{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-uniform-convergence-of-functions",
  "semantic_family_id": "sf-uniform-convergence-of-functions",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "函數列一致收斂"
  },
  "summary": {
    "zh-Hant": "一致收斂要求同一N(epsilon)同時控制domain全部x。"
  },
  "statement": {
    "zh-Hant": "使用有限domain及symbolic rational tails。逐點witness取有限最大值得該有限domain的一致witness；不可外推無限domain。另以sup gap與全稱tail核對，finite prefix不足。"
  },
  "formula": {
    "tex": "U(F)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>U</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "U",
        "glyph": "U"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "F",
          "glyph": "F"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b79690ffabf15d2a7a054c0b95bf9904e34a05a023563445f5fa3b12bedbbb95"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "共同有限domain/codomain與正有理epsilon。"
    },
    {
      "id": "a2",
      "zh-Hant": "同一N對所有x及所有n≥N。"
    }
  ],
  "symbols": [
    {
      "token": "U",
      "role": "uniform_function_convergence",
      "role_zh": "一致收斂",
      "scope": "representative_formula"
    },
    {
      "token": "F",
      "role": "function_sequence",
      "role_zh": "函數列",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-pointwise-convergence-of-functions",
      "title_zh": "逐點收斂",
      "reason_zh": "有限domain witness來源。"
    },
    {
      "id": "mko-metric-space",
      "title_zh": "度量空間",
      "reason_zh": "誤差度量。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "實值sup error。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "量詞順序。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "N不可依x。",
      "有限domain可取max pointwise witnesses。",
      "finite prefix與無限domain外推禁止。"
    ],
    "common_misconception_zh": "把每點各自N直接當同一N，或用有限測試點證無限domain一致。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "全稱證書限定明示有限domain symbolic tail。",
    "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uniform limit continuity theorem。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-uniform-convergence-of-functions.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-04 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"
  }
}