{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-cauchy-sequence",
  "semantic_family_id": "sf-cauchy-sequence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Cauchy序列"
  },
  "summary": {
    "zh-Hant": "度量空間序列為Cauchy，若每個正epsilon最終所有兩項彼此距離皆小於epsilon。"
  },
  "statement": {
    "zh-Hant": "序列(x_n)為Cauchy，若對每個epsilon>0存在N，使所有m,n>=N都有d(x_m,x_n)<epsilon。定義不先假設極限存在或位於空間內。"
  },
  "formula": {
    "tex": "C(x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5a2552b096f278ae774d19284d68417bf0014835c66087dc34487579cc1bb03f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "有限列出的尾段只作bounded audit，不能證一般Cauchy。"
    },
    {
      "id": "a2",
      "zh-Hant": "完整證書限明示最終常值模型、倒數序列界，或一個精確反例族。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "cauchy_record",
      "role_zh": "Cauchy紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "sequence",
      "role_zh": "序列",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-metric-space",
      "title_zh": "度量空間",
      "reason_zh": "兩項距離由度量給出。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "Cauchy責任量化序列索引。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "保留forall epsilon exists N forall m,n責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "Cauchy比較尾段中任兩項，不是只比較候選極限。",
      "最終常值序列對任意epsilon有直接N。",
      "交替正負常值可對每個N給相鄰反例。"
    ],
    "common_misconception_zh": "長有限尾段很靠近就宣稱整個無限序列Cauchy。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限尾段不作一般證明；只接受明示模量或反例族。",
    "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": "加入完備度量空間與Banach空間。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-cauchy-sequence.md 整理；2026-10-13 脈衝 R03（使用者手動接續 2026-10-26 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}