{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-sequence",
  "semantic_family_id": "sf-sequence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "數列"
  },
  "summary": {
    "zh-Hant": "本條目採從 0 起始的實數數列 a:N→R；數列項依索引排列，不能任意重排成集合。"
  },
  "statement": {
    "zh-Hant": "在 N={0,1,2,…} 的慣例下，實數數列為函數 a:N→R，第 n 項記為 a(n) 或 a_n。有限數列則明示長度 L 及索引 {0,…,L-1}。"
  },
  "formula": {
    "tex": "a:N\\to R",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>a</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>R</mi></mrow></math>",
    "semantic_ast": {
      "type": "mapping",
      "function": {
        "type": "symbol",
        "name": "a",
        "glyph": "a"
      },
      "domain": {
        "type": "symbol",
        "name": "N",
        "glyph": "N"
      },
      "codomain": {
        "type": "symbol",
        "name": "R",
        "glyph": "R"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "a563f2683c5f4c33551ed2e95b95604aa64fc7f27e80af21aad6509c35b1e4ca"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "無限數列採 N 包含 0，並把 a_0 稱為起始項。"
    },
    {
      "id": "a2",
      "zh-Hant": "有限數列的長度 L 為非負整數，只有 0≤n<L 可用；遞迴規則須附初始條件。"
    }
  ],
  "symbols": [
    {
      "token": "a",
      "role": "sequence",
      "role_zh": "實數數列",
      "scope": "representative_formula"
    },
    {
      "token": "N",
      "role": "zero_based_naturals",
      "role_zh": "包含 0 的自然數索引",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "real_codomain",
      "role_zh": "實數陪域",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "數列依索引為每個位置指定唯一項。"
    },
    {
      "id": "mko-natural-number",
      "title_zh": "自然數",
      "reason_zh": "自然數索引從 0 起始。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "本輪數列的陪域限定為實數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "數列的相同數值可以出現在不同索引；索引次序是函數資料的一部分，與無序集合不同。",
      "通項式直接指定 a_n；遞迴式以初始條件及先前項指定後續項。本轮有限計算用明示 a_n=u+dn 或 a_0=u、a_(n+1)=a_n+d。",
      "只有幾個初始項通常不能唯一推定後續規則。數列極限、單調性、無窮級數與一般遞迴存在唯一性保留為後續。"
    ],
    "common_misconception_zh": "混淆從 0 或 1 起始的索引，或僅憑一段前綴宣稱已找出唯一數列。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "指定公式与遞迴的有限項可精確重算；沒有宣稱數列收斂或无窮性質已證明。",
    "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": "綁定 Nat→Real、Fin L 有限索引與附初值遞迴的精確語義。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-sequence.md 整理；2026-09-30 脈衝 R05，英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-09-30"
  }
}