{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0061",
  "pilot_number": "R0053-27311",
  "number": 27311,
  "family_id": "qf-mod49-pump-anbn-all-splits",
  "title": {
    "zh-Hant": "第 27311 題：正規語言Pumping引理",
    "en": "Question 27311: Every Legal Split of a^p b^p Is Defeated by Pumping Down"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0061",
    "batch_id": "batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-pumping-lemma-regular-languages"
    ],
    "title_zh": "正規語言Pumping引理",
    "stem_zh": "exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={\"p\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(11, a^11b^11, [[0, 1, 10, a^10b^11, 否], [0, 2, 9, a^9b^11, 否], [0, 3, 8, a^8b^11, 否], [0, 4, 7, a^7b^11, 否], [0, 5, 6, a^6b^11, 否], [0, 6, 5, a^5b^11, 否], [0, 7, 4, a^4b^11, 否], [0, 8, 3, a^3b^11, 否], [0, 9, 2, a^2b^11, 否], [0, 10, 1, a^1b^11, 否], [0, 11, 0, a^0b^11, 否], [1, 1, 9, a^10b^11, 否], [1, 2, 8, a^9b^11, 否], [1, 3, 7, a^8b^11, 否], [1, 4, 6, a^7b^11, 否], [1, 5, 5, a^6b^11, 否], [1, 6, 4, a^5b^11, 否], [1, 7, 3, a^4b^11, 否], [1, 8, 2, a^3b^11, 否], [1, 9, 1, a^2b^11, 否], [1, 10, 0, a^1b^11, 否], [2, 1, 8, a^10b^11, 否], [2, 2, 7, a^9b^11, 否], [2, 3, 6, a^8b^11, 否], [2, 4, 5, a^7b^11, 否], [2, 5, 4, a^6b^11, 否], [2, 6, 3, a^5b^11, 否], [2, 7, 2, a^4b^11, 否], [2, 8, 1, a^3b^11, 否], [2, 9, 0, a^2b^11, 否], [3, 1, 7, a^10b^11, 否], [3, 2, 6, a^9b^11, 否], [3, 3, 5, a^8b^11, 否], [3, 4, 4, a^7b^11, 否], [3, 5, 3, a^6b^11, 否], [3, 6, 2, a^5b^11, 否], [3, 7, 1, a^4b^11, 否], [3, 8, 0, a^3b^11, 否], [4, 1, 6, a^10b^11, 否], [4, 2, 5, a^9b^11, 否], [4, 3, 4, a^8b^11, 否], [4, 4, 3, a^7b^11, 否], [4, 5, 2, a^6b^11, 否], [4, 6, 1, a^5b^11, 否], [4, 7, 0, a^4b^11, 否], [5, 1, 5, a^10b^11, 否], [5, 2, 4, a^9b^11, 否], [5, 3, 3, a^8b^11, 否], [5, 4, 2, a^7b^11, 否], [5, 5, 1, a^6b^11, 否], [5, 6, 0, a^5b^11, 否], [6, 1, 4, a^10b^11, 否], [6, 2, 3, a^9b^11, 否], [6, 3, 2, a^8b^11, 否], [6, 4, 1, a^7b^11, 否], [6, 5, 0, a^6b^11, 否], [7, 1, 3, a^10b^11, 否], [7, 2, 2, a^9b^11, 否], [7, 3, 1, a^8b^11, 否], [7, 4, 0, a^7b^11, 否], [8, 1, 2, a^10b^11, 否], [8, 2, 1, a^9b^11, 否], [8, 3, 0, a^8b^11, 否], [9, 1, 1, a^10b^11, 否], [9, 2, 0, a^9b^11, 否], [10, 1, 0, a^10b^11, 否]], 66, 66, 是, every_legal_split_is_defeated_by_i=0)",
      "explanation_zh": "Wave6第十二層核對=(11, a^11b^11, [[0, 1, 10, a^10b^11, 否], [0, 2, 9, a^9b^11, 否], [0, 3, 8, a^8b^11, 否], [0, 4, 7, a^7b^11, 否], [0, 5, 6, a^6b^11, 否], [0, 6, 5, a^5b^11, 否], [0, 7, 4, a^4b^11, 否], [0, 8, 3, a^3b^11, 否], [0, 9, 2, a^2b^11, 否], [0, 10, 1, a^1b^11, 否], [0, 11, 0, a^0b^11, 否], [1, 1, 9, a^10b^11, 否], [1, 2, 8, a^9b^11, 否], [1, 3, 7, a^8b^11, 否], [1, 4, 6, a^7b^11, 否], [1, 5, 5, a^6b^11, 否], [1, 6, 4, a^5b^11, 否], [1, 7, 3, a^4b^11, 否], [1, 8, 2, a^3b^11, 否], [1, 9, 1, a^2b^11, 否], [1, 10, 0, a^1b^11, 否], [2, 1, 8, a^10b^11, 否], [2, 2, 7, a^9b^11, 否], [2, 3, 6, a^8b^11, 否], [2, 4, 5, a^7b^11, 否], [2, 5, 4, a^6b^11, 否], [2, 6, 3, a^5b^11, 否], [2, 7, 2, a^4b^11, 否], [2, 8, 1, a^3b^11, 否], [2, 9, 0, a^2b^11, 否], [3, 1, 7, a^10b^11, 否], [3, 2, 6, a^9b^11, 否], [3, 3, 5, a^8b^11, 否], [3, 4, 4, a^7b^11, 否], [3, 5, 3, a^6b^11, 否], [3, 6, 2, a^5b^11, 否], [3, 7, 1, a^4b^11, 否], [3, 8, 0, a^3b^11, 否], [4, 1, 6, a^10b^11, 否], [4, 2, 5, a^9b^11, 否], [4, 3, 4, a^8b^11, 否], [4, 4, 3, a^7b^11, 否], [4, 5, 2, a^6b^11, 否], [4, 6, 1, a^5b^11, 否], [4, 7, 0, a^4b^11, 否], [5, 1, 5, a^10b^11, 否], [5, 2, 4, a^9b^11, 否], [5, 3, 3, a^8b^11, 否], [5, 4, 2, a^7b^11, 否], [5, 5, 1, a^6b^11, 否], [5, 6, 0, a^5b^11, 否], [6, 1, 4, a^10b^11, 否], [6, 2, 3, a^9b^11, 否], [6, 3, 2, a^8b^11, 否], [6, 4, 1, a^7b^11, 否], [6, 5, 0, a^6b^11, 否], [7, 1, 3, a^10b^11, 否], [7, 2, 2, a^9b^11, 否], [7, 3, 1, a^8b^11, 否], [7, 4, 0, a^7b^11, 否], [8, 1, 2, a^10b^11, 否], [8, 2, 1, a^9b^11, 否], [8, 3, 0, a^8b^11, 否], [9, 1, 1, a^10b^11, 否], [9, 2, 0, a^9b^11, 否], [10, 1, 0, a^10b^11, 否]], 66, 66, 是, every_legal_split_is_defeated_by_i=0)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "正規語言Pumping引理"
      ]
    },
    "variant": {
      "family_id": "qf-mod49-pump-anbn-all-splits",
      "seed": 202611187061,
      "parameters": {
        "p": 11,
        "domain": "Exact_regular_pumping_lemma",
        "operation": "pump-anbn-all-splits",
        "semantic_template": "pump-anbn-all-splits|{p:11}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0061",
    "batch_id": "batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "33ccbdd10927719a77028b365df41173e0c7253f2ce72232edba2c9cf6197b2f",
      "family_id": "qf-mod49-pump-anbn-all-splits",
      "seed": 202611187061,
      "semantic_template": "pump-anbn-all-splits|{p:11}"
    },
    "fields": {
      "title": "Every Legal Split of a^p b^p Is Defeated by Pumping Down",
      "stem": "Use positive pumping length p=11 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).",
      "answer": {
        "kind": "expression",
        "value": "(11, a^11b^11, [[0, 1, 10, a^10b^11, No], [0, 2, 9, a^9b^11, No], [0, 3, 8, a^8b^11, No], [0, 4, 7, a^7b^11, No], [0, 5, 6, a^6b^11, No], [0, 6, 5, a^5b^11, No], [0, 7, 4, a^4b^11, No], [0, 8, 3, a^3b^11, No], [0, 9, 2, a^2b^11, No], [0, 10, 1, a^1b^11, No], [0, 11, 0, a^0b^11, No], [1, 1, 9, a^10b^11, No], [1, 2, 8, a^9b^11, No], [1, 3, 7, a^8b^11, No], [1, 4, 6, a^7b^11, No], [1, 5, 5, a^6b^11, No], [1, 6, 4, a^5b^11, No], [1, 7, 3, a^4b^11, No], [1, 8, 2, a^3b^11, No], [1, 9, 1, a^2b^11, No], [1, 10, 0, a^1b^11, No], [2, 1, 8, a^10b^11, No], [2, 2, 7, a^9b^11, No], [2, 3, 6, a^8b^11, No], [2, 4, 5, a^7b^11, No], [2, 5, 4, a^6b^11, No], [2, 6, 3, a^5b^11, No], [2, 7, 2, a^4b^11, No], [2, 8, 1, a^3b^11, No], [2, 9, 0, a^2b^11, No], [3, 1, 7, a^10b^11, No], [3, 2, 6, a^9b^11, No], [3, 3, 5, a^8b^11, No], [3, 4, 4, a^7b^11, No], [3, 5, 3, a^6b^11, No], [3, 6, 2, a^5b^11, No], [3, 7, 1, a^4b^11, No], [3, 8, 0, a^3b^11, No], [4, 1, 6, a^10b^11, No], [4, 2, 5, a^9b^11, No], [4, 3, 4, a^8b^11, No], [4, 4, 3, a^7b^11, No], [4, 5, 2, a^6b^11, No], [4, 6, 1, a^5b^11, No], [4, 7, 0, a^4b^11, No], [5, 1, 5, a^10b^11, No], [5, 2, 4, a^9b^11, No], [5, 3, 3, a^8b^11, No], [5, 4, 2, a^7b^11, No], [5, 5, 1, a^6b^11, No], [5, 6, 0, a^5b^11, No], [6, 1, 4, a^10b^11, No], [6, 2, 3, a^9b^11, No], [6, 3, 2, a^8b^11, No], [6, 4, 1, a^7b^11, No], [6, 5, 0, a^6b^11, No], [7, 1, 3, a^10b^11, No], [7, 2, 2, a^9b^11, No], [7, 3, 1, a^8b^11, No], [7, 4, 0, a^7b^11, No], [8, 1, 2, a^10b^11, No], [8, 2, 1, a^9b^11, No], [8, 3, 0, a^8b^11, No], [9, 1, 1, a^10b^11, No], [9, 2, 0, a^9b^11, No], [10, 1, 0, a^10b^11, No]], 66, 66, Yes, every_legal_split_is_defeated_by_i=0)",
        "explanation": "Cover every x≥0,y≥1 with x+y≤p. Each y contains only a symbols; pumping with i=0 leaves fewer a symbols than b symbols. The split-row third entry is the remaining a count p−x−y, not the full z length, because z also contains all p b symbols. A fixed-p table supports this adversary pattern; the general nonregularity argument quantifies over every proposed p. The exact result, retaining the requested order, is (11, a^11b^11, [[0, 1, 10, a^10b^11, No], [0, 2, 9, a^9b^11, No], [0, 3, 8, a^8b^11, No], [0, 4, 7, a^7b^11, No], [0, 5, 6, a^6b^11, No], [0, 6, 5, a^5b^11, No], [0, 7, 4, a^4b^11, No], [0, 8, 3, a^3b^11, No], [0, 9, 2, a^2b^11, No], [0, 10, 1, a^1b^11, No], [0, 11, 0, a^0b^11, No], [1, 1, 9, a^10b^11, No], [1, 2, 8, a^9b^11, No], [1, 3, 7, a^8b^11, No], [1, 4, 6, a^7b^11, No], [1, 5, 5, a^6b^11, No], [1, 6, 4, a^5b^11, No], [1, 7, 3, a^4b^11, No], [1, 8, 2, a^3b^11, No], [1, 9, 1, a^2b^11, No], [1, 10, 0, a^1b^11, No], [2, 1, 8, a^10b^11, No], [2, 2, 7, a^9b^11, No], [2, 3, 6, a^8b^11, No], [2, 4, 5, a^7b^11, No], [2, 5, 4, a^6b^11, No], [2, 6, 3, a^5b^11, No], [2, 7, 2, a^4b^11, No], [2, 8, 1, a^3b^11, No], [2, 9, 0, a^2b^11, No], [3, 1, 7, a^10b^11, No], [3, 2, 6, a^9b^11, No], [3, 3, 5, a^8b^11, No], [3, 4, 4, a^7b^11, No], [3, 5, 3, a^6b^11, No], [3, 6, 2, a^5b^11, No], [3, 7, 1, a^4b^11, No], [3, 8, 0, a^3b^11, No], [4, 1, 6, a^10b^11, No], [4, 2, 5, a^9b^11, No], [4, 3, 4, a^8b^11, No], [4, 4, 3, a^7b^11, No], [4, 5, 2, a^6b^11, No], [4, 6, 1, a^5b^11, No], [4, 7, 0, a^4b^11, No], [5, 1, 5, a^10b^11, No], [5, 2, 4, a^9b^11, No], [5, 3, 3, a^8b^11, No], [5, 4, 2, a^7b^11, No], [5, 5, 1, a^6b^11, No], [5, 6, 0, a^5b^11, No], [6, 1, 4, a^10b^11, No], [6, 2, 3, a^9b^11, No], [6, 3, 2, a^8b^11, No], [6, 4, 1, a^7b^11, No], [6, 5, 0, a^6b^11, No], [7, 1, 3, a^10b^11, No], [7, 2, 2, a^9b^11, No], [7, 3, 1, a^8b^11, No], [7, 4, 0, a^7b^11, No], [8, 1, 2, a^10b^11, No], [8, 2, 1, a^9b^11, No], [8, 3, 0, a^8b^11, No], [9, 1, 1, a^10b^11, No], [9, 2, 0, a^9b^11, No], [10, 1, 0, a^10b^11, No]], 66, 66, Yes, every_legal_split_is_defeated_by_i=0)."
      },
      "skills": [
        "Every Legal Split of a^p b^p Is Defeated by Pumping Down"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "exists p forall w exists split forall i量詞與split constraints責任。操作=pump-anbn-all-splits；資料={\"p\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use positive pumping length p=11 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [pump-anbn-all-splits]",
      "after": "derived = [[p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。",
        "en": "List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [11]",
      "after": "derived = [a^11b^11, 22, ALL x>=0,y>=1,x+y<=p]",
      "relation": "implies",
      "rule_id": "Pumping_adversary_word_and_ALL_legal_split_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "假設regular可取得p；adversary選a^p b^p，長2p>=p。任意合法split的xy都在前p個a中，故y只有a且長度正；不是先挑一個方便的split。",
        "en": "Under assumed regularity take its pumping length p and choose a^p b^p of length2p>=p.Any legal xy lies in the initial p a-symbols,so y consists of a symbols and is nonempty.Do not preselect a convenient split."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split0,1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [11, 0, 1]",
      "after": "derived = [10, 21, [0, 1, 10, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split0,2"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [11, 0, 2]",
      "after": "derived = [9, 20, [0, 2, 9, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,3 := sub(11, 3)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,3 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,3，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "3"
        ],
        "output": "8",
        "label": "Pumping ORIGINAL remaining a after i0 split0,3"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [11, 0, 3]",
      "after": "derived = [8, 19, [0, 3, 8, a^8b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,4 := sub(11, 4)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,4 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,4，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "4"
        ],
        "output": "7",
        "label": "Pumping ORIGINAL remaining a after i0 split0,4"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [11, 0, 4]",
      "after": "derived = [7, 18, [0, 4, 7, a^7b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,5 := sub(11, 5)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,5 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,5，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "5"
        ],
        "output": "6",
        "label": "Pumping ORIGINAL remaining a after i0 split0,5"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [11, 0, 5]",
      "after": "derived = [6, 17, [0, 5, 6, a^6b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,6 := sub(11, 6)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,6 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,6，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "6"
        ],
        "output": "5",
        "label": "Pumping ORIGINAL remaining a after i0 split0,6"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [11, 0, 6]",
      "after": "derived = [5, 16, [0, 6, 5, a^5b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,7 := sub(11, 7)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,7 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,7，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "7"
        ],
        "output": "4",
        "label": "Pumping ORIGINAL remaining a after i0 split0,7"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "source roles = [11, 0, 7]",
      "after": "derived = [4, 15, [0, 7, 4, a^4b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,8 := sub(11, 8)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,8 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,8，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "8"
        ],
        "output": "3",
        "label": "Pumping ORIGINAL remaining a after i0 split0,8"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [11, 0, 8]",
      "after": "derived = [3, 14, [0, 8, 3, a^3b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,9 := sub(11, 9)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,9 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,9，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "9"
        ],
        "output": "2",
        "label": "Pumping ORIGINAL remaining a after i0 split0,9"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "source roles = [11, 0, 9]",
      "after": "derived = [2, 13, [0, 9, 2, a^2b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,10 := sub(11, 10)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,10 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,10，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,10 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "10"
        ],
        "output": "1",
        "label": "Pumping ORIGINAL remaining a after i0 split0,10"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "source roles = [11, 0, 10]",
      "after": "derived = [1, 12, [0, 10, 1, a^1b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split0,11 := sub(11, 11)",
      "after": "Pumping ORIGINAL remaining a after i0 split0,11 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split0,11，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split0,11 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "11"
        ],
        "output": "0",
        "label": "Pumping ORIGINAL remaining a after i0 split0,11"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "source roles = [11, 0, 11]",
      "after": "derived = [0, 11, [0, 11, 0, a^0b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split1,1"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "source roles = [11, 1, 1]",
      "after": "derived = [9, 20, [1, 1, 9, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split1,2"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "source roles = [11, 1, 2]",
      "after": "derived = [8, 19, [1, 2, 8, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,3 := sub(11, 3)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,3 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,3，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "3"
        ],
        "output": "8",
        "label": "Pumping ORIGINAL remaining a after i0 split1,3"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "source roles = [11, 1, 3]",
      "after": "derived = [7, 18, [1, 3, 7, a^8b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,4 := sub(11, 4)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,4 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,4，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "4"
        ],
        "output": "7",
        "label": "Pumping ORIGINAL remaining a after i0 split1,4"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "source roles = [11, 1, 4]",
      "after": "derived = [6, 17, [1, 4, 6, a^7b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,5 := sub(11, 5)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,5 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,5，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "5"
        ],
        "output": "6",
        "label": "Pumping ORIGINAL remaining a after i0 split1,5"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "source roles = [11, 1, 5]",
      "after": "derived = [5, 16, [1, 5, 5, a^6b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,6 := sub(11, 6)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,6 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,6，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "6"
        ],
        "output": "5",
        "label": "Pumping ORIGINAL remaining a after i0 split1,6"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "source roles = [11, 1, 6]",
      "after": "derived = [4, 15, [1, 6, 4, a^5b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,7 := sub(11, 7)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,7 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,7，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "7"
        ],
        "output": "4",
        "label": "Pumping ORIGINAL remaining a after i0 split1,7"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "source roles = [11, 1, 7]",
      "after": "derived = [3, 14, [1, 7, 3, a^4b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,8 := sub(11, 8)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,8 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,8，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "8"
        ],
        "output": "3",
        "label": "Pumping ORIGINAL remaining a after i0 split1,8"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "source roles = [11, 1, 8]",
      "after": "derived = [2, 13, [1, 8, 2, a^3b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,9 := sub(11, 9)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,9 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,9，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "9"
        ],
        "output": "2",
        "label": "Pumping ORIGINAL remaining a after i0 split1,9"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "source roles = [11, 1, 9]",
      "after": "derived = [1, 12, [1, 9, 1, a^2b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split1,10 := sub(11, 10)",
      "after": "Pumping ORIGINAL remaining a after i0 split1,10 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split1,10，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split1,10 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "10"
        ],
        "output": "1",
        "label": "Pumping ORIGINAL remaining a after i0 split1,10"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "source roles = [11, 1, 10]",
      "after": "derived = [0, 11, [1, 10, 0, a^1b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split2,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split2,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split2,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split2,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split2,1"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "source roles = [11, 2, 1]",
      "after": "derived = [8, 19, [2, 1, 8, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split2,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split2,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split2,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split2,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split2,2"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "source roles = [11, 2, 2]",
      "after": "derived = [7, 18, [2, 2, 7, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split2,3 := sub(11, 3)",
      "after": "Pumping ORIGINAL remaining a after i0 split2,3 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split2,3，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split2,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "3"
        ],
        "output": "8",
        "label": "Pumping ORIGINAL remaining a after i0 split2,3"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "source roles = [11, 2, 3]",
      "after": "derived = [6, 17, [2, 3, 6, a^8b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split2,4 := sub(11, 4)",
      "after": "Pumping ORIGINAL remaining a after i0 split2,4 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split2,4，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split2,4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "4"
        ],
        "output": "7",
        "label": "Pumping ORIGINAL remaining a after i0 split2,4"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "source roles = [11, 2, 4]",
      "after": "derived = [5, 16, [2, 4, 5, a^7b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split2,5 := sub(11, 5)",
      "after": "Pumping ORIGINAL remaining a after i0 split2,5 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split2,5，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split2,5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "5"
        ],
        "output": "6",
        "label": "Pumping ORIGINAL remaining a after i0 split2,5"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "source roles = [11, 2, 5]",
      "after": "derived = [4, 15, [2, 5, 4, a^6b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split2,6 := sub(11, 6)",
      "after": "Pumping ORIGINAL remaining a after i0 split2,6 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split2,6，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split2,6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "6"
        ],
        "output": "5",
        "label": "Pumping ORIGINAL remaining a after i0 split2,6"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "source roles = [11, 2, 6]",
      "after": "derived = [3, 14, [2, 6, 3, a^5b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split2,7 := sub(11, 7)",
      "after": "Pumping ORIGINAL remaining a after i0 split2,7 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split2,7，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split2,7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "7"
        ],
        "output": "4",
        "label": "Pumping ORIGINAL remaining a after i0 split2,7"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "source roles = [11, 2, 7]",
      "after": "derived = [2, 13, [2, 7, 2, a^4b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split2,8 := sub(11, 8)",
      "after": "Pumping ORIGINAL remaining a after i0 split2,8 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split2,8，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split2,8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "8"
        ],
        "output": "3",
        "label": "Pumping ORIGINAL remaining a after i0 split2,8"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "source roles = [11, 2, 8]",
      "after": "derived = [1, 12, [2, 8, 1, a^3b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split2,9 := sub(11, 9)",
      "after": "Pumping ORIGINAL remaining a after i0 split2,9 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split2,9，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split2,9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "9"
        ],
        "output": "2",
        "label": "Pumping ORIGINAL remaining a after i0 split2,9"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "source roles = [11, 2, 9]",
      "after": "derived = [0, 11, [2, 9, 0, a^2b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split3,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split3,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split3,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split3,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split3,1"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "source roles = [11, 3, 1]",
      "after": "derived = [7, 18, [3, 1, 7, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split3,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split3,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split3,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split3,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split3,2"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "source roles = [11, 3, 2]",
      "after": "derived = [6, 17, [3, 2, 6, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split3,3 := sub(11, 3)",
      "after": "Pumping ORIGINAL remaining a after i0 split3,3 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split3,3，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split3,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "3"
        ],
        "output": "8",
        "label": "Pumping ORIGINAL remaining a after i0 split3,3"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "source roles = [11, 3, 3]",
      "after": "derived = [5, 16, [3, 3, 5, a^8b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split3,4 := sub(11, 4)",
      "after": "Pumping ORIGINAL remaining a after i0 split3,4 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split3,4，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split3,4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "4"
        ],
        "output": "7",
        "label": "Pumping ORIGINAL remaining a after i0 split3,4"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "source roles = [11, 3, 4]",
      "after": "derived = [4, 15, [3, 4, 4, a^7b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split3,5 := sub(11, 5)",
      "after": "Pumping ORIGINAL remaining a after i0 split3,5 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split3,5，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split3,5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "5"
        ],
        "output": "6",
        "label": "Pumping ORIGINAL remaining a after i0 split3,5"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "source roles = [11, 3, 5]",
      "after": "derived = [3, 14, [3, 5, 3, a^6b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split3,6 := sub(11, 6)",
      "after": "Pumping ORIGINAL remaining a after i0 split3,6 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split3,6，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split3,6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "6"
        ],
        "output": "5",
        "label": "Pumping ORIGINAL remaining a after i0 split3,6"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "source roles = [11, 3, 6]",
      "after": "derived = [2, 13, [3, 6, 2, a^5b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split3,7 := sub(11, 7)",
      "after": "Pumping ORIGINAL remaining a after i0 split3,7 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split3,7，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split3,7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "7"
        ],
        "output": "4",
        "label": "Pumping ORIGINAL remaining a after i0 split3,7"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "source roles = [11, 3, 7]",
      "after": "derived = [1, 12, [3, 7, 1, a^4b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split3,8 := sub(11, 8)",
      "after": "Pumping ORIGINAL remaining a after i0 split3,8 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split3,8，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split3,8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "8"
        ],
        "output": "3",
        "label": "Pumping ORIGINAL remaining a after i0 split3,8"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "source roles = [11, 3, 8]",
      "after": "derived = [0, 11, [3, 8, 0, a^3b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split4,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split4,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split4,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split4,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split4,1"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "source roles = [11, 4, 1]",
      "after": "derived = [6, 17, [4, 1, 6, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split4,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split4,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split4,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split4,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split4,2"
      }
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "source roles = [11, 4, 2]",
      "after": "derived = [5, 16, [4, 2, 5, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split4,3 := sub(11, 3)",
      "after": "Pumping ORIGINAL remaining a after i0 split4,3 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split4,3，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split4,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "3"
        ],
        "output": "8",
        "label": "Pumping ORIGINAL remaining a after i0 split4,3"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "source roles = [11, 4, 3]",
      "after": "derived = [4, 15, [4, 3, 4, a^8b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split4,4 := sub(11, 4)",
      "after": "Pumping ORIGINAL remaining a after i0 split4,4 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split4,4，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split4,4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "4"
        ],
        "output": "7",
        "label": "Pumping ORIGINAL remaining a after i0 split4,4"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "source roles = [11, 4, 4]",
      "after": "derived = [3, 14, [4, 4, 3, a^7b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split4,5 := sub(11, 5)",
      "after": "Pumping ORIGINAL remaining a after i0 split4,5 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split4,5，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split4,5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "5"
        ],
        "output": "6",
        "label": "Pumping ORIGINAL remaining a after i0 split4,5"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "source roles = [11, 4, 5]",
      "after": "derived = [2, 13, [4, 5, 2, a^6b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split4,6 := sub(11, 6)",
      "after": "Pumping ORIGINAL remaining a after i0 split4,6 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split4,6，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split4,6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "6"
        ],
        "output": "5",
        "label": "Pumping ORIGINAL remaining a after i0 split4,6"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "source roles = [11, 4, 6]",
      "after": "derived = [1, 12, [4, 6, 1, a^5b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split4,7 := sub(11, 7)",
      "after": "Pumping ORIGINAL remaining a after i0 split4,7 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split4,7，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split4,7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "7"
        ],
        "output": "4",
        "label": "Pumping ORIGINAL remaining a after i0 split4,7"
      }
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "source roles = [11, 4, 7]",
      "after": "derived = [0, 11, [4, 7, 0, a^4b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split5,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split5,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split5,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split5,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split5,1"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "source roles = [11, 5, 1]",
      "after": "derived = [5, 16, [5, 1, 5, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split5,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split5,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split5,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split5,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split5,2"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "source roles = [11, 5, 2]",
      "after": "derived = [4, 15, [5, 2, 4, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split5,3 := sub(11, 3)",
      "after": "Pumping ORIGINAL remaining a after i0 split5,3 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split5,3，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split5,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "3"
        ],
        "output": "8",
        "label": "Pumping ORIGINAL remaining a after i0 split5,3"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "source roles = [11, 5, 3]",
      "after": "derived = [3, 14, [5, 3, 3, a^8b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split5,4 := sub(11, 4)",
      "after": "Pumping ORIGINAL remaining a after i0 split5,4 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split5,4，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split5,4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "4"
        ],
        "output": "7",
        "label": "Pumping ORIGINAL remaining a after i0 split5,4"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "source roles = [11, 5, 4]",
      "after": "derived = [2, 13, [5, 4, 2, a^7b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split5,5 := sub(11, 5)",
      "after": "Pumping ORIGINAL remaining a after i0 split5,5 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split5,5，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split5,5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "5"
        ],
        "output": "6",
        "label": "Pumping ORIGINAL remaining a after i0 split5,5"
      }
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "source roles = [11, 5, 5]",
      "after": "derived = [1, 12, [5, 5, 1, a^6b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split5,6 := sub(11, 6)",
      "after": "Pumping ORIGINAL remaining a after i0 split5,6 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split5,6，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split5,6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "6"
        ],
        "output": "5",
        "label": "Pumping ORIGINAL remaining a after i0 split5,6"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "source roles = [11, 5, 6]",
      "after": "derived = [0, 11, [5, 6, 0, a^5b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split6,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split6,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split6,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split6,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split6,1"
      }
    },
    {
      "id": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "source roles = [11, 6, 1]",
      "after": "derived = [4, 15, [6, 1, 4, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split6,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split6,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split6,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split6,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split6,2"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "source roles = [11, 6, 2]",
      "after": "derived = [3, 14, [6, 2, 3, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split6,3 := sub(11, 3)",
      "after": "Pumping ORIGINAL remaining a after i0 split6,3 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split6,3，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split6,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "3"
        ],
        "output": "8",
        "label": "Pumping ORIGINAL remaining a after i0 split6,3"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "source roles = [11, 6, 3]",
      "after": "derived = [2, 13, [6, 3, 2, a^8b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split6,4 := sub(11, 4)",
      "after": "Pumping ORIGINAL remaining a after i0 split6,4 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split6,4，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split6,4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "4"
        ],
        "output": "7",
        "label": "Pumping ORIGINAL remaining a after i0 split6,4"
      }
    },
    {
      "id": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "source roles = [11, 6, 4]",
      "after": "derived = [1, 12, [6, 4, 1, a^7b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split6,5 := sub(11, 5)",
      "after": "Pumping ORIGINAL remaining a after i0 split6,5 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split6,5，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split6,5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "5"
        ],
        "output": "6",
        "label": "Pumping ORIGINAL remaining a after i0 split6,5"
      }
    },
    {
      "id": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "source roles = [11, 6, 5]",
      "after": "derived = [0, 11, [6, 5, 0, a^6b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split7,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split7,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split7,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split7,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split7,1"
      }
    },
    {
      "id": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "source roles = [11, 7, 1]",
      "after": "derived = [3, 14, [7, 1, 3, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split7,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split7,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split7,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split7,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split7,2"
      }
    },
    {
      "id": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "source roles = [11, 7, 2]",
      "after": "derived = [2, 13, [7, 2, 2, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split7,3 := sub(11, 3)",
      "after": "Pumping ORIGINAL remaining a after i0 split7,3 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split7,3，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split7,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "3"
        ],
        "output": "8",
        "label": "Pumping ORIGINAL remaining a after i0 split7,3"
      }
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "source roles = [11, 7, 3]",
      "after": "derived = [1, 12, [7, 3, 1, a^8b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split7,4 := sub(11, 4)",
      "after": "Pumping ORIGINAL remaining a after i0 split7,4 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split7,4，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split7,4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "4"
        ],
        "output": "7",
        "label": "Pumping ORIGINAL remaining a after i0 split7,4"
      }
    },
    {
      "id": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "source roles = [11, 7, 4]",
      "after": "derived = [0, 11, [7, 4, 0, a^7b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split8,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split8,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split8,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split8,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split8,1"
      }
    },
    {
      "id": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "source roles = [11, 8, 1]",
      "after": "derived = [2, 13, [8, 1, 2, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split8,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split8,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split8,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split8,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split8,2"
      }
    },
    {
      "id": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "source roles = [11, 8, 2]",
      "after": "derived = [1, 12, [8, 2, 1, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split8,3 := sub(11, 3)",
      "after": "Pumping ORIGINAL remaining a after i0 split8,3 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split8,3，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split8,3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "3"
        ],
        "output": "8",
        "label": "Pumping ORIGINAL remaining a after i0 split8,3"
      }
    },
    {
      "id": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "source roles = [11, 8, 3]",
      "after": "derived = [0, 11, [8, 3, 0, a^8b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split9,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split9,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split9,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split9,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split9,1"
      }
    },
    {
      "id": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "source roles = [11, 9, 1]",
      "after": "derived = [1, 12, [9, 1, 1, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split9,2 := sub(11, 2)",
      "after": "Pumping ORIGINAL remaining a after i0 split9,2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split9,2，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split9,2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "2"
        ],
        "output": "9",
        "label": "Pumping ORIGINAL remaining a after i0 split9,2"
      }
    },
    {
      "id": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "source roles = [11, 9, 2]",
      "after": "derived = [0, 11, [9, 2, 0, a^9b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "Pumping ORIGINAL remaining a after i0 split10,1 := sub(11, 1)",
      "after": "Pumping ORIGINAL remaining a after i0 split10,1 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL remaining a after i0 split10,1，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL remaining a after i0 split10,1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "11",
          "1"
        ],
        "output": "10",
        "label": "Pumping ORIGINAL remaining a after i0 split10,1"
      }
    },
    {
      "id": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "source roles = [11, 10, 1]",
      "after": "derived = [0, 11, [10, 1, 0, a^10b^11, 否], 是]",
      "relation": "implies",
      "rule_id": "Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第三欄p-x-y只計z中的剩餘a，完整|z|尚加p個b。i=0刪掉正長y，a數p-y<p而b數仍p，所以這個合法split的pump-down不在語言。",
        "en": "The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "Pumping ORIGINAL triangular numerator := mul(11, 12)",
      "after": "Pumping ORIGINAL triangular numerator = 132",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL triangular numerator，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL triangular numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "12"
        ],
        "output": "132",
        "label": "Pumping ORIGINAL triangular numerator"
      }
    },
    {
      "id": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "Pumping ORIGINAL triangular all-splits count := div(132, 2)",
      "after": "Pumping ORIGINAL triangular all-splits count = 66",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Pumping ORIGINAL triangular all-splits count，不以小數近似替代。",
          "en": "Recompute Pumping ORIGINAL triangular all-splits count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "132",
          "2"
        ],
        "output": "66",
        "label": "Pumping ORIGINAL triangular all-splits count"
      }
    },
    {
      "id": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "source roles = [11, [[0, 1, 10, a^10b^11, 否], [0, 2, 9, a^9b^11, 否], [0, 3, 8, a^8b^11, 否], [0, 4, 7, a^7b^11, 否], [0, 5, 6, a^6b^11, 否], [0, 6, 5, a^5b^11, 否], [0, 7, 4, a^4b^11, 否], [0, 8, 3, a^3b^11, 否], [0, 9, 2, a^2b^11, 否], [0, 10, 1, a^1b^11, 否], [0, 11, 0, a^0b^11, 否], [1, 1, 9, a^10b^11, 否], [1, 2, 8, a^9b^11, 否], [1, 3, 7, a^8b^11, 否], [1, 4, 6, a^7b^11, 否], [1, 5, 5, a^6b^11, 否], [1, 6, 4, a^5b^11, 否], [1, 7, 3, a^4b^11, 否], [1, 8, 2, a^3b^11, 否], [1, 9, 1, a^2b^11, 否], [1, 10, 0, a^1b^11, 否], [2, 1, 8, a^10b^11, 否], [2, 2, 7, a^9b^11, 否], [2, 3, 6, a^8b^11, 否], [2, 4, 5, a^7b^11, 否], [2, 5, 4, a^6b^11, 否], [2, 6, 3, a^5b^11, 否], [2, 7, 2, a^4b^11, 否], [2, 8, 1, a^3b^11, 否], [2, 9, 0, a^2b^11, 否], [3, 1, 7, a^10b^11, 否], [3, 2, 6, a^9b^11, 否], [3, 3, 5, a^8b^11, 否], [3, 4, 4, a^7b^11, 否], [3, 5, 3, a^6b^11, 否], [3, 6, 2, a^5b^11, 否], [3, 7, 1, a^4b^11, 否], [3, 8, 0, a^3b^11, 否], [4, 1, 6, a^10b^11, 否], [4, 2, 5, a^9b^11, 否], [4, 3, 4, a^8b^11, 否], [4, 4, 3, a^7b^11, 否], [4, 5, 2, a^6b^11, 否], [4, 6, 1, a^5b^11, 否], [4, 7, 0, a^4b^11, 否], [5, 1, 5, a^10b^11, 否], [5, 2, 4, a^9b^11, 否], [5, 3, 3, a^8b^11, 否], [5, 4, 2, a^7b^11, 否], [5, 5, 1, a^6b^11, 否], [5, 6, 0, a^5b^11, 否], [6, 1, 4, a^10b^11, 否], [6, 2, 3, a^9b^11, 否], [6, 3, 2, a^8b^11, 否], [6, 4, 1, a^7b^11, 否], [6, 5, 0, a^6b^11, 否], [7, 1, 3, a^10b^11, 否], [7, 2, 2, a^9b^11, 否], [7, 3, 1, a^8b^11, 否], [7, 4, 0, a^7b^11, 否], [8, 1, 2, a^10b^11, 否], [8, 2, 1, a^9b^11, 否], [8, 3, 0, a^8b^11, 否], [9, 1, 1, a^10b^11, 否], [9, 2, 0, a^9b^11, 否], [10, 1, 0, a^10b^11, 否]]]",
      "after": "derived = [66, 66, 是]",
      "relation": "implies",
      "rule_id": "Pumping_ALL_splits_and_GENERAL_every_proposed_p_contradiction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個x有p-x個y，總數p+(p-1)+...+1=p(p+1)/2，表未遺漏任何合法split。一般論證對任意proposed p重複相同理由，故forall split exists i0反駁pumping結論；不把僅一個固定p表冒充全部證明。",
        "en": "For each x there are p-x choices of y,total p(p+1)/2,so every legal split is included.The general argument repeats for ANY proposed p,with i=0 defeating every split.A table at one fixed p alone is not the general proof."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      11,
      "a^11b^11",
      [
        [
          0,
          1,
          10,
          "a^10b^11",
          false
        ],
        [
          0,
          2,
          9,
          "a^9b^11",
          false
        ],
        [
          0,
          3,
          8,
          "a^8b^11",
          false
        ],
        [
          0,
          4,
          7,
          "a^7b^11",
          false
        ],
        [
          0,
          5,
          6,
          "a^6b^11",
          false
        ],
        [
          0,
          6,
          5,
          "a^5b^11",
          false
        ],
        [
          0,
          7,
          4,
          "a^4b^11",
          false
        ],
        [
          0,
          8,
          3,
          "a^3b^11",
          false
        ],
        [
          0,
          9,
          2,
          "a^2b^11",
          false
        ],
        [
          0,
          10,
          1,
          "a^1b^11",
          false
        ],
        [
          0,
          11,
          0,
          "a^0b^11",
          false
        ],
        [
          1,
          1,
          9,
          "a^10b^11",
          false
        ],
        [
          1,
          2,
          8,
          "a^9b^11",
          false
        ],
        [
          1,
          3,
          7,
          "a^8b^11",
          false
        ],
        [
          1,
          4,
          6,
          "a^7b^11",
          false
        ],
        [
          1,
          5,
          5,
          "a^6b^11",
          false
        ],
        [
          1,
          6,
          4,
          "a^5b^11",
          false
        ],
        [
          1,
          7,
          3,
          "a^4b^11",
          false
        ],
        [
          1,
          8,
          2,
          "a^3b^11",
          false
        ],
        [
          1,
          9,
          1,
          "a^2b^11",
          false
        ],
        [
          1,
          10,
          0,
          "a^1b^11",
          false
        ],
        [
          2,
          1,
          8,
          "a^10b^11",
          false
        ],
        [
          2,
          2,
          7,
          "a^9b^11",
          false
        ],
        [
          2,
          3,
          6,
          "a^8b^11",
          false
        ],
        [
          2,
          4,
          5,
          "a^7b^11",
          false
        ],
        [
          2,
          5,
          4,
          "a^6b^11",
          false
        ],
        [
          2,
          6,
          3,
          "a^5b^11",
          false
        ],
        [
          2,
          7,
          2,
          "a^4b^11",
          false
        ],
        [
          2,
          8,
          1,
          "a^3b^11",
          false
        ],
        [
          2,
          9,
          0,
          "a^2b^11",
          false
        ],
        [
          3,
          1,
          7,
          "a^10b^11",
          false
        ],
        [
          3,
          2,
          6,
          "a^9b^11",
          false
        ],
        [
          3,
          3,
          5,
          "a^8b^11",
          false
        ],
        [
          3,
          4,
          4,
          "a^7b^11",
          false
        ],
        [
          3,
          5,
          3,
          "a^6b^11",
          false
        ],
        [
          3,
          6,
          2,
          "a^5b^11",
          false
        ],
        [
          3,
          7,
          1,
          "a^4b^11",
          false
        ],
        [
          3,
          8,
          0,
          "a^3b^11",
          false
        ],
        [
          4,
          1,
          6,
          "a^10b^11",
          false
        ],
        [
          4,
          2,
          5,
          "a^9b^11",
          false
        ],
        [
          4,
          3,
          4,
          "a^8b^11",
          false
        ],
        [
          4,
          4,
          3,
          "a^7b^11",
          false
        ],
        [
          4,
          5,
          2,
          "a^6b^11",
          false
        ],
        [
          4,
          6,
          1,
          "a^5b^11",
          false
        ],
        [
          4,
          7,
          0,
          "a^4b^11",
          false
        ],
        [
          5,
          1,
          5,
          "a^10b^11",
          false
        ],
        [
          5,
          2,
          4,
          "a^9b^11",
          false
        ],
        [
          5,
          3,
          3,
          "a^8b^11",
          false
        ],
        [
          5,
          4,
          2,
          "a^7b^11",
          false
        ],
        [
          5,
          5,
          1,
          "a^6b^11",
          false
        ],
        [
          5,
          6,
          0,
          "a^5b^11",
          false
        ],
        [
          6,
          1,
          4,
          "a^10b^11",
          false
        ],
        [
          6,
          2,
          3,
          "a^9b^11",
          false
        ],
        [
          6,
          3,
          2,
          "a^8b^11",
          false
        ],
        [
          6,
          4,
          1,
          "a^7b^11",
          false
        ],
        [
          6,
          5,
          0,
          "a^6b^11",
          false
        ],
        [
          7,
          1,
          3,
          "a^10b^11",
          false
        ],
        [
          7,
          2,
          2,
          "a^9b^11",
          false
        ],
        [
          7,
          3,
          1,
          "a^8b^11",
          false
        ],
        [
          7,
          4,
          0,
          "a^7b^11",
          false
        ],
        [
          8,
          1,
          2,
          "a^10b^11",
          false
        ],
        [
          8,
          2,
          1,
          "a^9b^11",
          false
        ],
        [
          8,
          3,
          0,
          "a^8b^11",
          false
        ],
        [
          9,
          1,
          1,
          "a^10b^11",
          false
        ],
        [
          9,
          2,
          0,
          "a^9b^11",
          false
        ],
        [
          10,
          1,
          0,
          "a^10b^11",
          false
        ]
      ],
      66,
      66,
      true,
      "every_legal_split_is_defeated_by_i=0"
    ]
  },
  "conclusion": {
    "canonical_value": "(11, a^11b^11, [[0, 1, 10, a^10b^11, 否], [0, 2, 9, a^9b^11, 否], [0, 3, 8, a^8b^11, 否], [0, 4, 7, a^7b^11, 否], [0, 5, 6, a^6b^11, 否], [0, 6, 5, a^5b^11, 否], [0, 7, 4, a^4b^11, 否], [0, 8, 3, a^3b^11, 否], [0, 9, 2, a^2b^11, 否], [0, 10, 1, a^1b^11, 否], [0, 11, 0, a^0b^11, 否], [1, 1, 9, a^10b^11, 否], [1, 2, 8, a^9b^11, 否], [1, 3, 7, a^8b^11, 否], [1, 4, 6, a^7b^11, 否], [1, 5, 5, a^6b^11, 否], [1, 6, 4, a^5b^11, 否], [1, 7, 3, a^4b^11, 否], [1, 8, 2, a^3b^11, 否], [1, 9, 1, a^2b^11, 否], [1, 10, 0, a^1b^11, 否], [2, 1, 8, a^10b^11, 否], [2, 2, 7, a^9b^11, 否], [2, 3, 6, a^8b^11, 否], [2, 4, 5, a^7b^11, 否], [2, 5, 4, a^6b^11, 否], [2, 6, 3, a^5b^11, 否], [2, 7, 2, a^4b^11, 否], [2, 8, 1, a^3b^11, 否], [2, 9, 0, a^2b^11, 否], [3, 1, 7, a^10b^11, 否], [3, 2, 6, a^9b^11, 否], [3, 3, 5, a^8b^11, 否], [3, 4, 4, a^7b^11, 否], [3, 5, 3, a^6b^11, 否], [3, 6, 2, a^5b^11, 否], [3, 7, 1, a^4b^11, 否], [3, 8, 0, a^3b^11, 否], [4, 1, 6, a^10b^11, 否], [4, 2, 5, a^9b^11, 否], [4, 3, 4, a^8b^11, 否], [4, 4, 3, a^7b^11, 否], [4, 5, 2, a^6b^11, 否], [4, 6, 1, a^5b^11, 否], [4, 7, 0, a^4b^11, 否], [5, 1, 5, a^10b^11, 否], [5, 2, 4, a^9b^11, 否], [5, 3, 3, a^8b^11, 否], [5, 4, 2, a^7b^11, 否], [5, 5, 1, a^6b^11, 否], [5, 6, 0, a^5b^11, 否], [6, 1, 4, a^10b^11, 否], [6, 2, 3, a^9b^11, 否], [6, 3, 2, a^8b^11, 否], [6, 4, 1, a^7b^11, 否], [6, 5, 0, a^6b^11, 否], [7, 1, 3, a^10b^11, 否], [7, 2, 2, a^9b^11, 否], [7, 3, 1, a^8b^11, 否], [7, 4, 0, a^7b^11, 否], [8, 1, 2, a^10b^11, 否], [8, 2, 1, a^9b^11, 否], [8, 3, 0, a^8b^11, 否], [9, 1, 1, a^10b^11, 否], [9, 2, 0, a^9b^11, 否], [10, 1, 0, a^10b^11, 否]], 66, 66, 是, every_legal_split_is_defeated_by_i=0)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations/questions.jsonl",
      "batch_sha256": "657cec46a009582d6773652bd01d9e8b1a552cf28a6e0706bdf7b5813871f2ec",
      "original_line": 61,
      "original_line_sha256": "2d7900b209983ec560f2e9d0189388502731f0906f9c3f89e82ad8b3d547d278",
      "question_compact_sha256": "33ccbdd10927719a77028b365df41173e0c7253f2ce72232edba2c9cf6197b2f",
      "english_row_compact_sha256": "11df2f4345f2db0f69c1b672c232edfc28b365374eb94335c615538ed76c7301",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-pumping-lemma-regular-languages": {
        "sha256": "7466612ae549a8cda880e2921d8c9163ab7681371f543eb905240583b66e6ed4",
        "english_sha256": "b98a8e7b0fb61714b4bea18947593dabe418b1c724d8240eaba1b40c826bd9ec",
        "original_path": "public/data/mko/pumping-lemma-regular-languages.json",
        "english_path": "public/data/i18n/editions/mko-pumping-lemma-regular-languages.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a54409c0bc3f68730e1072c6d3d772970462c1bc7ffc203b4030a6136a610283",
    "original_candidate_sha256": "fe07284d6680c780d1aa137d80965eb5e98ae614cc769ca8dc6b4fc91452327a",
    "original_candidate_line_sha256": "fe07284d6680c780d1aa137d80965eb5e98ae614cc769ca8dc6b4fc91452327a"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-010-p54",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62",
      "question_first_witness_sha256": "79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 137,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "650d51bec962e9008e9f788fcca38f83f04abdef7bbc364c2f4cc5810398258f",
      "arithmetic_steps": 68
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}