{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r04-set-membership-0001","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1ffaaed51b7c91c12a431b4180c6b2da6e90320ea626bd59acca18d105ffdb1","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124001,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"untyped\",\"untyped\",\"untyped\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(0)=∅. Complete characteristic table bits=[] uses 0/1 membership from index 0; ordered queries=[-1, 0, 1]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([No, No, No], [ill-typed, ill-typed, ill-typed])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, No, No], [ill-typed, ill-typed, ill-typed])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0002","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb334b73db9c1751c3c579442349ad81388e21171f13abe5217c60cd29cc4c61","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124002,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"untyped\",\"untyped\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 101, 102]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, No], [True, ill-typed, ill-typed])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, No], [True, ill-typed, ill-typed])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0003","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f27e0c0c6b42d3381f4ccc6db008e531c1568152cf76c4193775dabc9924d57","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124003,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"nonmember\",\"untyped\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 1, 102]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, No], [False, False, ill-typed])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, False, ill-typed])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0004","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e8b7a2df8554c73f0ddb1c95c360f8c0be13951fa5a97148c974393a46499fd","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124004,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"nonmember\",\"untyped\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 1, 102]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, No], [True, False, ill-typed])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, False, ill-typed])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0005","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7929d8590699d33cb8b083d3351659129c22a6d803de8917e737d4fdcbe85dd0","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124005,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"untyped\",\"member\",\"untyped\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 4, 102]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, No], [ill-typed, True, ill-typed])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, No], [ill-typed, True, ill-typed])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0006","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec6cdb6b53d933c9ad394fbaa85a888660c47d7536fbcbe238deb7b339b2d004","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124006,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"member\",\"untyped\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 4, 102]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, No], [False, True, ill-typed])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, True, ill-typed])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0007","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40153c1cf6756ff6d2e705c74d4d84a4ecc8b38fa31ec14c43d5c7772d15c501","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124007,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"member\",\"untyped\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 4, 102]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, No], [True, True, ill-typed])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, True, ill-typed])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0008","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7252553f70572193468768a547683512bc9bfe97a61aad86ae7099f6588c294b","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124008,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"untyped\",\"untyped\",\"nonmember\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 101, 2]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([No, No, Yes], [ill-typed, ill-typed, False])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, No, Yes], [ill-typed, ill-typed, False])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0009","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b16cff401c76321d1c98267bcd5e7ec211078f2ba83b36fbaa71ad61d013b8ba","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124009,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"untyped\",\"nonmember\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 101, 2]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, Yes], [False, ill-typed, False])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, ill-typed, False])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0010","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"179f9f016d05fc658f8f8239151675053efb68347f466d144be2810c4fc35d71","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124010,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"untyped\",\"nonmember\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 101, 2]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, Yes], [True, ill-typed, False])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, ill-typed, False])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0011","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b9fdb1a7f3574c6add4b6ed04947e04983893ac6740861a5b145e840d400a32","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124011,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"untyped\",\"nonmember\",\"nonmember\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 1, 2]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, Yes], [ill-typed, False, False])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, Yes], [ill-typed, False, False])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0012","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dfbd5a715cae57adf1f11860a8201178b3ef70684292dee961dd926cf9eb51f","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124012,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"nonmember\",\"nonmember\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 1, 2]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [False, False, False])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, False])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0013","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7c7061ef479f4647552fcb04f342cacf17186f2c0e7f2a6e1ee93d12874108a","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124013,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"nonmember\",\"nonmember\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 1, 2]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [True, False, False])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, False, False])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0014","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"925444c85c26ff6aeb2a0f2245f4624c5ab4e0bda60d5ad2320da95b24137130","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124014,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"untyped\",\"member\",\"nonmember\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 4, 2]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, Yes], [ill-typed, True, False])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, Yes], [ill-typed, True, False])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0015","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14e5af3e221ea2426564218234833507db5dd74bca5ba2523556340f8e1627e2","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124015,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"member\",\"nonmember\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 4, 2]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [False, True, False])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, True, False])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0016","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6b90a9bdfea0edb9b56143ad6478da0376ff56a69f7abc9696c53afbfa4c87e","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124016,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"member\",\"nonmember\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 4, 2]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [True, True, False])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, False])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0017","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bbb45b23d8262a9e67e90f3aa4913568547b42e2a7a2c08cacf0ea0002a7f3c","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124017,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"untyped\",\"untyped\",\"member\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 101, 5]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([No, No, Yes], [ill-typed, ill-typed, True])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, No, Yes], [ill-typed, ill-typed, True])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0018","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64d0f164259ba146fdc14aafad42e6dc6f0fc0fedbcb0dda4296037fd8c6be2b","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124018,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"untyped\",\"member\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 101, 5]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, Yes], [False, ill-typed, True])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, ill-typed, True])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0019","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37fc7b7cde181096bfbe7bb1a856997a8411d06495a4b6da7c75f3d0dbe048f7","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124019,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"untyped\",\"member\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 101, 5]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, Yes], [True, ill-typed, True])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, ill-typed, True])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0020","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"681cf8ac3500832cb21b510e14060fd4b40ed99146d4be7d843eb8366fbd8f0a","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124020,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"untyped\",\"nonmember\",\"member\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 1, 5]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, Yes], [ill-typed, False, True])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, Yes], [ill-typed, False, True])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0021","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a685bc9818ed7dc70050dfb5a9dce8b6cdf7ea3adf60e6cb83259afb0c2441f0","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124021,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"nonmember\",\"member\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 1, 5]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [False, False, True])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, True])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0022","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5fff8a65c99cba100b3f5d93c40649333b0e483eb1040cd2144ba6dd44033f3","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124022,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"nonmember\",\"member\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 1, 5]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [True, False, True])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, False, True])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0023","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc064ee767f928f0e29a0d83cfd134a1e4f99f1f8bb114521867f7c049c9cecc","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124023,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"untyped\",\"member\",\"member\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 4, 5]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, Yes], [ill-typed, True, True])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, Yes], [ill-typed, True, True])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0024","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"284efe52a3a9e77974422ca9e2a09b33ec2186f34b032871bea8a17cf521f3b9","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124024,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"member\",\"member\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 4, 5]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [False, True, True])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, True, True])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0025","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"720c436ef8dd4d6a5d9fcaea74ac9933ea72661c47aed914971bca73800161c3","family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124025,"semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"member\",\"member\"]"},"fields":{"title":"Typed Membership Queries and Truth Values","stem":"The finite type is Fin(1)={0,…,0}. Complete characteristic table bits=[1] uses 0/1 membership from index 0; ordered queries=[0, 0, 0]. Give (type-eligibility mask, membership truth values or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [True, True, True])","explanation":"Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, True])."},"skills":["Typed Membership Queries and Truth Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0026","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c05b4722d1b86dbdd0ac73fc48526af81ca92b5cd9d2aed4d16313767d80b6a2","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124026,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[], x=0. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0027","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21cff01006305124903e2abfd55cee8051937e4f41cfb15c45eb30304df36bbc","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124027,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[0], x=0. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0028","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1770b4d8a40bfa78b485770de49327ebe9468530ddf5eeea89b55a460758e49","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124028,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-2,0,2]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-2, 0], x=2. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-2, 0, 2])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-2, 0, 2])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0029","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eac44995ca57a384480f87bcae7bc6935214c4cd29698ee2e590d8d468ca8d9e","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124029,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-3,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-3, 0], x=-3. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-3, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-3, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0030","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b0fc989305e856f544bcf823db7fc6b14dbb8fffd8d370930a8d89ff6f63489","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124030,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-4,0,4]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-4, 0], x=4. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-4, 0, 4])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-4, 0, 4])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0031","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb0c93af2ca182b544df1bfa3640192bc3ed67f2bce79d8357283e29b57da16b","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124031,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-5,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-5, 0], x=-5. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-5, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-5, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0032","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c3591a4edb9c2bd910a2d0fb25c58a4eac23d643405f9563b0f9f5e8b939061","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124032,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-6,0,6]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-6, 0], x=6. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-6, 0, 6])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-6, 0, 6])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0033","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08f0e7d62823a5edd7248b463d248467f5a3e34ee7c09d48966856e7ffbf7111","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124033,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-7,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-7, 0], x=-7. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-7, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-7, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0034","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b45d1b19b4f9c7e6f31cbf5899a9aebc6dbcda574587af74e53253881baf33c","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124034,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-8,0,8]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-8, 0], x=8. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-8, 0, 8])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-8, 0, 8])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0035","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e06858a5580e92079f83871c66f127ac97d9fe4611a06c7baa3d2cd34cbdb9a","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124035,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-9,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-9, 0], x=-9. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-9, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-9, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0036","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eddcc887992bcd0a4758eb7f568da696646c6de02900cafad89a3355fab73558","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124036,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-10,0,10]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-10, 0], x=10. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-10, 0, 10])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-10, 0, 10])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0037","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30cd6242ce337425d481e8a84c71c9b986a9cb3627b7ad8d04b7fe0d01477a41","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124037,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-11,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-11, 0], x=-11. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-11, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-11, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0038","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9156ed3e98911f9498b4b12f9c247843bab8aa1e4c837de08032569375a438c1","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124038,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-12,0,12]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-12, 0], x=12. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-12, 0, 12])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-12, 0, 12])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0039","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7576d14c1c2b8971b7439dacaa813ffbe685d35c0fdc56df037b9128ad4b1f5f","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124039,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-13,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-13, 0], x=-13. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-13, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-13, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0040","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cbc8e5d3825c3da78a5db13917ae98f8a49a404976411b8235f4dc065f6b886","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124040,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-14,0,14]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-14, 0], x=14. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-14, 0, 14])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-14, 0, 14])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0041","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84cb37b8e015f52f3896e0f5d683c8fc7dae2554836120eba0d3007c245d3b6b","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124041,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-15,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-15, 0], x=-15. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-15, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-15, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0042","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"333b035e3551b701c23da37390f95209df8ae0b0ca8ac0654b82c1630863e414","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124042,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-16,0,16]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-16, 0], x=16. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-16, 0, 16])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-16, 0, 16])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0043","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e175fea53baa18eb025ad72efc03d6e5155bd2b7b28d708090020d28b5ab4a23","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124043,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-17,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-17, 0], x=-17. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-17, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-17, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0044","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd5354e02fd47c21dbe559d5183c80058b69539b549f6c6d16f595ce3b61e1c2","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124044,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-18,0,18]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-18, 0], x=18. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-18, 0, 18])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-18, 0, 18])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0045","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fad3dbc212b66d1753291aab4b3e0e2dd8c6f94717c93f2ce4d7020a8cede7d","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124045,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-19,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-19, 0], x=-19. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-19, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-19, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0046","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"130e971bbd1d95f971e85964e2a984df8a8cd240b2072d61c8adb104dd5ed1fb","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124046,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-20,0,20]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-20, 0], x=20. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-20, 0, 20])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-20, 0, 20])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0047","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aef60b6504b6fd54ee32a1e155715c6d318ade0f3f40273bab9d6ca917908b6","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124047,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-21,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-21, 0], x=-21. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-21, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-21, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0048","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b85b9687d1405501d08c69c107192a1a4d5585ce8fec332c4521e23cc64cf520","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124048,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-22,0,22]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-22, 0], x=22. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-22, 0, 22])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-22, 0, 22])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0049","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87d9e904783fc8a3047cef69a13451d32d343762ad73d12931f13cafcf5893e1","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124049,"semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-23,0]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-23, 0], x=-23. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(True, [-23, 0])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-23, 0])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0050","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54b0dfd38d205de0f218fcb25dd25f37993a9c11dd814258f0a40a7a76cc2587","family_id":"qf-density12-membership-set-insertion-delta","seed":202610124050,"semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-24,0,24]]"},"fields":{"title":"Original Membership and Set Insertion","stem":"Finite integer set A=[-24, 0], x=24. Give (original truth of x∈A, sorted deduplicated A∪{x}).","answer":{"kind":"expression","value":"(False, [-24, 0, 24])","explanation":"Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-24, 0, 24])."},"skills":["Original Membership and Set Insertion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0051","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0eb5eadd1ad682088d2e30db057ac2ce12fa78e7f05231e6f14a610465125e4","family_id":"qf-density12-membership-deletion-residual","seed":202610124051,"semantic_template":"qf-density12-membership-deletion-residual/[true,[]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[0], x=0. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0052","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e182c4b916c2fbb561192957d4598795ad9a241db6c312222864265e1b1a01eb","family_id":"qf-density12-membership-deletion-residual","seed":202610124052,"semantic_template":"qf-density12-membership-deletion-residual/[false,[]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[], x=99. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0053","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2efb91a01848bb2c9eccc5b1b1e0eb36848dcb8a7a9ec9963612a51b713bd5a","family_id":"qf-density12-membership-deletion-residual","seed":202610124053,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-2,0,2]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-2, 0, 2], x=3. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-2, 0, 2])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-2, 0, 2])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0054","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b60b586e7a25a74f3e00c0191599dae7ffb63c564c8d344c4dac0a54d667e7e","family_id":"qf-density12-membership-deletion-residual","seed":202610124054,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,3]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-3, 0, 3], x=-3. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 3])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 3])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0055","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f10302751ee9d591fb6dbcab180043f2a3881b0ca56ede0701318891ac95c3d","family_id":"qf-density12-membership-deletion-residual","seed":202610124055,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-4,0,4]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-4, 0, 4], x=5. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-4, 0, 4])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-4, 0, 4])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0056","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cb27ea035dd7b03edfd50846beccf92a392be624caab42edec5e9d3d422675b","family_id":"qf-density12-membership-deletion-residual","seed":202610124056,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,5]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-5, 0, 5], x=-5. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 5])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 5])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0057","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f876518250a36daa9d24a662bea4487302c444bfe2dbc0c9c254c906a074ee6","family_id":"qf-density12-membership-deletion-residual","seed":202610124057,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-6,0,6]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-6, 0, 6], x=7. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-6, 0, 6])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-6, 0, 6])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0058","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f5c0ecf56b2d99796493fd1d117fbf1d00fbf47e0661e26ec49d6dd5d46482c","family_id":"qf-density12-membership-deletion-residual","seed":202610124058,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,7]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-7, 0, 7], x=-7. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 7])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 7])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0059","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"428218f31fba008c9406aa94b673a969fd73901f9477af8896f9428891b35799","family_id":"qf-density12-membership-deletion-residual","seed":202610124059,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-8,0,8]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-8, 0, 8], x=9. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-8, 0, 8])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-8, 0, 8])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0060","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b835c9573c47528bc1fb2ec6d30c739d0a4b424bcea93d30f866775fedaf3cfd","family_id":"qf-density12-membership-deletion-residual","seed":202610124060,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,9]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-9, 0, 9], x=-9. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 9])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 9])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0061","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93e863be76c49821cbeb9e4d17b393529ae1f0037e2368037f027472c6a6729e","family_id":"qf-density12-membership-deletion-residual","seed":202610124061,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-10,0,10]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-10, 0, 10], x=11. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-10, 0, 10])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-10, 0, 10])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0062","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f5e68388e90e743562c7e176d1f8721cf861118ad4ca906bf2b1acbdb478192","family_id":"qf-density12-membership-deletion-residual","seed":202610124062,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,11]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-11, 0, 11], x=-11. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 11])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 11])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0063","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16cb5f2e6ab77e37695f53e06cb7b03d4038b1e03f0060ce698f3e842647239f","family_id":"qf-density12-membership-deletion-residual","seed":202610124063,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-12,0,12]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-12, 0, 12], x=13. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-12, 0, 12])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-12, 0, 12])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0064","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"782397c808c6d89b8881f62d6dec518f2b6e9ca5917d00a520b9fc485b1fa9bc","family_id":"qf-density12-membership-deletion-residual","seed":202610124064,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,13]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-13, 0, 13], x=-13. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 13])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 13])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0065","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8645a68606ed671aa0ab17ba562fef29ec232ad0690259156895923a0fca53d1","family_id":"qf-density12-membership-deletion-residual","seed":202610124065,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-14,0,14]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-14, 0, 14], x=15. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-14, 0, 14])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-14, 0, 14])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0066","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a402297f6b020debb0212fe7b6747922ae69f0d5a55d80ba50fa21465b8a0586","family_id":"qf-density12-membership-deletion-residual","seed":202610124066,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,15]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-15, 0, 15], x=-15. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 15])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 15])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0067","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2abd041317f6b95cc101f10f7adbbe43a6659c72bcbf653f75377aab88510ea","family_id":"qf-density12-membership-deletion-residual","seed":202610124067,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-16,0,16]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-16, 0, 16], x=17. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-16, 0, 16])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-16, 0, 16])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0068","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50b1c86f2edc3f4bf5baf2b5ff84d60d5341357eb320c2e9e1846d15ebc3d9cc","family_id":"qf-density12-membership-deletion-residual","seed":202610124068,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,17]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-17, 0, 17], x=-17. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 17])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 17])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0069","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbb20ea05e9e133a2576c3576d880d0005d54ba5c00d88cfc2f6eb69c4128cd2","family_id":"qf-density12-membership-deletion-residual","seed":202610124069,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-18,0,18]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-18, 0, 18], x=19. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-18, 0, 18])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-18, 0, 18])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0070","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c45b0a503cf1022c99e63817707c593034228da3d0058eb171f5187e216f75d","family_id":"qf-density12-membership-deletion-residual","seed":202610124070,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,19]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-19, 0, 19], x=-19. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 19])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 19])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0071","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f01674b07a043cb3b1ee559692ecb06a27188e4236e48fa1aad50398f874e904","family_id":"qf-density12-membership-deletion-residual","seed":202610124071,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-20,0,20]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-20, 0, 20], x=21. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-20, 0, 20])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-20, 0, 20])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0072","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3a167766d67d7b5f673c65f7a49bcb1b8b144aff34c44fb3fe811e2e841c458","family_id":"qf-density12-membership-deletion-residual","seed":202610124072,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,21]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-21, 0, 21], x=-21. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 21])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 21])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0073","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"750f94e4251645c2c269a19621aeb07145c0fb349f4a31cdb20ca69063b46b11","family_id":"qf-density12-membership-deletion-residual","seed":202610124073,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-22,0,22]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-22, 0, 22], x=23. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-22, 0, 22])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-22, 0, 22])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0074","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a5361d1c74c662f05a6c7dd12bf6011e08721427733991f5bdbba466c61b90c","family_id":"qf-density12-membership-deletion-residual","seed":202610124074,"semantic_template":"qf-density12-membership-deletion-residual/[true,[0,23]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-23, 0, 23], x=-23. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(True, [0, 23])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 23])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0075","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62cf86921d9aa1c867336426ac74c365470256b06802fca43cab7ec922e3da4e","family_id":"qf-density12-membership-deletion-residual","seed":202610124075,"semantic_template":"qf-density12-membership-deletion-residual/[false,[-24,0,24]]"},"fields":{"title":"Original Membership and Set Deletion","stem":"Finite integer set A=[-24, 0, 24], x=25. Give (original truth of x∈A, increasing A\\{x}), retaining [].","answer":{"kind":"expression","value":"(False, [-24, 0, 24])","explanation":"Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-24, 0, 24])."},"skills":["Original Membership and Set Deletion"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0076","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42b8e12de1db9e57065a4f5a46ad394fb4ba80f8e8b7dd69c8bd3deb78af3338","family_id":"qf-density12-membership-predicate-extension","seed":202610124076,"semantic_template":"qf-density12-membership-predicate-extension/[[],[]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([], [])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([], [])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0077","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a3b7f00f2fc106aa8f1c1785f635b92dac4179478824461bf6af077a0224d6d","family_id":"qf-density12-membership-predicate-extension","seed":202610124077,"semantic_template":"qf-density12-membership-predicate-extension/[[-2,0,2],[true,true,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-2, 0, 2]; P(x) means 2 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([-2, 0, 2], [True, True, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-2, 0, 2], [True, True, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0078","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5361e5948ae6d921fc944e544197bc53b8fc91c3194e5bc37c135608f2895b58","family_id":"qf-density12-membership-predicate-extension","seed":202610124078,"semantic_template":"qf-density12-membership-predicate-extension/[[0],[false,true,false]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-1, 0, 1]; P(x) means x=0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([0], [False, True, False])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0], [False, True, False])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0079","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e202eed15310076f57adfd680cf845e1c8c6e4ea4bac987b637991da9f8ece0","family_id":"qf-density12-membership-predicate-extension","seed":202610124079,"semantic_template":"qf-density12-membership-predicate-extension/[[0,1,3],[false,false,true,true,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-3, -1, 0, 1, 3]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([0, 1, 3], [False, False, True, True, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 3], [False, False, True, True, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0080","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f7b6cefa8ce142159f5f6229f0007036911bd48b8b7ee422630028c7576139f","family_id":"qf-density12-membership-predicate-extension","seed":202610124080,"semantic_template":"qf-density12-membership-predicate-extension/[[4],[false,false,false,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-4, -1, 0, 1, 4]; P(x) means x=4. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([4], [False, False, False, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([4], [False, False, False, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0081","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e275a36c053797b241b2178f7e5668a66ba28754e364bbc01263cf942f358381","family_id":"qf-density12-membership-predicate-extension","seed":202610124081,"semantic_template":"qf-density12-membership-predicate-extension/[[-5,0,5],[true,false,true,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-5, -1, 0, 1, 5]; P(x) means 5 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([-5, 0, 5], [True, False, True, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-5, 0, 5], [True, False, True, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0082","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b831c7eed8a3178f2648885c98b6118ebda8f681b32137d9edfb126484740d6","family_id":"qf-density12-membership-predicate-extension","seed":202610124082,"semantic_template":"qf-density12-membership-predicate-extension/[[0,1,6],[false,false,true,true,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-6, -1, 0, 1, 6]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([0, 1, 6], [False, False, True, True, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 6], [False, False, True, True, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0083","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f41bf6615cf07c89e6edc71993db1a358f39e499ddd560c986852e420259f9e9","family_id":"qf-density12-membership-predicate-extension","seed":202610124083,"semantic_template":"qf-density12-membership-predicate-extension/[[7],[false,false,false,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-7, -1, 0, 1, 7]; P(x) means x=7. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([7], [False, False, False, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([7], [False, False, False, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0084","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"755d5454872016b02f717dfd8cdbf1ab64463b48c23489052eb85ebabccfe6bb","family_id":"qf-density12-membership-predicate-extension","seed":202610124084,"semantic_template":"qf-density12-membership-predicate-extension/[[-8,0,8],[true,false,true,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-8, -1, 0, 1, 8]; P(x) means 8 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([-8, 0, 8], [True, False, True, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-8, 0, 8], [True, False, True, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0085","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73b3b70e4dba630e53235fb9b6dc28e7043e78f232d957d5230bc6812fa7915e","family_id":"qf-density12-membership-predicate-extension","seed":202610124085,"semantic_template":"qf-density12-membership-predicate-extension/[[0,1,9],[false,false,true,true,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-9, -1, 0, 1, 9]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([0, 1, 9], [False, False, True, True, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 9], [False, False, True, True, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0086","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54ca1a92dbc4173e19556611d93fa450e195964650f313b64b85cedb09cff24a","family_id":"qf-density12-membership-predicate-extension","seed":202610124086,"semantic_template":"qf-density12-membership-predicate-extension/[[10],[false,false,false,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-10, -1, 0, 1, 10]; P(x) means x=10. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([10], [False, False, False, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([10], [False, False, False, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0087","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b4a1fb337d86c6d942903989d740bd1d6019a7511184365f7d474a920eff15e","family_id":"qf-density12-membership-predicate-extension","seed":202610124087,"semantic_template":"qf-density12-membership-predicate-extension/[[-11,0,11],[true,false,true,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-11, -1, 0, 1, 11]; P(x) means 11 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([-11, 0, 11], [True, False, True, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-11, 0, 11], [True, False, True, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0088","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d3c0d275c1ece98b593c4518521f3deefc68af94993950c11919c3d377f9291","family_id":"qf-density12-membership-predicate-extension","seed":202610124088,"semantic_template":"qf-density12-membership-predicate-extension/[[0,1,12],[false,false,true,true,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-12, -1, 0, 1, 12]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([0, 1, 12], [False, False, True, True, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 12], [False, False, True, True, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0089","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e97c9652cced6d4226a208f12b4ec2ca3b63284c236abc9273527572c7bd7ae2","family_id":"qf-density12-membership-predicate-extension","seed":202610124089,"semantic_template":"qf-density12-membership-predicate-extension/[[13],[false,false,false,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-13, -1, 0, 1, 13]; P(x) means x=13. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([13], [False, False, False, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([13], [False, False, False, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0090","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3b2ef8ee34bcd1dc0804da8ab9605a839e9878e31c716c0699f645db5234f51","family_id":"qf-density12-membership-predicate-extension","seed":202610124090,"semantic_template":"qf-density12-membership-predicate-extension/[[-14,0,14],[true,false,true,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-14, -1, 0, 1, 14]; P(x) means 14 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([-14, 0, 14], [True, False, True, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-14, 0, 14], [True, False, True, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0091","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07ae317c19dc2572c80ba1439b872c491b9453198469d345ec05e3c17b9d9034","family_id":"qf-density12-membership-predicate-extension","seed":202610124091,"semantic_template":"qf-density12-membership-predicate-extension/[[0,1,15],[false,false,true,true,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-15, -1, 0, 1, 15]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([0, 1, 15], [False, False, True, True, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 15], [False, False, True, True, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0092","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed56c555d4ab161bead87026e0d26d158cd95a15ac9693c14443c4a6280fe18a","family_id":"qf-density12-membership-predicate-extension","seed":202610124092,"semantic_template":"qf-density12-membership-predicate-extension/[[16],[false,false,false,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-16, -1, 0, 1, 16]; P(x) means x=16. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([16], [False, False, False, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([16], [False, False, False, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0093","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f4b1e81a5e3bdd251ccde9ab30cea73d0219a73e0e721d1105503921ad8beaa","family_id":"qf-density12-membership-predicate-extension","seed":202610124093,"semantic_template":"qf-density12-membership-predicate-extension/[[-17,0,17],[true,false,true,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-17, -1, 0, 1, 17]; P(x) means 17 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([-17, 0, 17], [True, False, True, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-17, 0, 17], [True, False, True, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0094","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6513b0a3cb2d806c74dfed03f11d5ccf1b1ebf4975c147f7fbdc05263bae8abf","family_id":"qf-density12-membership-predicate-extension","seed":202610124094,"semantic_template":"qf-density12-membership-predicate-extension/[[0,1,18],[false,false,true,true,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-18, -1, 0, 1, 18]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([0, 1, 18], [False, False, True, True, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 18], [False, False, True, True, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0095","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c59e62a2373c96605443e8e01839cb1352b238fb102c66e141773df3e4bfa70","family_id":"qf-density12-membership-predicate-extension","seed":202610124095,"semantic_template":"qf-density12-membership-predicate-extension/[[19],[false,false,false,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-19, -1, 0, 1, 19]; P(x) means x=19. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([19], [False, False, False, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([19], [False, False, False, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0096","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"851e644a67bd49409294934c4ead9bf5102932aa6d4309710cdc603280d08ede","family_id":"qf-density12-membership-predicate-extension","seed":202610124096,"semantic_template":"qf-density12-membership-predicate-extension/[[-20,0,20],[true,false,true,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-20, -1, 0, 1, 20]; P(x) means 20 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([-20, 0, 20], [True, False, True, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-20, 0, 20], [True, False, True, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0097","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3173f7c1d32f768f7420e6bf3fc32608909db7ee27c57e582222856d145b3462","family_id":"qf-density12-membership-predicate-extension","seed":202610124097,"semantic_template":"qf-density12-membership-predicate-extension/[[0,1,21],[false,false,true,true,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-21, -1, 0, 1, 21]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([0, 1, 21], [False, False, True, True, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 21], [False, False, True, True, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0098","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72cb63877fe3d79e090ca0f2f48622b0d62e0b7b6f673afd53f06c7956171395","family_id":"qf-density12-membership-predicate-extension","seed":202610124098,"semantic_template":"qf-density12-membership-predicate-extension/[[22],[false,false,false,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-22, -1, 0, 1, 22]; P(x) means x=22. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([22], [False, False, False, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([22], [False, False, False, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0099","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7bf478e84e915ae171efabe1bd179dc61d3732986cf486a8fd6b9c9d3303a81","family_id":"qf-density12-membership-predicate-extension","seed":202610124099,"semantic_template":"qf-density12-membership-predicate-extension/[[-23,0,23],[true,false,true,false,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-23, -1, 0, 1, 23]; P(x) means 23 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([-23, 0, 23], [True, False, True, False, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-23, 0, 23], [True, False, True, False, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261012-r04-set-membership-0100","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f5ef65d87e9e25eabf7a88aebe217902cf16d500974735e8707e5e5e36cb029","family_id":"qf-density12-membership-predicate-extension","seed":202610124100,"semantic_template":"qf-density12-membership-predicate-extension/[[0,1,24],[false,false,true,true,true]]"},"fields":{"title":"Finite Predicate Extension and Aligned Truths","stem":"Finite integer set U=[-24, -1, 0, 1, 24]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).","answer":{"kind":"expression","value":"([0, 1, 24], [False, False, True, True, True])","explanation":"Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 24], [False, False, True, True, True])."},"skills":["Finite Predicate Extension and Aligned Truths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}