{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-cardinality-comparison",
  "semantic_family_id": "sf-cardinality-comparison",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "集合基數比較"
  },
  "summary": {
    "zh-Hant": "明示source與target間的單射、滿射、雙射分別提供有限集合基數比較證書。"
  },
  "statement": {
    "zh-Hant": "本輪限完整有限集合。A到B單射證|A|<=|B|，滿射證|A|>=|B|，雙射證相等；若|A|>|B|則不存在單射。無限集合的surjection逆向比較需要額外choice／section責任，不在本輪。"
  },
  "formula": {
    "tex": "C(A,B)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        },
        {
          "type": "symbol",
          "name": "B",
          "glyph": "B"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "1b0cba0c4e560067f331e0c6fcd8cfa4344f70f4c8b983a9ccb9b52f42693388"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "source與target為互異元素完整列表，mapping表長度等於source。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有像屬target；證書只涵蓋指定有限集合。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "cardinality_comparison",
      "role_zh": "基數比較",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "finite_set",
      "role_zh": "來源集合",
      "scope": "representative_formula"
    },
    {
      "token": "B",
      "role": "finite_set",
      "role_zh": "目標集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-injective",
      "title_zh": "單射",
      "reason_zh": "單射提供一方向比較。"
    },
    {
      "id": "mko-surjective",
      "title_zh": "滿射",
      "reason_zh": "有限集合滿射提供反方向比較。"
    },
    {
      "id": "mko-bijective",
      "title_zh": "雙射",
      "reason_zh": "雙射給基數相等。"
    },
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "mapping綁定明示source與target。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先核mapping總且像在target。",
      "單射與滿射責任分開。",
      "pigeonhole為有限集合否證；不外推無限基數。"
    ],
    "common_misconception_zh": "不綁source／target就談injective，或把有限surjection規則無條件外推無限集合。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理明示有限集合；無限比較與choice不在本輪。",
    "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": "加入至多可數集。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-cardinality-comparison.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-10-30 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"
  }
}