{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0001","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=0，Fin(0)=∅，facts=[]，rules=[]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[]","explanation_zh":"有限MP最短深度=[]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134001,"parameters":{"n":0,"facts":[],"rules":[],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0002","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=2，facts=[]，rules=[[0, 1], [1, 0]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 未得]","explanation_zh":"有限MP最短深度=[未得, 未得]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134002,"parameters":{"n":2,"facts":[],"rules":[[0,1],[1,0]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,null]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0003","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=1，facts=[0]，rules=[]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0]","explanation_zh":"有限MP最短深度=[0]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134003,"parameters":{"n":1,"facts":[0],"rules":[],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0004","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 1, 2, 3, 4, 5, 6, 7]","explanation_zh":"有限MP最短深度=[0, 1, 2, 3, 4, 5, 6, 7]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134004,"parameters":{"n":8,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,1,2,3,4,5,6,7]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0005","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[1]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 0, 1, 2, 3, 4, 5, 6]","explanation_zh":"有限MP最短深度=[未得, 0, 1, 2, 3, 4, 5, 6]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134005,"parameters":{"n":8,"facts":[1],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,0,1,2,3,4,5,6]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0006","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 1]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 0, 1, 2, 3, 4, 5, 6]","explanation_zh":"有限MP最短深度=[0, 0, 1, 2, 3, 4, 5, 6]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134006,"parameters":{"n":8,"facts":[0,1],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,0,1,2,3,4,5,6]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0007","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[2]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 未得, 0, 1, 2, 3, 4, 5]","explanation_zh":"有限MP最短深度=[未得, 未得, 0, 1, 2, 3, 4, 5]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134007,"parameters":{"n":8,"facts":[2],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,null,0,1,2,3,4,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0008","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 2]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 1, 0, 1, 2, 3, 4, 5]","explanation_zh":"有限MP最短深度=[0, 1, 0, 1, 2, 3, 4, 5]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134008,"parameters":{"n":8,"facts":[0,2],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,1,0,1,2,3,4,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0009","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[1, 2]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 0, 0, 1, 2, 3, 4, 5]","explanation_zh":"有限MP最短深度=[未得, 0, 0, 1, 2, 3, 4, 5]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134009,"parameters":{"n":8,"facts":[1,2],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,0,0,1,2,3,4,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0010","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 1, 2]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 0, 0, 1, 2, 3, 4, 5]","explanation_zh":"有限MP最短深度=[0, 0, 0, 1, 2, 3, 4, 5]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134010,"parameters":{"n":8,"facts":[0,1,2],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,0,0,1,2,3,4,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0011","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[3]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 未得, 未得, 0, 1, 2, 3, 4]","explanation_zh":"有限MP最短深度=[未得, 未得, 未得, 0, 1, 2, 3, 4]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134011,"parameters":{"n":8,"facts":[3],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,null,null,0,1,2,3,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0012","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 3]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 1, 2, 0, 1, 2, 3, 4]","explanation_zh":"有限MP最短深度=[0, 1, 2, 0, 1, 2, 3, 4]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134012,"parameters":{"n":8,"facts":[0,3],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,1,2,0,1,2,3,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0013","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[1, 3]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 0, 1, 0, 1, 2, 3, 4]","explanation_zh":"有限MP最短深度=[未得, 0, 1, 0, 1, 2, 3, 4]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134013,"parameters":{"n":8,"facts":[1,3],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,0,1,0,1,2,3,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0014","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 1, 3]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 0, 1, 0, 1, 2, 3, 4]","explanation_zh":"有限MP最短深度=[0, 0, 1, 0, 1, 2, 3, 4]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134014,"parameters":{"n":8,"facts":[0,1,3],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,0,1,0,1,2,3,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0015","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[2, 3]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 未得, 0, 0, 1, 2, 3, 4]","explanation_zh":"有限MP最短深度=[未得, 未得, 0, 0, 1, 2, 3, 4]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134015,"parameters":{"n":8,"facts":[2,3],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,null,0,0,1,2,3,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0016","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 2, 3]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 1, 0, 0, 1, 2, 3, 4]","explanation_zh":"有限MP最短深度=[0, 1, 0, 0, 1, 2, 3, 4]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134016,"parameters":{"n":8,"facts":[0,2,3],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,1,0,0,1,2,3,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0017","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[1, 2, 3]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 0, 0, 0, 1, 2, 3, 4]","explanation_zh":"有限MP最短深度=[未得, 0, 0, 0, 1, 2, 3, 4]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134017,"parameters":{"n":8,"facts":[1,2,3],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,0,0,0,1,2,3,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0018","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 1, 2, 3]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 0, 0, 0, 1, 2, 3, 4]","explanation_zh":"有限MP最短深度=[0, 0, 0, 0, 1, 2, 3, 4]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134018,"parameters":{"n":8,"facts":[0,1,2,3],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,0,0,0,1,2,3,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0019","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[4]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 未得, 未得, 未得, 0, 1, 2, 3]","explanation_zh":"有限MP最短深度=[未得, 未得, 未得, 未得, 0, 1, 2, 3]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134019,"parameters":{"n":8,"facts":[4],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,null,null,null,0,1,2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0020","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 4]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 1, 2, 3, 0, 1, 2, 3]","explanation_zh":"有限MP最短深度=[0, 1, 2, 3, 0, 1, 2, 3]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134020,"parameters":{"n":8,"facts":[0,4],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,1,2,3,0,1,2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0021","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[1, 4]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 0, 1, 2, 0, 1, 2, 3]","explanation_zh":"有限MP最短深度=[未得, 0, 1, 2, 0, 1, 2, 3]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134021,"parameters":{"n":8,"facts":[1,4],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,0,1,2,0,1,2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0022","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 1, 4]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 0, 1, 2, 0, 1, 2, 3]","explanation_zh":"有限MP最短深度=[0, 0, 1, 2, 0, 1, 2, 3]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134022,"parameters":{"n":8,"facts":[0,1,4],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,0,1,2,0,1,2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0023","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[2, 4]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 未得, 0, 1, 0, 1, 2, 3]","explanation_zh":"有限MP最短深度=[未得, 未得, 0, 1, 0, 1, 2, 3]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134023,"parameters":{"n":8,"facts":[2,4],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,null,0,1,0,1,2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0024","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0, 2, 4]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[0, 1, 0, 1, 0, 1, 2, 3]","explanation_zh":"有限MP最短深度=[0, 1, 0, 1, 0, 1, 2, 3]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134024,"parameters":{"n":8,"facts":[0,2,4],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[0,1,0,1,0,1,2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0025","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"有限MP的最短推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[1, 2, 4]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]]（i→j為單步規則）。事實深度0，每條MP加1。按全部索引列最短深度，不能推得則寫未得。","answer":{"kind":"expression","value":"[未得, 0, 0, 1, 0, 1, 2, 3]","explanation_zh":"有限MP最短深度=[未得, 0, 0, 1, 0, 1, 2, 3]。由事實做多源BFS取得最少邊數；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限MP的最短推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-min-depths","seed":202610134025,"parameters":{"n":8,"facts":[1,2,4],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]],"operation":"density13_proof_horn_min_depths","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-min-depths/[null,0,0,1,0,1,2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0026","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=0，Fin(0)=∅，facts=[]，rules=[]（i→j為單步規則）。queries=[-1, 0, 1]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([否, 否, 否], [否, 否, 否], [不合型別, 不合型別, 不合型別])","explanation_zh":"目標型別與推導深度=([否, 否, 否], [否, 否, 否], [不合型別, 不合型別, 不合型別])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134026,"parameters":{"n":0,"facts":[],"rules":[],"queries":[-1,0,1],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"untyped\"],[\"untyped\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0027","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[0, 101, 102]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 否, 否], [是, 否, 否], [0, 不合型別, 不合型別])","explanation_zh":"目標型別與推導深度=([是, 否, 否], [是, 否, 否], [0, 不合型別, 不合型別])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134027,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[0,101,102],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"untyped\"],[\"untyped\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0028","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[3, 3, 102]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 否], [否, 否, 否], [未得, 未得, 不合型別])","explanation_zh":"目標型別與推導深度=([是, 是, 否], [否, 否, 否], [未得, 未得, 不合型別])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134028,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[3,3,102],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"unreachable\"],[\"untyped\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0029","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[0, 3, 102]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 否], [是, 否, 否], [0, 未得, 不合型別])","explanation_zh":"目標型別與推導深度=([是, 是, 否], [是, 否, 否], [0, 未得, 不合型別])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134029,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[0,3,102],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"unreachable\"],[\"untyped\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0030","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[100, 1, 102]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([否, 是, 否], [否, 是, 否], [不合型別, 1, 不合型別])","explanation_zh":"目標型別與推導深度=([否, 是, 否], [否, 是, 否], [不合型別, 1, 不合型別])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134030,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[100,1,102],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"depth\",1],[\"untyped\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0031","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[3, 1, 102]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 否], [否, 是, 否], [未得, 1, 不合型別])","explanation_zh":"目標型別與推導深度=([是, 是, 否], [否, 是, 否], [未得, 1, 不合型別])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134031,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[3,1,102],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"depth\",1],[\"untyped\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0032","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[0, 1, 102]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 否], [是, 是, 否], [0, 1, 不合型別])","explanation_zh":"目標型別與推導深度=([是, 是, 否], [是, 是, 否], [0, 1, 不合型別])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134032,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[0,1,102],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"depth\",1],[\"untyped\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0033","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[100, 101, 3]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([否, 否, 是], [否, 否, 否], [不合型別, 不合型別, 未得])","explanation_zh":"目標型別與推導深度=([否, 否, 是], [否, 否, 否], [不合型別, 不合型別, 未得])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134033,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[100,101,3],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"untyped\"],[\"unreachable\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0034","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[3, 101, 3]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 否, 是], [否, 否, 否], [未得, 不合型別, 未得])","explanation_zh":"目標型別與推導深度=([是, 否, 是], [否, 否, 否], [未得, 不合型別, 未得])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134034,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[3,101,3],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"untyped\"],[\"unreachable\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0035","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[0, 101, 3]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 否, 是], [是, 否, 否], [0, 不合型別, 未得])","explanation_zh":"目標型別與推導深度=([是, 否, 是], [是, 否, 否], [0, 不合型別, 未得])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134035,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[0,101,3],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"untyped\"],[\"unreachable\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0036","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[100, 3, 3]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([否, 是, 是], [否, 否, 否], [不合型別, 未得, 未得])","explanation_zh":"目標型別與推導深度=([否, 是, 是], [否, 否, 否], [不合型別, 未得, 未得])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134036,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[100,3,3],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"unreachable\"],[\"unreachable\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0037","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[3, 3, 3]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 是], [否, 否, 否], [未得, 未得, 未得])","explanation_zh":"目標型別與推導深度=([是, 是, 是], [否, 否, 否], [未得, 未得, 未得])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134037,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[3,3,3],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"unreachable\"],[\"unreachable\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0038","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[0, 3, 3]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 否, 否], [0, 未得, 未得])","explanation_zh":"目標型別與推導深度=([是, 是, 是], [是, 否, 否], [0, 未得, 未得])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134038,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[0,3,3],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"unreachable\"],[\"unreachable\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0039","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[100, 1, 3]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([否, 是, 是], [否, 是, 否], [不合型別, 1, 未得])","explanation_zh":"目標型別與推導深度=([否, 是, 是], [否, 是, 否], [不合型別, 1, 未得])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134039,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[100,1,3],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"depth\",1],[\"unreachable\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0040","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[3, 1, 3]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 是], [否, 是, 否], [未得, 1, 未得])","explanation_zh":"目標型別與推導深度=([是, 是, 是], [否, 是, 否], [未得, 1, 未得])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134040,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[3,1,3],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"depth\",1],[\"unreachable\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0041","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[0, 1, 3]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 是, 否], [0, 1, 未得])","explanation_zh":"目標型別與推導深度=([是, 是, 是], [是, 是, 否], [0, 1, 未得])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134041,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[0,1,3],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"depth\",1],[\"unreachable\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0042","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[100, 101, 2]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([否, 否, 是], [否, 否, 是], [不合型別, 不合型別, 2])","explanation_zh":"目標型別與推導深度=([否, 否, 是], [否, 否, 是], [不合型別, 不合型別, 2])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134042,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[100,101,2],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"untyped\"],[\"depth\",2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0043","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[3, 101, 2]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 否, 是], [否, 否, 是], [未得, 不合型別, 2])","explanation_zh":"目標型別與推導深度=([是, 否, 是], [否, 否, 是], [未得, 不合型別, 2])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134043,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[3,101,2],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"untyped\"],[\"depth\",2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0044","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[0, 101, 2]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 否, 是], [是, 否, 是], [0, 不合型別, 2])","explanation_zh":"目標型別與推導深度=([是, 否, 是], [是, 否, 是], [0, 不合型別, 2])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134044,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[0,101,2],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"untyped\"],[\"depth\",2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0045","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[100, 3, 2]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([否, 是, 是], [否, 否, 是], [不合型別, 未得, 2])","explanation_zh":"目標型別與推導深度=([否, 是, 是], [否, 否, 是], [不合型別, 未得, 2])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134045,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[100,3,2],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"unreachable\"],[\"depth\",2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0046","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[3, 3, 2]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 是], [否, 否, 是], [未得, 未得, 2])","explanation_zh":"目標型別與推導深度=([是, 是, 是], [否, 否, 是], [未得, 未得, 2])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134046,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[3,3,2],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"unreachable\"],[\"depth\",2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0047","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[0, 3, 2]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 否, 是], [0, 未得, 2])","explanation_zh":"目標型別與推導深度=([是, 是, 是], [是, 否, 是], [0, 未得, 2])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134047,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[0,3,2],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"unreachable\"],[\"depth\",2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0048","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[100, 1, 2]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([否, 是, 是], [否, 是, 是], [不合型別, 1, 2])","explanation_zh":"目標型別與推導深度=([否, 是, 是], [否, 是, 是], [不合型別, 1, 2])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134048,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[100,1,2],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"depth\",1],[\"depth\",2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0049","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[3, 1, 2]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 是], [否, 是, 是], [未得, 1, 2])","explanation_zh":"目標型別與推導深度=([是, 是, 是], [否, 是, 是], [未得, 1, 2])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134049,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[3,1,2],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"depth\",1],[\"depth\",2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0050","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"三個目標的型別與推導深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。queries=[0, 1, 2]。依序列 (目標型別資格, 依規則可得資格, 最短深度／未得／不合型別列表)。","answer":{"kind":"expression","value":"([是, 是, 是], [是, 是, 是], [0, 1, 2])","explanation_zh":"目標型別與推導深度=([是, 是, 是], [是, 是, 是], [0, 1, 2])。先核型別再查由事實得到的最短深度，未得不等於命題為假；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["三個目標的型別與推導深度"]},"variant":{"family_id":"qf-density13-proof-horn-goal-check","seed":202610134050,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2]],"queries":[0,1,2],"operation":"density13_proof_horn_goal_check","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"depth\",1],[\"depth\",2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0051","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=0，Fin(0)=∅，facts=[]，rules=[]（i→j為單步規則）。有序trace=[]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([], 無)","explanation_zh":"候選逐行資格與首錯=([], 無)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134051,"parameters":{"n":0,"facts":[],"rules":[],"trace":[],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0052","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=2，facts=[]，rules=[[0, 1], [1, 0]]（i→j為單步規則）。有序trace=[0, 1, 0]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 否, 否], 0)","explanation_zh":"候選逐行資格與首錯=([否, 否, 否], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134052,"parameters":{"n":2,"facts":[],"rules":[[0,1],[1,0]],"trace":[0,1,0],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0053","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 2, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 否, 否], 1)","explanation_zh":"候選逐行資格與首錯=([是, 否, 否], 1)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134053,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,2,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0054","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 1, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 是, 否], 0)","explanation_zh":"候選逐行資格與首錯=([否, 是, 否], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134054,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,1,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0055","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 1, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 是, 否], 2)","explanation_zh":"候選逐行資格與首錯=([是, 是, 否], 2)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134055,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,1,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0056","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 2, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 否, 是], 0)","explanation_zh":"候選逐行資格與首錯=([否, 否, 是], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134056,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,2,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0057","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 2, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 否, 是], 1)","explanation_zh":"候選逐行資格與首錯=([是, 否, 是], 1)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134057,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,2,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0058","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。有序trace=[2, 1, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 是, 是], 0)","explanation_zh":"候選逐行資格與首錯=([否, 是, 是], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134058,"parameters":{"n":3,"facts":[0],"rules":[[0,1],[1,2]],"trace":[2,1,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0059","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 1, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 是, 是], 無)","explanation_zh":"候選逐行資格與首錯=([是, 是, 是], 無)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134059,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,1,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0060","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 2, 2, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 否, 否, 否], 0)","explanation_zh":"候選逐行資格與首錯=([否, 否, 否, 否], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134060,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,2,2,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,false,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0061","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 2, 2, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 否, 否, 否], 1)","explanation_zh":"候選逐行資格與首錯=([是, 否, 否, 否], 1)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134061,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,2,2,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,false,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0062","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 1, 2, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 是, 否, 否], 0)","explanation_zh":"候選逐行資格與首錯=([否, 是, 否, 否], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134062,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,1,2,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,false,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0063","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 1, 2, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 是, 否, 否], 2)","explanation_zh":"候選逐行資格與首錯=([是, 是, 否, 否], 2)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134063,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,1,2,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,false,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0064","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 2, 1, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 否, 是, 否], 0)","explanation_zh":"候選逐行資格與首錯=([否, 否, 是, 否], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134064,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,2,1,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,true,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0065","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 2, 1, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 否, 是, 否], 1)","explanation_zh":"候選逐行資格與首錯=([是, 否, 是, 否], 1)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134065,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,2,1,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,true,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0066","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 1, 1, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 是, 是, 否], 0)","explanation_zh":"候選逐行資格與首錯=([否, 是, 是, 否], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134066,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,1,1,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,true,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0067","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 1, 1, 2]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 是, 是, 否], 3)","explanation_zh":"候選逐行資格與首錯=([是, 是, 是, 否], 3)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134067,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,1,1,2],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,true,false]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0068","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 2, 2, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 否, 否, 是], 0)","explanation_zh":"候選逐行資格與首錯=([否, 否, 否, 是], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134068,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,2,2,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,false,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0069","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 2, 2, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 否, 否, 是], 1)","explanation_zh":"候選逐行資格與首錯=([是, 否, 否, 是], 1)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134069,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,2,2,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,false,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0070","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 1, 2, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 是, 否, 是], 0)","explanation_zh":"候選逐行資格與首錯=([否, 是, 否, 是], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134070,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,1,2,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,false,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0071","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 1, 2, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 是, 否, 是], 2)","explanation_zh":"候選逐行資格與首錯=([是, 是, 否, 是], 2)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134071,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,1,2,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,false,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0072","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 2, 1, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 否, 是, 是], 0)","explanation_zh":"候選逐行資格與首錯=([否, 否, 是, 是], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134072,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,2,1,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,true,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0073","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 2, 1, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 否, 是, 是], 1)","explanation_zh":"候選逐行資格與首錯=([是, 否, 是, 是], 1)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134073,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,2,1,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,true,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0074","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[2, 1, 1, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([否, 是, 是, 是], 0)","explanation_zh":"候選逐行資格與首錯=([否, 是, 是, 是], 0)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134074,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[2,1,1,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,true,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0075","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"候選MP序列的逐行稽核","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。有序trace=[1, 1, 1, 1]。每列只能用facts及此前合法列作單前提MP，依序答 (逐行是否合法, 第一個不合法列的0起索引或無)，錯後繼續稽核。","answer":{"kind":"expression","value":"([是, 是, 是, 是], 無)","explanation_zh":"候選逐行資格與首錯=([是, 是, 是, 是], 無)。空序列不證明目標；無效列不加入前提，只保留已接受列；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選MP序列的逐行稽核"]},"variant":{"family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134075,"parameters":{"n":3,"facts":[0],"rules":[[0,1]],"trace":[1,1,1,1],"operation":"density13_proof_horn_candidate_trace","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,true,true]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0076","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=2，facts=[]，rules=[[0, 1], [1, 0]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[0]，goal=1。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(未得, 未得, 否)","explanation_zh":"刪規則前後深度=(未得, 未得, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134076,"parameters":{"n":2,"facts":[],"rules":[[0,1],[1,0]],"remove_indices":[0],"goal":1,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[null,null]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0077","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=1，facts=[0]，rules=[[0, 0]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[0]，goal=0。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(0, 0, 否)","explanation_zh":"刪規則前後深度=(0, 0, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134077,"parameters":{"n":1,"facts":[0],"rules":[[0,0]],"remove_indices":[0],"goal":0,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[0,0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0078","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=2，facts=[0]，rules=[[0, 1], [0, 1]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[0]，goal=1。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(1, 1, 否)","explanation_zh":"刪規則前後深度=(1, 1, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134078,"parameters":{"n":2,"facts":[0],"rules":[[0,1],[0,1]],"remove_indices":[0],"goal":1,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[1,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0079","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1], [1, 2], [0, 2]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[2]，goal=2。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(1, 2, 否)","explanation_zh":"刪規則前後深度=(1, 2, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134079,"parameters":{"n":3,"facts":[0],"rules":[[0,1],[1,2],[0,2]],"remove_indices":[2],"goal":2,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[1,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0080","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [0, 3]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[3]，goal=3。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(1, 3, 否)","explanation_zh":"刪規則前後深度=(1, 3, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134080,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2],[2,3],[0,3]],"remove_indices":[3],"goal":3,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[1,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0081","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=5，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 4]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[4]，goal=4。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(1, 4, 否)","explanation_zh":"刪規則前後深度=(1, 4, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134081,"parameters":{"n":5,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[0,4]],"remove_indices":[4],"goal":4,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[1,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0082","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=6，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 5]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[5]，goal=5。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(1, 5, 否)","explanation_zh":"刪規則前後深度=(1, 5, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134082,"parameters":{"n":6,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[0,5]],"remove_indices":[5],"goal":5,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[1,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0083","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=7，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [0, 6]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[6]，goal=6。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(1, 6, 否)","explanation_zh":"刪規則前後深度=(1, 6, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134083,"parameters":{"n":7,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[0,6]],"remove_indices":[6],"goal":6,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[1,6]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0084","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [0, 7]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[7]，goal=7。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(1, 7, 否)","explanation_zh":"刪規則前後深度=(1, 7, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134084,"parameters":{"n":8,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[0,7]],"remove_indices":[7],"goal":7,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[1,7]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0085","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=2，facts=[0]，rules=[[0, 1]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[0]，goal=1。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(1, 未得, 是)","explanation_zh":"刪規則前後深度=(1, 未得, 是)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134085,"parameters":{"n":2,"facts":[0],"rules":[[0,1]],"remove_indices":[0],"goal":1,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[1,null]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0086","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[]，goal=2。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(2, 2, 否)","explanation_zh":"刪規則前後深度=(2, 2, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134086,"parameters":{"n":3,"facts":[0],"rules":[[0,1],[1,2]],"remove_indices":[],"goal":2,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[2,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0087","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [0, 2]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[3]，goal=3。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(2, 3, 否)","explanation_zh":"刪規則前後深度=(2, 3, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134087,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2],[2,3],[0,2]],"remove_indices":[3],"goal":3,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0088","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=5，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 3]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[4]，goal=4。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(2, 4, 否)","explanation_zh":"刪規則前後深度=(2, 4, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134088,"parameters":{"n":5,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[0,3]],"remove_indices":[4],"goal":4,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[2,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0089","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=6，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 4]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[5]，goal=5。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(2, 5, 否)","explanation_zh":"刪規則前後深度=(2, 5, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134089,"parameters":{"n":6,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[0,4]],"remove_indices":[5],"goal":5,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[2,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0090","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=7，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [0, 5]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[6]，goal=6。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(2, 6, 否)","explanation_zh":"刪規則前後深度=(2, 6, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134090,"parameters":{"n":7,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[0,5]],"remove_indices":[6],"goal":6,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[2,6]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0091","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [0, 6]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[7]，goal=7。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(2, 7, 否)","explanation_zh":"刪規則前後深度=(2, 7, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134091,"parameters":{"n":8,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[0,6]],"remove_indices":[7],"goal":7,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[2,7]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0092","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=3，facts=[0]，rules=[[0, 1], [1, 2]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[1]，goal=2。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(2, 未得, 是)","explanation_zh":"刪規則前後深度=(2, 未得, 是)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134092,"parameters":{"n":3,"facts":[0],"rules":[[0,1],[1,2]],"remove_indices":[1],"goal":2,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[2,null]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0093","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2], [2, 3]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[]，goal=3。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(3, 3, 否)","explanation_zh":"刪規則前後深度=(3, 3, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134093,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2],[2,3]],"remove_indices":[],"goal":3,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[3,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0094","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=5，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 2]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[4]，goal=4。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(3, 4, 否)","explanation_zh":"刪規則前後深度=(3, 4, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134094,"parameters":{"n":5,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[0,2]],"remove_indices":[4],"goal":4,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[3,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0095","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=6，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 3]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[5]，goal=5。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(3, 5, 否)","explanation_zh":"刪規則前後深度=(3, 5, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134095,"parameters":{"n":6,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[0,3]],"remove_indices":[5],"goal":5,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[3,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0096","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=7，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [0, 4]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[6]，goal=6。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(3, 6, 否)","explanation_zh":"刪規則前後深度=(3, 6, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134096,"parameters":{"n":7,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[0,4]],"remove_indices":[6],"goal":6,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[3,6]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0097","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=8，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [0, 5]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[7]，goal=7。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(3, 7, 否)","explanation_zh":"刪規則前後深度=(3, 7, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134097,"parameters":{"n":8,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[0,5]],"remove_indices":[7],"goal":7,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[3,7]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0098","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=4，facts=[0]，rules=[[0, 1], [1, 2], [2, 3]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[2]，goal=3。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(3, 未得, 是)","explanation_zh":"刪規則前後深度=(3, 未得, 是)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134098,"parameters":{"n":4,"facts":[0],"rules":[[0,1],[1,2],[2,3]],"remove_indices":[2],"goal":3,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[3,null]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0099","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=5，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[]，goal=4。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(4, 4, 否)","explanation_zh":"刪規則前後深度=(4, 4, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134099,"parameters":{"n":5,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4]],"remove_indices":[],"goal":4,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[4,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261009-advance-20261013-r04-proof-0100","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"刪除原規則列前後的深度","stem_zh":"給定事實與單前提MP；索引從0，n=6，facts=[0]，rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 2]]（i→j為單步規則）。原始規則列索引從0，remove_indices=[5]，goal=5。依序答 (刪前深度或未得, 刪後深度或未得, 可得性是否改變)。","answer":{"kind":"expression","value":"(4, 5, 否)","explanation_zh":"刪規則前後深度=(4, 5, 否)。先驗全部原列，再按原索引刪除並分別BFS；深度增長但仍可得時可得性未變；僅依給定規則，循環無事實不能自生，有限稽核非一般邏輯完備性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["刪除原規則列前後的深度"]},"variant":{"family_id":"qf-density13-proof-horn-rule-removal","seed":202610134100,"parameters":{"n":6,"facts":[0],"rules":[[0,1],[1,2],[2,3],[3,4],[4,5],[0,2]],"remove_indices":[5],"goal":5,"operation":"density13_proof_horn_rule_removal","domain":"finite_given_MP_protocol","semantic_template":"qf-density13-proof-horn-rule-removal/[4,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-09","contract_version":"ocme-question-factory-v0.1"}}