{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-asymptotic-resource-bound",
  "semantic_family_id": "sf-asymptotic-resource-bound",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "漸近資源界"
  },
  "summary": {
    "zh-Hant": "漸近上界、下界與緊界必須相對輸入大小、成本模型及全尾段量化定義。"
  },
  "statement": {
    "zh-Hant": "對非負函數 f,g:N→R，f∈O(g) 表示存在 c>0、N，使所有 n≥N 均有 f(n)≤c·g(n)；f∈Ω(g) 反向比較；f∈Θ(g) 表示兩者同時成立。輸入大小與成本模型是聲明的一部分。"
  },
  "formula": {
    "tex": "f(n)=g(n)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mrow><mi>g</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "n",
            "glyph": "n"
          }
        ]
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "g",
          "glyph": "g"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "n",
            "glyph": "n"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "121ac3768a973b4b4c757d7875398221b55faeca2911cfabc098bb622eed51c3"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "自然數輸入 n 從 0 起；題目明示函數族與非負係數。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有漸近結論都需全尾段 witness 或明示函數族定理；有限 benchmark 只作觀察。"
    }
  ],
  "symbols": [
    {
      "token": "f",
      "role": "resource_function",
      "role_zh": "待界定資源函數",
      "scope": "representative_formula"
    },
    {
      "token": "g",
      "role": "comparison_function",
      "role_zh": "比較函數",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "input_size",
      "role_zh": "輸入大小",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-natural-number",
      "title_zh": "自然數",
      "reason_zh": "輸入大小以自然數索引。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "成本函數與 witness 使用精確運算。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "上界與下界是全尾段不等式。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "資源量是由輸入大小到非負數的函數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "大 O 不是等號，也不等於只比較某個 n；常數因子與有限前綴被吸收到 witness 中。",
      "對非負係數多項式，最高非零次數決定其 Θ 次數；題目會保留全零多項式與常數邊界。",
      "不同成本模型可以給同一算法不同函數。沒有先固定基本操作、輸入表示與 size，就不能比較資源界。"
    ],
    "common_misconception_zh": "用幾個 benchmark 點證明漸近關係，或把 O、Ω、Θ 當成可互換符號。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只在明示非負單項式／多項式族建立全尾段結論；不從有限測量外推一般算法複雜度。",
    "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": "綁定標準 asymptotic predicates、eventually filter 與具 witness 的算法成本函數。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-asymptotic-resource-bound.md 整理；2026-10-13 脈衝 R06（使用者手動接續 2026-10-19 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"
  }
}