{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-bolzano-weierstrass-theorem-real",
  "semantic_family_id": "sf-bolzano-weierstrass-theorem-real",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實數列Bolzano－Weierstrass定理"
  },
  "summary": {
    "zh-Hant": "每個整體有界的實數列都有一個以嚴格遞增無限索引取得的收斂子數列。"
  },
  "statement": {
    "zh-Hant": "bounded須為exists M forall n；subsequence index genuinely strictly increasing且無限，並有forall-tail epsilon witness。週期列與oscillatory rational tails提供exact slices。有限prefix/cluster search不足；定理不保證原數列收斂或limit unique。"
  },
  "formula": {
    "tex": "a_k\\to L",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>a</mi><mi>k</mi></msub><mo>→</mo><mi>L</mi></mrow></math>",
    "semantic_ast": {
      "type": "tends_to",
      "expression": {
        "type": "subscript",
        "base": {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        },
        "subscript": {
          "type": "symbol",
          "name": "k",
          "glyph": "k"
        }
      },
      "target": {
        "type": "symbol",
        "name": "L",
        "glyph": "L"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "cd62533d2e467ddd3dd740be8d677bacd6cf093d22c2cdd638d2be0932cfbcdb"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "主families使用完整periodic law或symbolic oscillatory tail。"
    },
    {
      "id": "a2",
      "zh-Hant": "epsilon為正有理且subsequence index為affine strictly increasing。"
    }
  ],
  "symbols": [
    {
      "token": "a_k",
      "role": "convergent_subsequence",
      "role_zh": "收斂子數列",
      "scope": "representative_formula"
    },
    {
      "token": "L",
      "role": "subsequence_limit",
      "role_zh": "子數列極限",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-subsequence",
      "title_zh": "子數列",
      "reason_zh": "嚴格遞增索引。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "epsilon tail。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "bounded sequence carrier。"
    },
    {
      "id": "mko-supremum-and-infimum",
      "title_zh": "上確界與下確界",
      "reason_zh": "存在性支撐。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "whole sequence boundedness不可由prefix取代。",
      "subsequence index須infinite且strict。",
      "只保證某個convergent subsequence。"
    ],
    "common_misconception_zh": "有限觀察到重複值便宣稱一般BW theorem已證。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "exact題族只證明宣告symbolic sequences。",
    "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sequential compactness。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-bolzano-weierstrass-theorem-real.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-09 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"
  }
}