{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-at-most-countable-set",
  "semantic_family_id": "sf-at-most-countable-set",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "至多可數集"
  },
  "summary": {
    "zh-Hant": "集合若有限或存在到自然數的明示單射，則為至多可數；不等同可數無限。"
  },
  "statement": {
    "zh-Hant": "本輪以完整有限集合證書或整數仿射編碼的全稱單射證書處理。某個編碼失敗只否證該witness，不證集合不可數。"
  },
  "formula": {
    "tex": "A(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>A</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "dde799663ba218245a9b35d2d1baa486043c6b4b87197f3f6293eaa6ac210367"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "有限集合完整列出；symbolic family使用明示整數參數與全稱注入公式。"
    },
    {
      "id": "a2",
      "zh-Hant": "有限、可數無限與至多可數三者不混同。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "at_most_countable",
      "role_zh": "至多可數證書",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "set",
      "role_zh": "集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-cardinality-comparison",
      "title_zh": "集合基數比較",
      "reason_zh": "注入自然數提供比較。"
    },
    {
      "id": "mko-natural-number",
      "title_zh": "自然數",
      "reason_zh": "codomain為N。"
    },
    {
      "id": "mko-injective",
      "title_zh": "單射",
      "reason_zh": "證書須全稱單射。"
    },
    {
      "id": "mko-surjective",
      "title_zh": "滿射",
      "reason_zh": "可數無限需額外雙射／滿射責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "有限集合自動至多可數。",
      "仿射整數編碼以非零斜率給全稱單射。",
      "失敗witness不證不可數。"
    ],
    "common_misconception_zh": "把至多可數等同可數無限，或因一個encoding碰撞就宣稱不可數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只接受完整有限或明示全稱注入證書；不作一般不可數性聲明。",
    "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": "加入可數無限、對角論證與可數聯集。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-at-most-countable-set.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-10-31 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"
  }
}