{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261012-advance-20261016-r06-injective-0043", "pilot_number": "R0013-07468", "number": 7468, "family_id": "qf-density16-inj-remove-input-collision-change", "title": { "zh-Hant": "第 7468 題：刪一輸入的碰撞變化", "en": "Question 7468: Input Deletion with Original Collision Labels Retained" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261012-advance-20261016-r06-injective-0043", "batch_id": "batch-20261012-advance-20261016-r06-injective-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-injective" ], "title_zh": "刪一輸入的碰撞變化", "stem_zh": "Fin型別從0起，完整總表；Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]，images=[2, 1, 1, 0, 1]。刪原輸入at=4，陪域不變。求 (原images, 原碰撞, 保留原始輸入標籤, 刪後碰撞, 刪後單射資格)，不重編輸入索引。", "answer": { "kind": "expression", "value": "([2, 1, 1, 0, 1], [[1, 2], [1, 4], [2, 4]], [0, 1, 2, 3], [[1, 2]], 否)", "explanation_zh": "有限結構核對=([2, 1, 1, 0, 1], [[1, 2], [1, 4], [2, 4]], [0, 1, 2, 3], [[1, 2]], 否)。只移去該原始標籤，完整碰撞比較仍使用保留的原始輸入標籤。有限精確核對不是一般形式證明。先資格後求值，空函數條件依宣告陪域。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "刪一輸入的碰撞變化" ] }, "variant": { "family_id": "qf-density16-inj-remove-input-collision-change", "seed": 202610166043, "parameters": { "n": 5, "m": 3, "images": [ 2, 1, 1, 0, 1 ], "at": 4, "domain": "Fin_total_table", "operation": "inj-remove-input-collision-change", "semantic_template": "qf-density16-inj-remove-input-collision-change/{\"at\":4,\"images\":[2,1,1,0,1],\"m\":3,\"n\":5}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-12", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261012-advance-20261016-r06-injective-0043", "batch_id": "batch-20261012-advance-20261016-r06-injective-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "887b5cac7d226ff24db0053bdc28f9715108b3955030e1ae93e0be5b7b243e92", "family_id": "qf-density16-inj-remove-input-collision-change", "seed": 202610166043, "semantic_template": "qf-density16-inj-remove-input-collision-change/{\"at\":4,\"images\":[2,1,1,0,1],\"m\":3,\"n\":5}" }, "fields": { "title": "Input Deletion with Original Collision Labels Retained", "stem": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.", "answer": { "kind": "expression", "value": "([2, 1, 1, 0, 1], [[1, 2], [1, 4], [2, 4]], [0, 1, 2, 3], [[1, 2]], No)", "explanation": "Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [[1, 2], [1, 4], [2, 4]], [0, 1, 2, 3], [[1, 2]], No)." }, "skills": [ "Input Deletion with Original Collision Labels Retained" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "Fin型別從0起，完整總表；Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]，images=[2, 1, 1, 0, 1]。刪原輸入at=4，陪域不變。求 (原images, 原碰撞, 保留原始輸入標籤, 刪後碰撞, 刪後單射資格)，不重編輸入索引。", "en": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source index=0∈Fin(5)", "after": "image[0]=2∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source index=1∈Fin(5)", "after": "image[1]=1∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source index=2∈Fin(5)", "after": "image[2]=1∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source index=3∈Fin(5)", "after": "image[3]=0∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source index=4∈Fin(5)", "after": "image[4]=1∈Fin(3)", "relation": "implies", "rule_id": "injective_complete_original_table_row", "conditions": [], "explanation": { "zh-Hant": "完整原表的每格都先合型，不只核對稍後會查詢的格。", "en": "First type-check every cell of the complete original table,not only cells queried later." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Fin(5)→Fin(3);original images=[2, 1, 1, 0, 1]", "after": "injectivity forbids equal outputs from distinct original input labels", "relation": "implies", "rule_id": "finite_injectivity_label_scope", "conditions": [], "explanation": { "zh-Hant": "單射只禁止不同輸入同輸出；空函數可空真，不能混同滿射。所有原表格先合法，後續限制或刪除保留原始標籤。", "en": "Injectivity forbids equal outputs for distinct inputs;the empty function is vacuously injective,which is not surjectivity.Type-check the whole original table and retain original labels under restriction or deletion." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "phase=before_remove;inputs=(0,1);outputs=(2,1)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "phase=before_remove;inputs=(0,2);outputs=(2,1)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "phase=before_remove;inputs=(0,3);outputs=(2,0)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "phase=before_remove;inputs=(0,4);outputs=(2,1)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "phase=before_remove;inputs=(1,2);outputs=(1,1)", "after": "collision=是", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "phase=before_remove;inputs=(1,3);outputs=(1,0)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "phase=before_remove;inputs=(1,4);outputs=(1,1)", "after": "collision=是", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "phase=before_remove;inputs=(2,3);outputs=(1,0)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "phase=before_remove;inputs=(2,4);outputs=(1,1)", "after": "collision=是", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "phase=before_remove;inputs=(3,4);outputs=(0,1)", "after": "collision=否", "relation": "implies", "rule_id": "injective_collision_pair_audit", "conditions": [], "explanation": { "zh-Hant": "每一對不同原標籤只查一次；輸出值相同時兩個輸入仍不同，不能去重輸出而抹掉碰撞。", "en": "Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "phase=before_remove;all input labels=[0, 1, 2, 3, 4]", "after": "collisions=[[1, 2], [1, 4], [2, 4]];injective=否", "relation": "implies", "rule_id": "complete_pairwise_collision_inventory", "conditions": [], "explanation": { "zh-Hant": "完整比較全部x