{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261017-r03-relation-0065",
  "pilot_number": "R0015-08340",
  "number": 8340,
  "family_id": "qf-density17-relation-inverse-double",
  "title": {
    "zh-Hant": "第 8340 題：有序關係交換兩次",
    "en": "Question 8340: Typed Converse Relation and Double Converse"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261017-r03-relation-0065",
    "batch_id": "batch-20261012-advance-20261017-r03-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-relation"
    ],
    "title_zh": "有序關係交換兩次",
    "stem_zh": "明示Fin來源與目標，有序關係對。來源Fin(3)=[0, 1, 2]，目標Fin(3)=[0, 1, 2]，關係R的列項=[[0, 1], [1, 0]]。答 ([n,m], [m,n], 正規原關係對, 逆關係對, 再交換一次的對, 往返相等資格)。",
    "answer": {
      "kind": "expression",
      "value": "([3, 3], [3, 3], [[0, 1], [1, 0]], [[0, 1], [1, 0]], [[0, 1], [1, 0]], 是)",
      "explanation_zh": "離散結構核對=([3, 3], [3, 3], [[0, 1], [1, 0]], [[0, 1], [1, 0]], [[0, 1], [1, 0]], 是)；每次交換有序對分量並交換型別，再按字典序列集合。；有限精確核對不是一般形式證明；逆關係不自動是反函數；關係不必是總函數；重複列項按集合正規化。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "有序關係交換兩次"
      ]
    },
    "variant": {
      "family_id": "qf-density17-relation-inverse-double",
      "seed": 202610173065,
      "parameters": {
        "n": 3,
        "m": 3,
        "pairs": [
          [
            0,
            1
          ],
          [
            1,
            0
          ]
        ],
        "domain": "Fin_typed_relation",
        "operation": "relation-inverse-double",
        "semantic_template": "qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,0]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261017-r03-relation-0065",
    "batch_id": "batch-20261012-advance-20261017-r03-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "d2818ecac7684e8d0cb12306b05b79e254c0eb7dcfdbd003610e7724555fe1eb",
      "family_id": "qf-density17-relation-inverse-double",
      "seed": 202610173065,
      "semantic_template": "qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,0]]}"
    },
    "fields": {
      "title": "Typed Converse Relation and Double Converse",
      "stem": "The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).",
      "answer": {
        "kind": "expression",
        "value": "([3, 3], [3, 3], [[0, 1], [1, 0]], [[0, 1], [1, 0]], [[0, 1], [1, 0]], Yes)",
        "explanation": "Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1], [1, 0]], [[0, 1], [1, 0]], [[0, 1], [1, 0]], Yes)."
      },
      "skills": [
        "Typed Converse Relation and Double Converse"
      ],
      "verification_warning": "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": "明示Fin來源與目標，有序關係對。來源Fin(3)=[0, 1, 2]，目標Fin(3)=[0, 1, 2]，關係R的列項=[[0, 1], [1, 0]]。答 ([n,m], [m,n], 正規原關係對, 逆關係對, 再交換一次的對, 往返相等資格)。",
        "en": "The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "types=(3,3);raw ordered pairs=[[0, 1], [1, 0]]",
      "after": "normalized relation=[[0, 1], [1, 0]]",
      "relation": "implies",
      "rule_id": "typed_relation_set_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核每一有序對的來源與目標型別，再把重複的同一有序對去重。交換座標是逆關係，不是同一對的排序。關係可非單值或非總。",
        "en": "Validate both types of every ordered pair,then deduplicate identical pairs.Swapping coordinates forms the converse,not a sorting of one pair.A relation may be multivalued or partial."
      },
      "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": "phase=converse;pair=[0, 1]",
      "after": "swapped=[1, 0]",
      "relation": "implies",
      "rule_id": "relation_swap_ordered_coordinates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時交換有序對兩座標與來源／目標型別；逆關係不自動成反函數。",
        "en": "Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function."
      },
      "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": "phase=converse;pair=[1, 0]",
      "after": "swapped=[0, 1]",
      "relation": "implies",
      "rule_id": "relation_swap_ordered_coordinates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時交換有序對兩座標與來源／目標型別；逆關係不自動成反函數。",
        "en": "Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function."
      },
      "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": "phase=double converse;pair=[0, 1]",
      "after": "swapped=[1, 0]",
      "relation": "implies",
      "rule_id": "relation_swap_ordered_coordinates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時交換有序對兩座標與來源／目標型別；逆關係不自動成反函數。",
        "en": "Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function."
      },
      "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": "phase=double converse;pair=[1, 0]",
      "after": "swapped=[0, 1]",
      "relation": "implies",
      "rule_id": "relation_swap_ordered_coordinates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時交換有序對兩座標與來源／目標型別；逆關係不自動成反函數。",
        "en": "Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function."
      },
      "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": "old types=[3, 3];converse types=[3, 3];converse=[[0, 1], [1, 0]]",
      "after": "double converse=[[0, 1], [1, 0]];equal=是",
      "relation": "implies",
      "rule_id": "double_converse_typed_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "交換兩次回到原型別與原集合，即使原關係不是函數也成立。",
        "en": "Swapping twice returns the original types and pair set even when the relation is not a function."
      },
      "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
    }
  ],
  "math_model": {
    "result": [
      [
        3,
        3
      ],
      [
        3,
        3
      ],
      [
        [
          0,
          1
        ],
        [
          1,
          0
        ]
      ],
      [
        [
          0,
          1
        ],
        [
          1,
          0
        ]
      ],
      [
        [
          0,
          1
        ],
        [
          1,
          0
        ]
      ],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([3, 3], [3, 3], [[0, 1], [1, 0]], [[0, 1], [1, 0]], [[0, 1], [1, 0]], 是)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261012-advance-20261017-r03-relation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261017-r03-relation-foundations/questions.jsonl",
      "batch_sha256": "fa93b079fe3332713b9e2dc835c033d2610769938b4950f23e6348ae0d11f894",
      "original_line": 65,
      "original_line_sha256": "d7539a866ed7a5b8881068f7c895986f2fae138446d7a28be1b924b77195e88d",
      "question_compact_sha256": "d2818ecac7684e8d0cb12306b05b79e254c0eb7dcfdbd003610e7724555fe1eb",
      "english_row_compact_sha256": "f57193b9e0b7cdf628647180966c36881cc2602007690f6084bda717b2d3eec2",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261017-r03-relation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261017-r03-relation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-relation": {
        "sha256": "3788215401bc9a5610691985883d04b506880cc9bd5e2ce0a98054cd5a5f0774",
        "english_sha256": "7d0700fb988776905f5d6ce3bf7399bc7a370ea89a38fe4e6b4c9e8509acdfdb",
        "original_path": "public/data/mko/relation.json",
        "english_path": "public/data/i18n/editions/mko-relation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "ca1078dc467a0f3769865919aadc46c87712d098f1dfbde90f787471e22bcfec",
    "original_candidate_sha256": "881cb1cd684807b9a8671408cc5be5b84323327dba81230607e333350a03b18a",
    "original_candidate_line_sha256": "881cb1cd684807b9a8671408cc5be5b84323327dba81230607e333350a03b18a"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-005-p16",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18",
      "question_first_witness_sha256": "e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4872f1cd81a7a7050aa1946453d6c0bc886b32fec81344d2aa2223ef438a77c4",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}