{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-bijective",
  "semantic_family_id": "sf-bijective",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "雙射與唯一原像"
  },
  "summary": {
    "zh-Hant": "雙射同時單射與滿射，等價於陪域每個元素恰有一個合法原像。"
  },
  "statement": {
    "zh-Hant": "總函數 f:A→B 為雙射，當且僅當它同時單射與滿射；等價地，每個 b∈B 的纖維恰有一個元素。對有限集合，存在某個 A→B 的雙射當且僅當 |A|=|B|；基數相等不保證任意給定 f 都是雙射。"
  },
  "formula": {
    "tex": "f:A\\to B",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow></math>",
    "semantic_ast": {
      "type": "mapping",
      "function": {
        "type": "symbol",
        "name": "f",
        "glyph": "f"
      },
      "domain": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      },
      "codomain": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5a60a2d0dc6d3fe1eee2b4f6bcd1ad8a4c7f29d3622d3feffede72f213ed426f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "給定函數性質與是否存在某個雙射是不同問題。"
    },
    {
      "id": "a2",
      "zh-Hant": "基數存在性題限定有限集合，大小是非負整數；零到零包含唯一空雙射。"
    }
  ],
  "symbols": [
    {
      "token": "f",
      "role": "candidate_bijection",
      "role_zh": "待檢查總函數",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "domain",
      "role_zh": "定義域",
      "scope": "representative_formula"
    },
    {
      "token": "B",
      "role": "codomain",
      "role_zh": "陪域",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-injective",
      "title_zh": "單射與不同輸入的辨識",
      "reason_zh": "雙射包含單射要求。"
    },
    {
      "id": "mko-surjective",
      "title_zh": "滿射與陪域覆蓋",
      "reason_zh": "雙射包含滿射要求。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "有限函數可分為雙射、僅單射、僅滿射、兩者皆非。不能只檢查兩個集合大小而忽略實際映射資料。",
      "完整纖維概況按陪域目標排序，包含沒有原像的零纖維；全部纖維大小都為一才是雙射。",
      "有限集合相同基數允許逐一配對形成某個雙射，但可能另有非雙射的函數。基數題只問存在可能性，不把它冒充給定函數的審定。"
    ],
    "common_misconception_zh": "由 |A|=|B| 直接判給定 f 雙射，或只查其中一個條件。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限纖維與基數切片可重算；一般雙射、等勢及逆函數存在性尚未附形式 Evidence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定 Function.Bijective、existsUnique 與 Fintype.card／Equiv 的有限橋。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-bijective.md 整理；2026-10-04 脈衝 R03（使用者手動接續 2026-10-05 工作序列；實際日期仍為 2026-10-04），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-04"
  }
}