{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-pumping-lemma-regular-languages",
  "semantic_family_id": "sf-pumping-lemma-regular-languages",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "正規語言Pumping引理"
  },
  "summary": {
    "zh-Hant": "每個正規語言都有p，使每個長度至少p的成員word存在可反覆pumping且仍留在語言的分解。"
  },
  "statement": {
    "zh-Hant": "固定量詞順序exists p forall w in L with |w|≥p exists x,y,z forall i≥0，且w=xyz、|xy|≤p、|y|>0、xy^i z∈L。regular positive family由完整DFA run中的repeated-state loop提供對所有i的symbolic closure，不只測若干i。對L={a^n b^n}選w=a^p b^p，完整枚舉所有合法splits；每個y皆只含a，選i=0即離開L。單一decomposition、有限words或bounded checks不能證regular，且不得倒置對手選word與證明者選decomposition的量詞。"
  },
  "formula": {
    "tex": "P(L,p,w)=b",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>P</mi><mo>(</mo><mi>L</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mo>=</mo><mi>b</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "P",
          "glyph": "P"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "L",
            "glyph": "L"
          },
          {
            "type": "symbol",
            "name": "p",
            "glyph": "p"
          },
          {
            "type": "symbol",
            "name": "w",
            "glyph": "w"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "b",
        "glyph": "b"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6359b7649677613321d982e7d148e6b9f812f5e591781e0cdc2d92ebbe4d5e91"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "p為正整數，word與split indices exact。"
    },
    {
      "id": "a2",
      "zh-Hant": "positive witness綁完整DFA state loop及對所有i的不變式。"
    },
    {
      "id": "a3",
      "zh-Hant": "negative adversary涵蓋全部符合|xy|≤p與|y|>0的splits。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "regular_pumping_operator",
      "role_zh": "正規Pumping引理算子",
      "scope": "representative_formula"
    },
    {
      "token": "L",
      "role": "formal_language",
      "role_zh": "形式語言",
      "scope": "representative_formula"
    },
    {
      "token": "p",
      "role": "pumping_length",
      "role_zh": "pumping長度",
      "scope": "representative_formula"
    },
    {
      "token": "w",
      "role": "long_language_word",
      "role_zh": "長成員word",
      "scope": "representative_formula"
    },
    {
      "token": "b",
      "role": "pumping_property_statement",
      "role_zh": "pumping property結論",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-regular-language",
      "title_zh": "正規語言",
      "reason_zh": "適用範圍。"
    },
    {
      "id": "mko-finite-automaton",
      "title_zh": "有限自動機",
      "reason_zh": "repeated-state loop。"
    },
    {
      "id": "mko-proof",
      "title_zh": "證明",
      "reason_zh": "反證與量詞對抗。"
    },
    {
      "id": "mko-pigeonhole-principle",
      "title_zh": "鴿籠原理",
      "reason_zh": "p+1 states positions產生loop。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞與明示論域",
      "reason_zh": "正確對手順序。"
    },
    {
      "id": "mko-mathematical-induction",
      "title_zh": "數學歸納法",
      "reason_zh": "對所有pump counts保持run。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整量詞次序與split constraints先核。",
      "DFA loop證所有i，不能只測有限pump counts。",
      "a^p b^p adversary完整涵蓋全部合法splits。"
    ],
    "common_misconception_zh": "找到一個可pump分解就證regular，或只挑一個不利分解便宣稱nonregular。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "Pumping property是regular的必要條件，不是充分條件。",
    "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Myhill–Nerode與context-free pumping boundaries。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-pumping-lemma-regular-languages.md 整理；2026-10-15 脈衝 R07（使用者手動接續 2026-11-18 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}