{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0001","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9ecb4cf01d5bdf1f728f86565faaf66f991cc82845594d4026ed977a48d5363","family_id":"qf-density13-proof-horn-min-depths","seed":202610134001,"semantic_template":"qf-density13-proof-horn-min-depths/[]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(0)=∅, facts=[], rules=[], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is []."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0002","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8adf3e26587eacb026b9848cb29faa8dad2a43398bd81429be5986645f8735c","family_id":"qf-density13-proof-horn-min-depths","seed":202610134002,"semantic_template":"qf-density13-proof-horn-min-depths/[null,null]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[], rules=[[0, 1], [1, 0]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, not-derived]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0003","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c07d8f03eb0f5efffacc20f67f2b068a2863931b2012519c77a98caf3cc2d00f","family_id":"qf-density13-proof-horn-min-depths","seed":202610134003,"semantic_template":"qf-density13-proof-horn-min-depths/[0]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(1)={0,…,0}, facts=[0], rules=[], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0004","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bffa989b18c7cc9bf0971b5e7b053efc036cf916cd1e213ba38b0d38a6ad248","family_id":"qf-density13-proof-horn-min-depths","seed":202610134004,"semantic_template":"qf-density13-proof-horn-min-depths/[0,1,2,3,4,5,6,7]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 1, 2, 3, 4, 5, 6, 7]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 2, 3, 4, 5, 6, 7]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0005","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac8350b1d7d64960f047bc02a5ae724bc1bbc381092dfb85f063774668a5e172","family_id":"qf-density13-proof-horn-min-depths","seed":202610134005,"semantic_template":"qf-density13-proof-horn-min-depths/[null,0,1,2,3,4,5,6]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, 0, 1, 2, 3, 4, 5, 6]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 1, 2, 3, 4, 5, 6]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0006","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dd14db47fd2007237f793cc0fee6a69a3cd1e22fe648244cfaaf9b3da59fe7e","family_id":"qf-density13-proof-horn-min-depths","seed":202610134006,"semantic_template":"qf-density13-proof-horn-min-depths/[0,0,1,2,3,4,5,6]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 0, 1, 2, 3, 4, 5, 6]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 1, 2, 3, 4, 5, 6]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0007","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd6cecdde741c1e3fb9b5ea9d10113742f559a8fc927b1fa3bcd71c3116b0b72","family_id":"qf-density13-proof-horn-min-depths","seed":202610134007,"semantic_template":"qf-density13-proof-horn-min-depths/[null,null,0,1,2,3,4,5]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[2], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, not-derived, 0, 1, 2, 3, 4, 5]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, 0, 1, 2, 3, 4, 5]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0008","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dae0fa12c38206abb21c9094d1c1e26d898d354866772593afdb9362bd1aa4ea","family_id":"qf-density13-proof-horn-min-depths","seed":202610134008,"semantic_template":"qf-density13-proof-horn-min-depths/[0,1,0,1,2,3,4,5]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 2], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 1, 0, 1, 2, 3, 4, 5]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 0, 1, 2, 3, 4, 5]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0009","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6a2a3cbe74f47844eb057b022503c49f95ddd5080c6767ec6df476119775960","family_id":"qf-density13-proof-horn-min-depths","seed":202610134009,"semantic_template":"qf-density13-proof-horn-min-depths/[null,0,0,1,2,3,4,5]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 2], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, 0, 0, 1, 2, 3, 4, 5]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 0, 1, 2, 3, 4, 5]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0010","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ddcf62fb51b9d89e5a737f317cecc0f7e874a39f04985418fd9245003993b22","family_id":"qf-density13-proof-horn-min-depths","seed":202610134010,"semantic_template":"qf-density13-proof-horn-min-depths/[0,0,0,1,2,3,4,5]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 2], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 0, 0, 1, 2, 3, 4, 5]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 0, 1, 2, 3, 4, 5]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0011","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c36131bccc3037b92b08c62c32ed9a776842cd67b3deb07093299a924212601","family_id":"qf-density13-proof-horn-min-depths","seed":202610134011,"semantic_template":"qf-density13-proof-horn-min-depths/[null,null,null,0,1,2,3,4]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, not-derived, not-derived, 0, 1, 2, 3, 4]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, not-derived, 0, 1, 2, 3, 4]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0012","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e68fee1ebaa214f69c9e747c8f7684ad2202bda0f7757eb72ec9eb88f0a95d4","family_id":"qf-density13-proof-horn-min-depths","seed":202610134012,"semantic_template":"qf-density13-proof-horn-min-depths/[0,1,2,0,1,2,3,4]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 1, 2, 0, 1, 2, 3, 4]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 2, 0, 1, 2, 3, 4]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0013","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d94aaff78b6826ffc5f483c80c6dd4b929b90c244aa5c2ff49c3e48c8b4c37b","family_id":"qf-density13-proof-horn-min-depths","seed":202610134013,"semantic_template":"qf-density13-proof-horn-min-depths/[null,0,1,0,1,2,3,4]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, 0, 1, 0, 1, 2, 3, 4]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 1, 0, 1, 2, 3, 4]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0014","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a10371673a93556332fb0857a405a96c871ab07d1079d36346c5b3847c3aad6","family_id":"qf-density13-proof-horn-min-depths","seed":202610134014,"semantic_template":"qf-density13-proof-horn-min-depths/[0,0,1,0,1,2,3,4]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 0, 1, 0, 1, 2, 3, 4]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 1, 0, 1, 2, 3, 4]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0015","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d4d8cb035013d10fe9fadad115f56143c36574d8abf79a5a8c511b57883b7b1","family_id":"qf-density13-proof-horn-min-depths","seed":202610134015,"semantic_template":"qf-density13-proof-horn-min-depths/[null,null,0,0,1,2,3,4]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[2, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, not-derived, 0, 0, 1, 2, 3, 4]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, 0, 0, 1, 2, 3, 4]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0016","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac417f3b47a7106f6fb4f6f086441cd66376c2ee358bde0448c017cc9966ff06","family_id":"qf-density13-proof-horn-min-depths","seed":202610134016,"semantic_template":"qf-density13-proof-horn-min-depths/[0,1,0,0,1,2,3,4]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 2, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 1, 0, 0, 1, 2, 3, 4]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 0, 0, 1, 2, 3, 4]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0017","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ced3400169ae3be78dd67d4005be6b9691859d5d3eb19175fc974207e1786012","family_id":"qf-density13-proof-horn-min-depths","seed":202610134017,"semantic_template":"qf-density13-proof-horn-min-depths/[null,0,0,0,1,2,3,4]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 2, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, 0, 0, 0, 1, 2, 3, 4]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 0, 0, 1, 2, 3, 4]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0018","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca1c10b86bcf5dca71aba63a0b9443e783be587809a658fd42cb803a0b4813d4","family_id":"qf-density13-proof-horn-min-depths","seed":202610134018,"semantic_template":"qf-density13-proof-horn-min-depths/[0,0,0,0,1,2,3,4]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 2, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 0, 0, 0, 1, 2, 3, 4]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 0, 0, 1, 2, 3, 4]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0019","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1acb2abbf792e708cdbd3c7bf98cdbf9b948b2ed73eac9165c7c82c805421f2","family_id":"qf-density13-proof-horn-min-depths","seed":202610134019,"semantic_template":"qf-density13-proof-horn-min-depths/[null,null,null,null,0,1,2,3]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, not-derived, not-derived, not-derived, 0, 1, 2, 3]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, not-derived, not-derived, 0, 1, 2, 3]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0020","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4605ce45bab21d05a847de5d139a36818ec002a0c75c23fdb82eb77d78bdaa18","family_id":"qf-density13-proof-horn-min-depths","seed":202610134020,"semantic_template":"qf-density13-proof-horn-min-depths/[0,1,2,3,0,1,2,3]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 1, 2, 3, 0, 1, 2, 3]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 2, 3, 0, 1, 2, 3]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0021","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49bcadc1f1142181ec4bc0454e5738c26e19d6862301bed78650e11ff2f89e13","family_id":"qf-density13-proof-horn-min-depths","seed":202610134021,"semantic_template":"qf-density13-proof-horn-min-depths/[null,0,1,2,0,1,2,3]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, 0, 1, 2, 0, 1, 2, 3]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 1, 2, 0, 1, 2, 3]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0022","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e826373b584c39bb69db12d60dd274c48825d680e3bdf31be5ebdd7500dc428","family_id":"qf-density13-proof-horn-min-depths","seed":202610134022,"semantic_template":"qf-density13-proof-horn-min-depths/[0,0,1,2,0,1,2,3]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 0, 1, 2, 0, 1, 2, 3]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 1, 2, 0, 1, 2, 3]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0023","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3155996303b1db7018db0a6f55f4c25c8c99a7464eb98780f9e2d86faf76af2","family_id":"qf-density13-proof-horn-min-depths","seed":202610134023,"semantic_template":"qf-density13-proof-horn-min-depths/[null,null,0,1,0,1,2,3]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[2, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, not-derived, 0, 1, 0, 1, 2, 3]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, 0, 1, 0, 1, 2, 3]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0024","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64be13dcf7b4cdfca49a658da124ed5583ab22a3bfe0458e28c4c57db8254049","family_id":"qf-density13-proof-horn-min-depths","seed":202610134024,"semantic_template":"qf-density13-proof-horn-min-depths/[0,1,0,1,0,1,2,3]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 2, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[0, 1, 0, 1, 0, 1, 2, 3]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 0, 1, 0, 1, 2, 3]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0025","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37351f9164c80317608bf4f925d0514110e00a2f36ce20a14962e0b0d745be0f","family_id":"qf-density13-proof-horn-min-depths","seed":202610134025,"semantic_template":"qf-density13-proof-horn-min-depths/[null,0,0,1,0,1,2,3]"},"fields":{"title":"All Minimum Depths in a Given MP Protocol","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 2, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.","answer":{"kind":"expression","value":"[not-derived, 0, 0, 1, 0, 1, 2, 3]","explanation":"Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 0, 1, 0, 1, 2, 3]."},"skills":["All Minimum Depths in a Given MP Protocol"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0026","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afc9533130ba2f0eaa29634509ae1289f6b3f524a61eb0981e97299c482f6973","family_id":"qf-density13-proof-horn-goal-check","seed":202610134026,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"untyped\"],[\"untyped\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(0)=∅, facts=[], rules=[], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[-1, 0, 1]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([No, No, No], [No, No, No], [ill-typed, ill-typed, ill-typed])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, No], [No, No, No], [ill-typed, ill-typed, ill-typed])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0027","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b03b33303456231a1aae057d9822476df804c23fdffdf8c1b2b91f6c1b3b477","family_id":"qf-density13-proof-horn-goal-check","seed":202610134027,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"untyped\"],[\"untyped\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 101, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, No], [Yes, No, No], [0, ill-typed, ill-typed])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, No], [Yes, No, No], [0, ill-typed, ill-typed])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0028","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"328ce1f457e5c7a2baa7508bbd5b5a94b211beaf25de59cfab4cc30b2562a0c2","family_id":"qf-density13-proof-horn-goal-check","seed":202610134028,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"unreachable\"],[\"untyped\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 3, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, No], [No, No, No], [not-derived, not-derived, ill-typed])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], [No, No, No], [not-derived, not-derived, ill-typed])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0029","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc297c5659c94f6c47372ad8ed2a1468cd05a8858ec5f0c5589ee4204ce68d03","family_id":"qf-density13-proof-horn-goal-check","seed":202610134029,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"unreachable\"],[\"untyped\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 3, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, No], [Yes, No, No], [0, not-derived, ill-typed])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, No, No], [0, not-derived, ill-typed])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0030","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b0a7cb0468edb5e5e5110ee2c2cb4db63a61c03e4b1e9893f9ecdcdb3cda222","family_id":"qf-density13-proof-horn-goal-check","seed":202610134030,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"depth\",1],[\"untyped\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 1, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, No], [No, Yes, No], [ill-typed, 1, ill-typed])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, No], [No, Yes, No], [ill-typed, 1, ill-typed])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0031","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fd0b4e185e10fe6f373d9297e74a829bde837657ed23dd7c0e9933bd66ca1ec","family_id":"qf-density13-proof-horn-goal-check","seed":202610134031,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"depth\",1],[\"untyped\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 1, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, No], [No, Yes, No], [not-derived, 1, ill-typed])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], [No, Yes, No], [not-derived, 1, ill-typed])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0032","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2025cd245d00628776b769cf68c9ceb77f93a8b174f261ca8f58e7df7373cb68","family_id":"qf-density13-proof-horn-goal-check","seed":202610134032,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"depth\",1],[\"untyped\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 1, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, No], [Yes, Yes, No], [0, 1, ill-typed])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, Yes, No], [0, 1, ill-typed])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0033","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c7227bcd4710fdd9f1e81d252a2efd474cfb03261df88691f855387425d751","family_id":"qf-density13-proof-horn-goal-check","seed":202610134033,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"untyped\"],[\"unreachable\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 101, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([No, No, Yes], [No, No, No], [ill-typed, ill-typed, not-derived])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes], [No, No, No], [ill-typed, ill-typed, not-derived])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0034","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"523ec0e69ef81149738308384c28447949f6b27d8a56b1080255ae7924b33b74","family_id":"qf-density13-proof-horn-goal-check","seed":202610134034,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"untyped\"],[\"unreachable\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 101, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, Yes], [No, No, No], [not-derived, ill-typed, not-derived])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], [No, No, No], [not-derived, ill-typed, not-derived])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0035","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"493818762534d2fec07734b486dfb4cd882503f604d1bf6f34160c7c373e23f5","family_id":"qf-density13-proof-horn-goal-check","seed":202610134035,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"untyped\"],[\"unreachable\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 101, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, Yes], [Yes, No, No], [0, ill-typed, not-derived])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, No, No], [0, ill-typed, not-derived])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0036","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4d82ae59858c5b198527c46e74b9b455d57eb633f388b78e52ff5dbd23dd67e","family_id":"qf-density13-proof-horn-goal-check","seed":202610134036,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"unreachable\"],[\"unreachable\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 3, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, Yes], [No, No, No], [ill-typed, not-derived, not-derived])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, No, No], [ill-typed, not-derived, not-derived])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0037","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50fed4846144ab0fbe44f86df17fd999652b81e16a1bf45c3ff7f3556f261fa6","family_id":"qf-density13-proof-horn-goal-check","seed":202610134037,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"unreachable\"],[\"unreachable\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 3, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, No, No], [not-derived, not-derived, not-derived])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, No, No], [not-derived, not-derived, not-derived])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0038","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d651a302798ffdb88a5c5acf8267750bdc5829b7aaaa3c6bd6105f518b8128c","family_id":"qf-density13-proof-horn-goal-check","seed":202610134038,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"unreachable\"],[\"unreachable\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 3, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, No, No], [0, not-derived, not-derived])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, No], [0, not-derived, not-derived])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0039","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd3dce7e412a65196a07e17198c60bbf2740230d668e6f6df20943a2baa1ba60","family_id":"qf-density13-proof-horn-goal-check","seed":202610134039,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"depth\",1],[\"unreachable\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 1, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, Yes], [No, Yes, No], [ill-typed, 1, not-derived])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, Yes, No], [ill-typed, 1, not-derived])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0040","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14cf72a8b5fb693136f577e07bea8b2a2f77186d528c1e5d3a7cef734183f05e","family_id":"qf-density13-proof-horn-goal-check","seed":202610134040,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"depth\",1],[\"unreachable\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 1, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, Yes, No], [not-derived, 1, not-derived])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, No], [not-derived, 1, not-derived])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0041","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71feaefdcf7baa297386239b73d77960d089ea71680275f7702a758e6a36694b","family_id":"qf-density13-proof-horn-goal-check","seed":202610134041,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"depth\",1],[\"unreachable\"]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 1, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, Yes, No], [0, 1, not-derived])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, No], [0, 1, not-derived])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0042","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e283ef9d7fe9092a515dbf33a527792e26fe35ecf58ac2a7d3eca80e779333ce","family_id":"qf-density13-proof-horn-goal-check","seed":202610134042,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"untyped\"],[\"depth\",2]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 101, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([No, No, Yes], [No, No, Yes], [ill-typed, ill-typed, 2])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes], [No, No, Yes], [ill-typed, ill-typed, 2])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0043","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efcd51d8763e7c5885fb177a1b33523d925c26ec83be3f92f6540fb227731926","family_id":"qf-density13-proof-horn-goal-check","seed":202610134043,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"untyped\"],[\"depth\",2]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 101, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, Yes], [No, No, Yes], [not-derived, ill-typed, 2])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], [No, No, Yes], [not-derived, ill-typed, 2])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0044","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcc287ba03848b0477ef3c7fa52fc442b93bdd219da91859493426bab7ec07ff","family_id":"qf-density13-proof-horn-goal-check","seed":202610134044,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"untyped\"],[\"depth\",2]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 101, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, No, Yes], [Yes, No, Yes], [0, ill-typed, 2])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, No, Yes], [0, ill-typed, 2])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0045","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e96576a90016a117d1901ea77c09f17d5ad70566e70654911b917b33b225ca8f","family_id":"qf-density13-proof-horn-goal-check","seed":202610134045,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"unreachable\"],[\"depth\",2]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 3, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, Yes], [No, No, Yes], [ill-typed, not-derived, 2])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, No, Yes], [ill-typed, not-derived, 2])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0046","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12cae3cacd1255eb2ad7fdb071e3d494d3f951884931e1878a66f541af3d9fac","family_id":"qf-density13-proof-horn-goal-check","seed":202610134046,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"unreachable\"],[\"depth\",2]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 3, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, No, Yes], [not-derived, not-derived, 2])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, No, Yes], [not-derived, not-derived, 2])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0047","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7831166beaf2052c2ec533d8e22dcf896dff7f563d9fed9d8dd3ac8d30c709aa","family_id":"qf-density13-proof-horn-goal-check","seed":202610134047,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"unreachable\"],[\"depth\",2]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 3, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, No, Yes], [0, not-derived, 2])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, Yes], [0, not-derived, 2])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0048","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3cab092cc3ea2aa9c4d92808c27ccc1435e5a3a090c74171ee5178fbd1f1dcb","family_id":"qf-density13-proof-horn-goal-check","seed":202610134048,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"untyped\"],[\"depth\",1],[\"depth\",2]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 1, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([No, Yes, Yes], [No, Yes, Yes], [ill-typed, 1, 2])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, Yes, Yes], [ill-typed, 1, 2])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0049","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd5254c201bb689b66d6aefc46174e2421af6b0b9c7b238f05ef281428a7ea91","family_id":"qf-density13-proof-horn-goal-check","seed":202610134049,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"unreachable\"],[\"depth\",1],[\"depth\",2]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 1, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, Yes, Yes], [not-derived, 1, 2])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, Yes], [not-derived, 1, 2])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0050","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0725c1d4ec5acf9aab51adc0b0a167b021953fbb94fbe9a5b52f9772c8b4f13f","family_id":"qf-density13-proof-horn-goal-check","seed":202610134050,"semantic_template":"qf-density13-proof-horn-goal-check/[[\"depth\",0],[\"depth\",1],[\"depth\",2]]"},"fields":{"title":"Typed MP Goals and Minimum Depths","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 1, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, Yes, Yes], [0, 1, 2])","explanation":"Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [0, 1, 2])."},"skills":["Typed MP Goals and Minimum Depths"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0051","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cd56d597c6c2ddf59f215cc3659c6b515e66c397571f5690bb93bf61e25cdd1","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134051,"semantic_template":"qf-density13-proof-horn-candidate-trace/[]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(0)=∅, facts=[], rules=[], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([], none)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([], none)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0052","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c890aae4cff6760dbdb02b477e03621a0739bc58ef40dc11cb92bbc8a269b26","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134052,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[], rules=[[0, 1], [1, 0]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[0, 1, 0]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, No, No], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, No], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0053","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c3225197a3620e978480ee15e8c0feede210f7e45d5aae7b4f17e222bf4a391","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134053,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, No, No], 1)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, No], 1)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0054","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2000166e9579f415df23e8d6ce7eecd809049dc6c7e5577b444e6973c87460fb","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134054,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, Yes, No], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, No], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0055","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf1c7420727b3ded8a52d6eed7f4fbffbc1215dd54ccf0b8a032084e1ac5b6fe","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134055,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, Yes, No], 2)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], 2)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0056","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5e2d96d502be3c93d75a641b55a17c254f59751cee987dc95f8124e4c4b9e9e","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134056,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, No, Yes], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0057","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4b5c727b3f37c38a8b9d10943f84e0f42f32f3fb4427375dab55fa2df5a1486","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134057,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, No, Yes], 1)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], 1)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0058","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd163d239e756b5a5003f55872eb9d5bc530934e3ea59a86220daa023a6c6936","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134058,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, Yes, Yes], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0059","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21d219463528fda8f4312fd618e1c21d4d45105b859a993eab4e91e0667b99de","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134059,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], none)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], none)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0060","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46aaaebef888382f73e1ff2e36209aeb1bb3c2342fe70ccc62c166dd0cd9ade8","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134060,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,false,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, No, No, No], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, No, No], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0061","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b510aa0223e591b21acaa35e322a25085398be87ca3591e3d09bc9f2cc9bb90","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134061,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,false,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, No, No, No], 1)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, No, No], 1)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0062","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91982f6fcd1e16d220c286285aa56289a4d6a7220a1c08bad04975905fe717d1","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134062,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,false,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, Yes, No, No], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, No, No], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0063","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87c3fb528ddeae9081a1fc5f8efa550f045bdca564d33f4f180f1242ea6f462e","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134063,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,false,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, Yes, No, No], 2)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No, No], 2)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0064","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2b73d79690a1cbd683a8a1e86a512882dce036b5a7fc85baa7b90b2f1da718d","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134064,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,true,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, No, Yes, No], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes, No], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0065","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30ef2a4778fe8a741b76c9c69c6eff708c14a79dd499511c5dbfa65d8231a984","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134065,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,true,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, No, Yes, No], 1)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes, No], 1)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0066","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"641c0d90affd8f1a3e109324e900f03054778d70dd9104b6d3bcfcc30daf3ea6","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134066,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,true,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, Yes, Yes, No], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes, No], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0067","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51bc0a3600e3ba029b395322ea7bf6d14e17d73a6f41d9a1690b867e1a775adf","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134067,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,true,false]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, Yes, Yes, No], 3)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes, No], 3)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0068","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3c957f6c1d5b3c9d1a3bc323a7766789dfd4e4b3278952318b845b9354ac193","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134068,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,false,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, No, No, Yes], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, No, Yes], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0069","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edd9b357e0221673911fc10a23d997d4e4145152de9cbc53143d1108e18576d4","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134069,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,false,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, No, No, Yes], 1)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, No, Yes], 1)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0070","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06aa74782aa98c3e7537930182c1b0717a075d2c049bd10481fa1203dc00c0db","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134070,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,false,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, Yes, No, Yes], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, No, Yes], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0071","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"488d4d49973c08f4ffb2b27bd4eda937478d1dc1381703f29282e396c9e17aa5","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134071,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,false,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, Yes, No, Yes], 2)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No, Yes], 2)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0072","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"178e37e26c9e103b8fb8f18f57fe647a6afe367c80827f855ad36446b286cf39","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134072,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,false,true,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, No, Yes, Yes], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes, Yes], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0073","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e854f78cae3b7418531b24921fff67e5935b06ec03b49fa0d78d540e9633100","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134073,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,false,true,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, No, Yes, Yes], 1)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes, Yes], 1)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0074","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dae6620721f86b8b753b68e613222867640fc94473064267a7ad5431dd57c256","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134074,"semantic_template":"qf-density13-proof-horn-candidate-trace/[false,true,true,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([No, Yes, Yes, Yes], 0)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes, Yes], 0)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0075","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1b746a5e2ff51060290c050defba725f2d88e73a44fc04c6592352aa970e727","family_id":"qf-density13-proof-horn-candidate-trace","seed":202610134075,"semantic_template":"qf-density13-proof-horn-candidate-trace/[true,true,true,true]"},"fields":{"title":"Sequential Candidate Trace and First Rejected Row","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.","answer":{"kind":"expression","value":"([Yes, Yes, Yes, Yes], none)","explanation":"Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes, Yes], none)."},"skills":["Sequential Candidate Trace and First Rejected Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0076","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6734483d5c67485d6ac218d823a8d470d2a6cd91bbaef3c7f69d7bd4abdc2d9b","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134076,"semantic_template":"qf-density13-proof-horn-rule-removal/[null,null]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[], rules=[[0, 1], [1, 0]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=1. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(not-derived, not-derived, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (not-derived, not-derived, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0077","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"646706cb52a8b681d0b4461cbaee4705777936f3e14f888f864b7c8b00f5cd14","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134077,"semantic_template":"qf-density13-proof-horn-rule-removal/[0,0]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(1)={0,…,0}, facts=[0], rules=[[0, 0]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=0. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(0, 0, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (0, 0, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0078","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"370c684f92e5abc9300722bbd3116945b837ce1799666181da807e8c2b2bc925","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134078,"semantic_template":"qf-density13-proof-horn-rule-removal/[1,1]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[0], rules=[[0, 1], [0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=1. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(1, 1, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 1, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0079","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0752696a4c9ebd76683a63c4f4558443129282ddd6c9d6ca702d9edca5502cf6","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134079,"semantic_template":"qf-density13-proof-horn-rule-removal/[1,2]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[2], goal=2. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(1, 2, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 2, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0080","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3929b709eb66ac610f7c356707e0786ed47fcea4d4abdd383071f9da3cee011d","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134080,"semantic_template":"qf-density13-proof-horn-rule-removal/[1,3]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [0, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[3], goal=3. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(1, 3, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 3, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0081","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a4c2d172587babb32d3fb7c1dd3e095292af79a9207d303655c2ea5346689c2","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134081,"semantic_template":"qf-density13-proof-horn-rule-removal/[1,4]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 4]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[4], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(1, 4, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 4, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0082","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41c06b0021a2bf823d1a4155f37c048cd5007ea3e8195641b83196203db19af2","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134082,"semantic_template":"qf-density13-proof-horn-rule-removal/[1,5]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(6)={0,…,5}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 5]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[5], goal=5. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(1, 5, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 5, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0083","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"354cb046f8116461c5a594be2e2189881d56096682491a29a9657cdf5b77dcf9","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134083,"semantic_template":"qf-density13-proof-horn-rule-removal/[1,6]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(7)={0,…,6}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [0, 6]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[6], goal=6. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(1, 6, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 6, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0084","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57fbf4735a4babc29df7ac43ca2124f01bad34844140b21bfb1b0e9c5235b773","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134084,"semantic_template":"qf-density13-proof-horn-rule-removal/[1,7]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [0, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[7], goal=7. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(1, 7, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 7, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0085","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea004b889c4a7487b2161fc6e6855cc69c7a161478c755212dc45cccbab3b1e6","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134085,"semantic_template":"qf-density13-proof-horn-rule-removal/[1,null]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=1. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(1, not-derived, Yes)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, not-derived, Yes)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0086","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b17bd58427c320ca5960ee8ba1831734bcd306586b60ea21cfa2e2dfc1f3b44","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134086,"semantic_template":"qf-density13-proof-horn-rule-removal/[2,2]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[], goal=2. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(2, 2, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 2, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0087","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db75d7845e1927a2f7ca894aa9206f6e562d9195100cb7acfd94e98ce636eb84","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134087,"semantic_template":"qf-density13-proof-horn-rule-removal/[2,3]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[3], goal=3. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(2, 3, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 3, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0088","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4a459bdb1cd2c5aa5c3054ece8aac451d0aa8dde9467905666e54867d9c0ec7","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134088,"semantic_template":"qf-density13-proof-horn-rule-removal/[2,4]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[4], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(2, 4, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 4, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0089","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d8fc65e061307ec95e2f6b4bc5646f9ec01aa7ef29c0cfd998a3d83f8014078","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134089,"semantic_template":"qf-density13-proof-horn-rule-removal/[2,5]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(6)={0,…,5}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 4]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[5], goal=5. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(2, 5, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 5, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0090","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67ae86f35b937424ad11d2e69e9590c8d012d7f8f6350fd0dd377af41d86a8c5","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134090,"semantic_template":"qf-density13-proof-horn-rule-removal/[2,6]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(7)={0,…,6}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [0, 5]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[6], goal=6. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(2, 6, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 6, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0091","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c12aad82d026ffc0fb107f6f1af8cb91608dc8e84882715270b0f8ec5429d85b","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134091,"semantic_template":"qf-density13-proof-horn-rule-removal/[2,7]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [0, 6]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[7], goal=7. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(2, 7, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 7, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0092","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f05386838683b5e5d3a9f4371c818f8d84e9ab86f278a9d043f6bc7efe7a8d85","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134092,"semantic_template":"qf-density13-proof-horn-rule-removal/[2,null]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[1], goal=2. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(2, not-derived, Yes)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, not-derived, Yes)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0093","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c899d08f6610f7fdde25e5dcc0b933ae255947e81f4f8e334172bd389f16ccc","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134093,"semantic_template":"qf-density13-proof-horn-rule-removal/[3,3]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2], [2, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[], goal=3. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(3, 3, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 3, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0094","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9df0b58f6917b5b174bcfd0dc1a5b9f5936fed9018823509fa4d529ec4941582","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134094,"semantic_template":"qf-density13-proof-horn-rule-removal/[3,4]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[4], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(3, 4, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 4, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0095","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f81a93b853244dc5cb5d9c7a923b4ece031ded97aed92e73fba36d3f83a8278d","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134095,"semantic_template":"qf-density13-proof-horn-rule-removal/[3,5]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(6)={0,…,5}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[5], goal=5. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(3, 5, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 5, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0096","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1254084eec059dbf1918f07e08a023cf0f415d2d8b8aede073bc5145fdf7fc88","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134096,"semantic_template":"qf-density13-proof-horn-rule-removal/[3,6]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(7)={0,…,6}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [0, 4]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[6], goal=6. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(3, 6, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 6, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0097","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"154221d7aebe830f5b6a47868f7e58aac17e7dc2fe7e41855edbe66f43c447cc","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134097,"semantic_template":"qf-density13-proof-horn-rule-removal/[3,7]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [0, 5]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[7], goal=7. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(3, 7, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 7, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0098","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd77a4bf566b2f141ded98c2c9a3aa328547336fe2c2a34451600e295339ef74","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134098,"semantic_template":"qf-density13-proof-horn-rule-removal/[3,null]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2], [2, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[2], goal=3. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(3, not-derived, Yes)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, not-derived, Yes)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0099","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40caf403640ceb1510da0f4e2eed2da8930dbc3c0313f5ba6aa705727d107533","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134099,"semantic_template":"qf-density13-proof-horn-rule-removal/[4,4]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(4, 4, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (4, 4, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r04-proof-0100","batch_id":"batch-20261009-advance-20261013-r04-proof-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2ae2c3f20b05c33fc38fb722419054bf99ff22addb3aeefd9e1e6b8709444c5","family_id":"qf-density13-proof-horn-rule-removal","seed":202610134100,"semantic_template":"qf-density13-proof-horn-rule-removal/[4,5]"},"fields":{"title":"Goal Depths Before and After Removing Original Rules","stem":"Given facts and single-premise MP only: Fin(6)={0,…,5}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[5], goal=5. Give (depth before or not-derived, depth after or not-derived, did derivability change?).","answer":{"kind":"expression","value":"(4, 5, No)","explanation":"Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (4, 5, No)."},"skills":["Goal Depths Before and After Removing Original Rules"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}