{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-pigeonhole-principle",
  "semantic_family_id": "sf-pigeonhole-principle",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "鴿籠原理"
  },
  "summary": {
    "zh-Hant": "有限total map的domain比codomain大時必有兩個不同元素共享像。"
  },
  "statement": {
    "zh-Hant": "對完整有限A、B與total f:A→B，|A|>|B|推出存在a1!=a2且f(a1)=f(a2)。|A|<=|B|不保證給定map單射；generalized版本的最大load至少ceil(|A|/|B|)須另行明示。"
  },
  "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": "A與B有限且mapping對A每個元素恰有一個B中值。"
    },
    {
      "id": "a2",
      "zh-Hant": "codomain非空；generalized負載用exact integer ceiling。"
    }
  ],
  "symbols": [
    {
      "token": "f",
      "role": "finite_total_map",
      "role_zh": "有限全函數",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "domain",
      "role_zh": "定義域",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-counting-principle",
      "title_zh": "基本計數原理",
      "reason_zh": "有限計數。"
    },
    {
      "id": "mko-cardinality-comparison",
      "title_zh": "集合基數比較",
      "reason_zh": "domain/codomain大小。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數與映射",
      "reason_zh": "total map與injectivity。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "collision witness包含兩個不同domain indices。",
      "complete map使有限實例核對是exact而非sample。",
      "converse不成立；小domain仍可有collision。"
    ],
    "common_misconception_zh": "由|A|<=|B|斷言任意f都是injective。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限maps；抽樣collision不證一般映射命題。",
    "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": "加入inclusion-exclusion與generalized applications。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-pigeonhole-principle.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-11-07 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}