd0768a7f29c276e112f7ad81b","family_id":"qf-density10-quantifier-nested-counts","seed":202610107076,"semantic_template":"qf-density10-quantifier-nested-counts/[0,0,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[], V=[] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0077","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f6273f5a89e93a69a88c30c8c658af7fece692ec12fcf11cb2c3d9aa4bfbc1a","family_id":"qf-density10-quantifier-nested-counts","seed":202610107077,"semantic_template":"qf-density10-quantifier-nested-counts/[[],[-1,1],\"le\"]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[], V=[-1, 1] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 2)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 2)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0078","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38729740b610606412bd87a95c973e749172c34af0b626c88316c4c16eec52a2","family_id":"qf-density10-quantifier-nested-counts","seed":202610107078,"semantic_template":"qf-density10-quantifier-nested-counts/[[1,2],[],\"le\"]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[1, 2], V=[] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0079","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae89d8b5ec78480b1222fdf41cec29b0056c75b90f09b9f0b5ecef70fe2ecb4c","family_id":"qf-density10-quantifier-nested-counts","seed":202610107079,"semantic_template":"qf-density10-quantifier-nested-counts/[[-1,0,1],[-2,0,2],\"le\"]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-1, 0, 1], V=[-2, 0, 2] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(3, 1)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (3, 1)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0080","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3f63e025c4c041b7be23ceec3df88f5deca49d8fcd8df14a0f457b5f07e5fa6","family_id":"qf-density10-quantifier-nested-counts","seed":202610107080,"semantic_template":"qf-density10-quantifier-nested-counts/[[-2,-1,0,1,2],[-1,0,1],\"le\"]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-2, -1, 0, 1, 2], V=[-1, 0, 1] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(4, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (4, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0081","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaf844a987fe9511783f8b065235a8760eed4d11c983cc589b7b102d05cdb334","family_id":"qf-density10-quantifier-nested-counts","seed":202610107081,"semantic_template":"qf-density10-quantifier-nested-counts/[1,1,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-1], V=[100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0082","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beadb5cd9dcba21eceea1bf96125553f072490d5bf9d49cd300247786b02e508","family_id":"qf-density10-quantifier-nested-counts","seed":202610107082,"semantic_template":"qf-density10-quantifier-nested-counts/[1,1,1]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0], V=[0] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(1, 1)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 1)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0083","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42a6950bd25cc29abc9b8f24a4f9377e4e6b7125560e898e5cd117eb318eb628","family_id":"qf-density10-quantifier-nested-counts","seed":202610107083,"semantic_template":"qf-density10-quantifier-nested-counts/[1,2,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-1], V=[100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0084","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b4899b498a01761330fb8db8047a7ca047882398896f9083e95ce25738a6883","family_id":"qf-density10-quantifier-nested-counts","seed":202610107084,"semantic_template":"qf-density10-quantifier-nested-counts/[1,2,1]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0], V=[0, 100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(1, 1)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 1)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0085","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6d3c9e0f517388e1f342e0093aadb84d316e65ff72c5537d7b222e35520ce50","family_id":"qf-density10-quantifier-nested-counts","seed":202610107085,"semantic_template":"qf-density10-quantifier-nested-counts/[1,3,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-1], V=[100, 101, 102] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0086","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03d99ddde566f6727325cdade97045890cff99b5a1f7397544e4b7f8fb5d3931","family_id":"qf-density10-quantifier-nested-counts","seed":202610107086,"semantic_template":"qf-density10-quantifier-nested-counts/[1,3,1]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0], V=[0, 100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(1, 1)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 1)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0087","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cf8b3f696da7cdd00d3e88ae9c13a1a41d5bc11601b90937287d5af8a82a2e6","family_id":"qf-density10-quantifier-nested-counts","seed":202610107087,"semantic_template":"qf-density10-quantifier-nested-counts/[1,4,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-1], V=[100, 101, 102, 103] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0088","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48c26d1b8babd741356251d0caa81837e40620bee94965193004652fdf39e477","family_id":"qf-density10-quantifier-nested-counts","seed":202610107088,"semantic_template":"qf-density10-quantifier-nested-counts/[1,4,1]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0], V=[0, 100, 101, 102] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(1, 1)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 1)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0089","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8163e3865ed414b77cb9f3bab5e95055a3e8be823d852665112978602208ba08","family_id":"qf-density10-quantifier-nested-counts","seed":202610107089,"semantic_template":"qf-density10-quantifier-nested-counts/[2,1,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-2, -1], V=[100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0090","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6136924fddbd794907d4e140f225547a687bd3578a1d5e2d1a0221095bc65ca1","family_id":"qf-density10-quantifier-nested-counts","seed":202610107090,"semantic_template":"qf-density10-quantifier-nested-counts/[2,1,1]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0, -1], V=[0] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(1, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0091","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cc6b6cc2f13343e6466381ffc3a899d88e2d61bf59eb5fdd445876c0aaff5df","family_id":"qf-density10-quantifier-nested-counts","seed":202610107091,"semantic_template":"qf-density10-quantifier-nested-counts/[2,2,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-2, -1], V=[100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0092","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20816128e671694d19eb6c77bcec6db60947168ff886914f91fb2a92c8586bca","family_id":"qf-density10-quantifier-nested-counts","seed":202610107092,"semantic_template":"qf-density10-quantifier-nested-counts/[2,2,1]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0, -1], V=[0, 100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(1, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0093","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aed71cb7fe539906bb4ea5c6d1476d24b50f6e90e8b9c3778dba049b01811504","family_id":"qf-density10-quantifier-nested-counts","seed":202610107093,"semantic_template":"qf-density10-quantifier-nested-counts/[2,2,2]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0, 1], V=[0, 1] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(2, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (2, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0094","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcb29ad2e3fb1de5171e3366c2443bb8c664a77e3f4463a40f0cd0688dd7f108","family_id":"qf-density10-quantifier-nested-counts","seed":202610107094,"semantic_template":"qf-density10-quantifier-nested-counts/[2,3,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-2, -1], V=[100, 101, 102] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0095","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dc587e9c80bb184d6e8bf74ad25ed4c7a42187acbf9b765659474caf37c4d3b","family_id":"qf-density10-quantifier-nested-counts","seed":202610107095,"semantic_template":"qf-density10-quantifier-nested-counts/[2,3,1]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0, -1], V=[0, 100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(1, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0096","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"246d92a50ced5944cb5aeb2985f469061d3261f92a48ab22da47e6cf5ded5adb","family_id":"qf-density10-quantifier-nested-counts","seed":202610107096,"semantic_template":"qf-density10-quantifier-nested-counts/[2,3,2]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0, 1], V=[0, 1, 100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(2, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (2, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0097","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a25ce55b5652cbf87f48bb15366ed5c89a0ee483da363076b5a1f0615fbbf279","family_id":"qf-density10-quantifier-nested-counts","seed":202610107097,"semantic_template":"qf-density10-quantifier-nested-counts/[2,4,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-2, -1], V=[100, 101, 102, 103] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0098","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eefa06022468017f22f887645fbaf75e5f9b54cc3cf2cc122c7fa7b92302c074","family_id":"qf-density10-quantifier-nested-counts","seed":202610107098,"semantic_template":"qf-density10-quantifier-nested-counts/[2,4,1]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0, -1], V=[0, 100, 101, 102] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(1, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0099","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2342c50f5f173e2a825286f2704dad91c22a49bd085299139f98dca2a3fe629c","family_id":"qf-density10-quantifier-nested-counts","seed":202610107099,"semantic_template":"qf-density10-quantifier-nested-counts/[2,4,2]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[0, 1], V=[0, 1, 100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(2, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (2, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r07-quantifier-0100","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c191ab95a48b95469c69495433ce10135bbe8df95e50220a9ced6d13e11cb4dd","family_id":"qf-density10-quantifier-nested-counts","seed":202610107100,"semantic_template":"qf-density10-quantifier-nested-counts/[3,1,0]"},"fields":{"title":"Two Ordered Nested-Quantifier Counts","stem":"U=[-3, -2, -1], V=[100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).","answer":{"kind":"expression","value":"(0, 0)","explanation":"Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0)."},"skills":["Two Ordered Nested-Quantifier Counts"],"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."}}