{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-pointwise-convergence-of-functions",
  "semantic_family_id": "sf-pointwise-convergence-of-functions",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "函數列逐點收斂"
  },
  "summary": {
    "zh-Hant": "同一domain上的函數列逐點收斂，若每個固定x的值序列收斂至f(x)，且N可依x與epsilon改變。"
  },
  "statement": {
    "zh-Hant": "本輪用完整有限domain及f_n(x)=f(x)+c_x/(n+s)的symbolic tail。逐x給exact epsilon witness；有限domain可取所有N的最大值推出該有限模型的一致收斂，但不可外推無限domain。有限prefix不足。"
  },
  "formula": {
    "tex": "P(F)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "F",
          "glyph": "F"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "87eb7397007e456976bc5d2c45cac36d8a9bf96c47a5fb98cf43b31ef30c7696"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "所有f_n與f綁定同一明示有限domain/codomain。"
    },
    {
      "id": "a2",
      "zh-Hant": "epsilon為正有理且tail對所有n≥N證明。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "pointwise_function_convergence",
      "role_zh": "逐點收斂",
      "scope": "representative_formula"
    },
    {
      "token": "F",
      "role": "function_sequence",
      "role_zh": "函數列",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "n索引。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "共同domain/codomain。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "每點值序列。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "forall x forall epsilon exists N(x,epsilon)。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "N可依x與epsilon。",
      "有限domain可取有限最大witness。",
      "有限prefix不證收斂且有限domain結論不外推無限domain。"
    ],
    "common_misconception_zh": "交換量詞把pointwise直接當uniform，或由有限項判收斂。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理明示有限domain與symbolic rational tails。",
    "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 convergence與極限交換條件。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-pointwise-convergence-of-functions.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-03 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"
  }
}