{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261027-r01-finite-unital-ring-0080",
  "pilot_number": "R0025-13230",
  "number": 13230,
  "family_id": "qf-mod27-ring-identities",
  "title": {
    "zh-Hant": "第 13230 題：有限含一環",
    "en": "Question 13230: Keep Additive Zero and Multiplicative One Distinct"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261027-r01-finite-unital-ring-0080",
    "batch_id": "batch-20261014-advance-20261027-r01-finite-unital-ring-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-unital-ring"
    ],
    "title_zh": "有限含一環",
    "stem_zh": "同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]]}。請依 [zero,one,distinct] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, 1, 是)",
      "explanation_zh": "Wave4根節點核對=(0, 1, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；全部環公理逐對或逐三元組核對；含一環不自動是交換環整環或域。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限含一環"
      ]
    },
    "variant": {
      "family_id": "qf-mod27-ring-identities",
      "seed": 202610271080,
      "parameters": {
        "n": 6,
        "addition": [
          [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            1,
            2,
            5,
            4,
            0,
            3
          ],
          [
            2,
            5,
            3,
            0,
            1,
            4
          ],
          [
            3,
            4,
            0,
            2,
            5,
            1
          ],
          [
            4,
            0,
            1,
            5,
            3,
            2
          ],
          [
            5,
            3,
            4,
            1,
            2,
            0
          ]
        ],
        "multiplication": [
          [
            0,
            0,
            0,
            0,
            0,
            0
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            0,
            2,
            3,
            2,
            3,
            0
          ],
          [
            0,
            3,
            2,
            3,
            2,
            0
          ],
          [
            0,
            4,
            3,
            2,
            1,
            5
          ],
          [
            0,
            5,
            0,
            0,
            5,
            5
          ]
        ],
        "domain": "Complete_finite_unital_ring_tables",
        "operation": "ring-identities",
        "semantic_template": "qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261027-r01-finite-unital-ring-0080",
    "batch_id": "batch-20261014-advance-20261027-r01-finite-unital-ring-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0c4f032e6fdb87aad7ccfb88fbef64d4e96aeab0616a9bf9ee150529347391ef",
      "family_id": "qf-mod27-ring-identities",
      "seed": 202610271080,
      "semantic_template": "qf-mod27-ring-identities/{\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]],\"n\":6}"
    },
    "fields": {
      "title": "Keep Additive Zero and Multiplicative One Distinct",
      "stem": "Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 3, 0], [0, 3, 2, 3, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?).",
      "answer": {
        "kind": "expression",
        "value": "(0, 1, Yes)",
        "explanation": "Find both two-sided identities from their respective operations. Labels denote carrier elements, not their assumed arithmetic values. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (0, 1, Yes)."
      },
      "skills": [
        "Keep Additive Zero and Multiplicative One Distinct"
      ],
      "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": "同一完整有限載體上的加法表與乘法表。操作=ring-identities；資料={\"n\":6,\"addition\":[[0,1,2,3,4,5],[1,2,5,4,0,3],[2,5,3,0,1,4],[3,4,0,2,5,1],[4,0,1,5,3,2],[5,3,4,1,2,0]],\"multiplication\":[[0,0,0,0,0,0],[0,1,2,3,4,5],[0,2,3,2,3,0],[0,3,2,3,2,0],[0,4,3,2,1,5],[0,5,0,0,5,5]]}。請依 [zero,one,distinct] 欄序回答。",
        "en": "Use the same carrier Fin(6)={0,…,5} for complete addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]] and multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 3, 0], [0, 3, 2, 3, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]] tables in label order. Check the additive abelian-group axioms, multiplicative associativity, both distributive laws and a two-sided multiplicative identity distinct from the additive identity. The two identity labels need not be 0 and 1. This protocol excludes the zero ring and does not require multiplicative commutativity or imply a field. Require fully qualified tables. Give (zero,one,distinct?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "same carrier Fin(6);addition=[[0, 1, 2, 3, 4, 5], [1, 2, 5, 4, 0, 3], [2, 5, 3, 0, 1, 4], [3, 4, 0, 2, 5, 1], [4, 0, 1, 5, 3, 2], [5, 3, 4, 1, 2, 0]];multiplication=[[0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5], [0, 2, 3, 2, 3, 0], [0, 3, 2, 3, 2, 0], [0, 4, 3, 2, 1, 5], [0, 5, 0, 0, 5, 5]]",
      "after": "both operations typed and closed;unital ring does not mean field",
      "relation": "implies",
      "rule_id": "ring_shared_typed_carrier",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩張表要綁同一完整有限載體。此協定要求含一且0≠1，不要求乘法交換或每個非零元素可逆，也不從少數實例判群環公理。",
        "en": "Bind both tables to the same complete finite carrier.This protocol requires a unit distinct from zero,not multiplicative commutativity or invertibility of every nonzero element.Neither group nor ring axioms are inferred from samples."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "axiom=addition_associativity;a=0;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "axiom=addition_associativity;a=0;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[1, 2, 5, 4, 0, 3];right values=[1, 2, 5, 4, 0, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "axiom=addition_associativity;a=0;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 5, 3, 0, 1, 4];right values=[2, 5, 3, 0, 1, 4];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "axiom=addition_associativity;a=0;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 4, 0, 2, 5, 1];right values=[3, 4, 0, 2, 5, 1];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "axiom=addition_associativity;a=0;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[4, 0, 1, 5, 3, 2];right values=[4, 0, 1, 5, 3, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "axiom=addition_associativity;a=0;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 3, 4, 1, 2, 0];right values=[5, 3, 4, 1, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "axiom=addition_associativity;a=1;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[1, 2, 5, 4, 0, 3];right values=[1, 2, 5, 4, 0, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "axiom=addition_associativity;a=1;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 5, 3, 0, 1, 4];right values=[2, 5, 3, 0, 1, 4];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "axiom=addition_associativity;a=1;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 3, 4, 1, 2, 0];right values=[5, 3, 4, 1, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "axiom=addition_associativity;a=1;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[4, 0, 1, 5, 3, 2];right values=[4, 0, 1, 5, 3, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "axiom=addition_associativity;a=1;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "axiom=addition_associativity;a=1;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 4, 0, 2, 5, 1];right values=[3, 4, 0, 2, 5, 1];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "axiom=addition_associativity;a=2;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 5, 3, 0, 1, 4];right values=[2, 5, 3, 0, 1, 4];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "axiom=addition_associativity;a=2;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 3, 4, 1, 2, 0];right values=[5, 3, 4, 1, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "axiom=addition_associativity;a=2;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 4, 0, 2, 5, 1];right values=[3, 4, 0, 2, 5, 1];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "axiom=addition_associativity;a=2;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "axiom=addition_associativity;a=2;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[1, 2, 5, 4, 0, 3];right values=[1, 2, 5, 4, 0, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "axiom=addition_associativity;a=2;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[4, 0, 1, 5, 3, 2];right values=[4, 0, 1, 5, 3, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "axiom=addition_associativity;a=3;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 4, 0, 2, 5, 1];right values=[3, 4, 0, 2, 5, 1];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "axiom=addition_associativity;a=3;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[4, 0, 1, 5, 3, 2];right values=[4, 0, 1, 5, 3, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "axiom=addition_associativity;a=3;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "axiom=addition_associativity;a=3;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 5, 3, 0, 1, 4];right values=[2, 5, 3, 0, 1, 4];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "axiom=addition_associativity;a=3;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 3, 4, 1, 2, 0];right values=[5, 3, 4, 1, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "axiom=addition_associativity;a=3;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[1, 2, 5, 4, 0, 3];right values=[1, 2, 5, 4, 0, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "axiom=addition_associativity;a=4;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[4, 0, 1, 5, 3, 2];right values=[4, 0, 1, 5, 3, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "axiom=addition_associativity;a=4;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "axiom=addition_associativity;a=4;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[1, 2, 5, 4, 0, 3];right values=[1, 2, 5, 4, 0, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "axiom=addition_associativity;a=4;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 3, 4, 1, 2, 0];right values=[5, 3, 4, 1, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "axiom=addition_associativity;a=4;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 4, 0, 2, 5, 1];right values=[3, 4, 0, 2, 5, 1];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "axiom=addition_associativity;a=4;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 5, 3, 0, 1, 4];right values=[2, 5, 3, 0, 1, 4];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "axiom=addition_associativity;a=5;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 3, 4, 1, 2, 0];right values=[5, 3, 4, 1, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "axiom=addition_associativity;a=5;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 4, 0, 2, 5, 1];right values=[3, 4, 0, 2, 5, 1];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "axiom=addition_associativity;a=5;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[4, 0, 1, 5, 3, 2];right values=[4, 0, 1, 5, 3, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "axiom=addition_associativity;a=5;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[1, 2, 5, 4, 0, 3];right values=[1, 2, 5, 4, 0, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "axiom=addition_associativity;a=5;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 5, 3, 0, 1, 4];right values=[2, 5, 3, 0, 1, 4];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "axiom=addition_associativity;a=5;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "additive unordered pair=[0, 1];a+b=1;b+a=1",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "additive unordered pair=[0, 2];a+b=2;b+a=2",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "additive unordered pair=[0, 3];a+b=3;b+a=3",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "additive unordered pair=[0, 4];a+b=4;b+a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "additive unordered pair=[0, 5];a+b=5;b+a=5",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "additive unordered pair=[1, 2];a+b=5;b+a=5",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "additive unordered pair=[1, 3];a+b=4;b+a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "additive unordered pair=[1, 4];a+b=0;b+a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "additive unordered pair=[1, 5];a+b=3;b+a=3",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "additive unordered pair=[2, 3];a+b=0;b+a=0",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "additive unordered pair=[2, 4];a+b=1;b+a=1",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "additive unordered pair=[2, 5];a+b=4;b+a=4",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "additive unordered pair=[3, 4];a+b=5;b+a=5",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "additive unordered pair=[3, 5];a+b=1;b+a=1",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "additive unordered pair=[4, 5];a+b=2;b+a=2",
      "after": "commute=是",
      "relation": "implies",
      "rule_id": "ring_additive_commutativity_all_pairs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法交換律是加法Abelian群責任，乘法沒有因此交換；對角對自動相等。",
        "en": "Additive commutativity belongs to the additive abelian-group obligation;it does not make multiplication commutative.Diagonal pairs agree automatically."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "operation=addition;candidate identity=0;left products=[0, 1, 2, 3, 4, 5];right products=[0, 1, 2, 3, 4, 5]",
      "after": "two-sided identity=是",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "operation=addition;candidate identity=1;left products=[1, 2, 5, 4, 0, 3];right products=[1, 2, 5, 4, 0, 3]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "operation=addition;candidate identity=2;left products=[2, 5, 3, 0, 1, 4];right products=[2, 5, 3, 0, 1, 4]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "operation=addition;candidate identity=3;left products=[3, 4, 0, 2, 5, 1];right products=[3, 4, 0, 2, 5, 1]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "operation=addition;candidate identity=4;left products=[4, 0, 1, 5, 3, 2];right products=[4, 0, 1, 5, 3, 2]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "operation=addition;candidate identity=5;left products=[5, 3, 4, 1, 2, 0];right products=[5, 3, 4, 1, 2, 0]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "element=0;actual additive zero=0;two-sided additive inverse candidates=[0]",
      "after": "additive inverse=0",
      "relation": "implies",
      "rule_id": "ring_additive_inverse_not_multiplicative_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這裡的逆元針對加法，以實際加法零為目標；缺逆元保留null並令加法資格失敗，不拿乘法表補救。",
        "en": "This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "element=1;actual additive zero=0;two-sided additive inverse candidates=[4]",
      "after": "additive inverse=4",
      "relation": "implies",
      "rule_id": "ring_additive_inverse_not_multiplicative_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這裡的逆元針對加法，以實際加法零為目標；缺逆元保留null並令加法資格失敗，不拿乘法表補救。",
        "en": "This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "element=2;actual additive zero=0;two-sided additive inverse candidates=[3]",
      "after": "additive inverse=3",
      "relation": "implies",
      "rule_id": "ring_additive_inverse_not_multiplicative_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這裡的逆元針對加法，以實際加法零為目標；缺逆元保留null並令加法資格失敗，不拿乘法表補救。",
        "en": "This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "element=3;actual additive zero=0;two-sided additive inverse candidates=[2]",
      "after": "additive inverse=2",
      "relation": "implies",
      "rule_id": "ring_additive_inverse_not_multiplicative_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這裡的逆元針對加法，以實際加法零為目標；缺逆元保留null並令加法資格失敗，不拿乘法表補救。",
        "en": "This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "element=4;actual additive zero=0;two-sided additive inverse candidates=[1]",
      "after": "additive inverse=1",
      "relation": "implies",
      "rule_id": "ring_additive_inverse_not_multiplicative_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這裡的逆元針對加法，以實際加法零為目標；缺逆元保留null並令加法資格失敗，不拿乘法表補救。",
        "en": "This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "element=5;actual additive zero=0;two-sided additive inverse candidates=[5]",
      "after": "additive inverse=5",
      "relation": "implies",
      "rule_id": "ring_additive_inverse_not_multiplicative_inverse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這裡的逆元針對加法，以實際加法零為目標；缺逆元保留null並令加法資格失敗，不拿乘法表補救。",
        "en": "This inverse is additive and targets the actual additive zero.A missing inverse remains null and fails additive qualification;the multiplication table cannot repair it."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "axiom=multiplication_associativity;a=0;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "axiom=multiplication_associativity;a=0;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "axiom=multiplication_associativity;a=0;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "axiom=multiplication_associativity;a=0;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "axiom=multiplication_associativity;a=0;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "axiom=multiplication_associativity;a=0;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "axiom=multiplication_associativity;a=1;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "axiom=multiplication_associativity;a=1;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "axiom=multiplication_associativity;a=1;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "axiom=multiplication_associativity;a=1;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "axiom=multiplication_associativity;a=1;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 4, 3, 2, 1, 5];right values=[0, 4, 3, 2, 1, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "axiom=multiplication_associativity;a=1;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "axiom=multiplication_associativity;a=2;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "axiom=multiplication_associativity;a=2;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "axiom=multiplication_associativity;a=2;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "axiom=multiplication_associativity;a=2;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "axiom=multiplication_associativity;a=2;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "axiom=multiplication_associativity;a=2;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "axiom=multiplication_associativity;a=3;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "axiom=multiplication_associativity;a=3;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "axiom=multiplication_associativity;a=3;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "axiom=multiplication_associativity;a=3;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "axiom=multiplication_associativity;a=3;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "axiom=multiplication_associativity;a=3;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "axiom=multiplication_associativity;a=4;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "axiom=multiplication_associativity;a=4;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 4, 3, 2, 1, 5];right values=[0, 4, 3, 2, 1, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "axiom=multiplication_associativity;a=4;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "axiom=multiplication_associativity;a=4;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "axiom=multiplication_associativity;a=4;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "axiom=multiplication_associativity;a=4;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "axiom=multiplication_associativity;a=5;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "axiom=multiplication_associativity;a=5;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "axiom=multiplication_associativity;a=5;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "axiom=multiplication_associativity;a=5;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "axiom=multiplication_associativity;a=5;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "axiom=multiplication_associativity;a=5;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "operation=multiplication;candidate identity=0;left products=[0, 0, 0, 0, 0, 0];right products=[0, 0, 0, 0, 0, 0]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "operation=multiplication;candidate identity=1;left products=[0, 1, 2, 3, 4, 5];right products=[0, 1, 2, 3, 4, 5]",
      "after": "two-sided identity=是",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "operation=multiplication;candidate identity=2;left products=[0, 2, 3, 2, 3, 0];right products=[0, 2, 3, 2, 3, 0]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "operation=multiplication;candidate identity=3;left products=[0, 3, 2, 3, 2, 0];right products=[0, 3, 2, 3, 2, 0]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "operation=multiplication;candidate identity=4;left products=[0, 4, 3, 2, 1, 5];right products=[0, 4, 3, 2, 1, 5]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "operation=multiplication;candidate identity=5;left products=[0, 5, 0, 0, 5, 5];right products=[0, 5, 0, 0, 5, 5]",
      "after": "two-sided identity=否",
      "relation": "implies",
      "rule_id": "ring_derive_operation_identity_labels",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側都回到原元素標籤才是該運算的單位元；加法與乘法分開找，沒有先指定0或1。",
        "en": "Both operation directions must return the original element label.Derive additive and multiplicative identities separately,without assigning labels0 or1 in advance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "axiom=left_distributivity;a=0;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "axiom=left_distributivity;a=0;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "axiom=left_distributivity;a=0;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "axiom=left_distributivity;a=0;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "axiom=left_distributivity;a=0;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "axiom=left_distributivity;a=0;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "axiom=left_distributivity;a=1;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "axiom=left_distributivity;a=1;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[1, 2, 5, 4, 0, 3];right values=[1, 2, 5, 4, 0, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "axiom=left_distributivity;a=1;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 5, 3, 0, 1, 4];right values=[2, 5, 3, 0, 1, 4];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "axiom=left_distributivity;a=1;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 4, 0, 2, 5, 1];right values=[3, 4, 0, 2, 5, 1];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "axiom=left_distributivity;a=1;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[4, 0, 1, 5, 3, 2];right values=[4, 0, 1, 5, 3, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "axiom=left_distributivity;a=1;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 3, 4, 1, 2, 0];right values=[5, 3, 4, 1, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "axiom=left_distributivity;a=2;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "axiom=left_distributivity;a=2;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 3, 0, 3, 0, 2];right values=[2, 3, 0, 3, 0, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "axiom=left_distributivity;a=2;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 0, 2, 0, 2, 3];right values=[3, 0, 2, 0, 2, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "axiom=left_distributivity;a=2;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 3, 0, 3, 0, 2];right values=[2, 3, 0, 3, 0, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "axiom=left_distributivity;a=2;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 0, 2, 0, 2, 3];right values=[3, 0, 2, 0, 2, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "axiom=left_distributivity;a=2;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "axiom=left_distributivity;a=3;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "axiom=left_distributivity;a=3;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 2, 0, 2, 0, 3];right values=[3, 2, 0, 2, 0, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "axiom=left_distributivity;a=3;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 0, 3, 0, 3, 2];right values=[2, 0, 3, 0, 3, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "axiom=left_distributivity;a=3;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 2, 0, 2, 0, 3];right values=[3, 2, 0, 2, 0, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "axiom=left_distributivity;a=3;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 0, 3, 0, 3, 2];right values=[2, 0, 3, 0, 3, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "axiom=left_distributivity;a=3;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "axiom=left_distributivity;a=4;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 4, 3, 2, 1, 5];right values=[0, 4, 3, 2, 1, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "axiom=left_distributivity;a=4;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[4, 3, 5, 1, 0, 2];right values=[4, 3, 5, 1, 0, 2];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "axiom=left_distributivity;a=4;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[3, 5, 2, 0, 4, 1];right values=[3, 5, 2, 0, 4, 1];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "axiom=left_distributivity;a=4;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[2, 1, 0, 3, 5, 4];right values=[2, 1, 0, 3, 5, 4];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "axiom=left_distributivity;a=4;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[1, 0, 4, 5, 2, 3];right values=[1, 0, 4, 5, 2, 3];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "axiom=left_distributivity;a=4;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 2, 1, 4, 3, 0];right values=[5, 2, 1, 4, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "axiom=left_distributivity;a=5;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "axiom=left_distributivity;a=5;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 0, 5, 5, 0, 0];right values=[5, 0, 5, 5, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S139",
      "uses": [
        "G001",
        "S138"
      ],
      "before": "axiom=left_distributivity;a=5;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S140",
      "uses": [
        "G001",
        "S139"
      ],
      "before": "axiom=left_distributivity;a=5;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S141",
      "uses": [
        "G001",
        "S140"
      ],
      "before": "axiom=left_distributivity;a=5;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 0, 5, 5, 0, 0];right values=[5, 0, 5, 5, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S142",
      "uses": [
        "G001",
        "S141"
      ],
      "before": "axiom=left_distributivity;a=5;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[5, 0, 5, 5, 0, 0];right values=[5, 0, 5, 5, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S143",
      "uses": [
        "G001",
        "S142"
      ],
      "before": "axiom=right_distributivity;a=0;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S144",
      "uses": [
        "G001",
        "S143"
      ],
      "before": "axiom=right_distributivity;a=0;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S145",
      "uses": [
        "G001",
        "S144"
      ],
      "before": "axiom=right_distributivity;a=0;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S146",
      "uses": [
        "G001",
        "S145"
      ],
      "before": "axiom=right_distributivity;a=0;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S147",
      "uses": [
        "G001",
        "S146"
      ],
      "before": "axiom=right_distributivity;a=0;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 4, 3, 2, 1, 5];right values=[0, 4, 3, 2, 1, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S148",
      "uses": [
        "G001",
        "S147"
      ],
      "before": "axiom=right_distributivity;a=0;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S149",
      "uses": [
        "G001",
        "S148"
      ],
      "before": "axiom=right_distributivity;a=1;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S150",
      "uses": [
        "G001",
        "S149"
      ],
      "before": "axiom=right_distributivity;a=1;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S151",
      "uses": [
        "G001",
        "S150"
      ],
      "before": "axiom=right_distributivity;a=1;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S152",
      "uses": [
        "G001",
        "S151"
      ],
      "before": "axiom=right_distributivity;a=1;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 4, 3, 2, 1, 5];right values=[0, 4, 3, 2, 1, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S153",
      "uses": [
        "G001",
        "S152"
      ],
      "before": "axiom=right_distributivity;a=1;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S154",
      "uses": [
        "G001",
        "S153"
      ],
      "before": "axiom=right_distributivity;a=1;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S155",
      "uses": [
        "G001",
        "S154"
      ],
      "before": "axiom=right_distributivity;a=2;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S156",
      "uses": [
        "G001",
        "S155"
      ],
      "before": "axiom=right_distributivity;a=2;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S157",
      "uses": [
        "G001",
        "S156"
      ],
      "before": "axiom=right_distributivity;a=2;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S158",
      "uses": [
        "G001",
        "S157"
      ],
      "before": "axiom=right_distributivity;a=2;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S159",
      "uses": [
        "G001",
        "S158"
      ],
      "before": "axiom=right_distributivity;a=2;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S160",
      "uses": [
        "G001",
        "S159"
      ],
      "before": "axiom=right_distributivity;a=2;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 4, 3, 2, 1, 5];right values=[0, 4, 3, 2, 1, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S161",
      "uses": [
        "G001",
        "S160"
      ],
      "before": "axiom=right_distributivity;a=3;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S162",
      "uses": [
        "G001",
        "S161"
      ],
      "before": "axiom=right_distributivity;a=3;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 4, 3, 2, 1, 5];right values=[0, 4, 3, 2, 1, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S163",
      "uses": [
        "G001",
        "S162"
      ],
      "before": "axiom=right_distributivity;a=3;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S164",
      "uses": [
        "G001",
        "S163"
      ],
      "before": "axiom=right_distributivity;a=3;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S165",
      "uses": [
        "G001",
        "S164"
      ],
      "before": "axiom=right_distributivity;a=3;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S166",
      "uses": [
        "G001",
        "S165"
      ],
      "before": "axiom=right_distributivity;a=3;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S167",
      "uses": [
        "G001",
        "S166"
      ],
      "before": "axiom=right_distributivity;a=4;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 4, 3, 2, 1, 5];right values=[0, 4, 3, 2, 1, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S168",
      "uses": [
        "G001",
        "S167"
      ],
      "before": "axiom=right_distributivity;a=4;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S169",
      "uses": [
        "G001",
        "S168"
      ],
      "before": "axiom=right_distributivity;a=4;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S170",
      "uses": [
        "G001",
        "S169"
      ],
      "before": "axiom=right_distributivity;a=4;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S171",
      "uses": [
        "G001",
        "S170"
      ],
      "before": "axiom=right_distributivity;a=4;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S172",
      "uses": [
        "G001",
        "S171"
      ],
      "before": "axiom=right_distributivity;a=4;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S173",
      "uses": [
        "G001",
        "S172"
      ],
      "before": "axiom=right_distributivity;a=5;b=0;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 5, 0, 0, 5, 5];right values=[0, 5, 0, 0, 5, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S174",
      "uses": [
        "G001",
        "S173"
      ],
      "before": "axiom=right_distributivity;a=5;b=1;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 3, 2, 3, 2, 0];right values=[0, 3, 2, 3, 2, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S175",
      "uses": [
        "G001",
        "S174"
      ],
      "before": "axiom=right_distributivity;a=5;b=2;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 4, 3, 2, 1, 5];right values=[0, 4, 3, 2, 1, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S176",
      "uses": [
        "G001",
        "S175"
      ],
      "before": "axiom=right_distributivity;a=5;b=3;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 1, 2, 3, 4, 5];right values=[0, 1, 2, 3, 4, 5];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S177",
      "uses": [
        "G001",
        "S176"
      ],
      "before": "axiom=right_distributivity;a=5;b=4;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 2, 3, 2, 3, 0];right values=[0, 2, 3, 2, 3, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S178",
      "uses": [
        "G001",
        "S177"
      ],
      "before": "axiom=right_distributivity;a=5;b=5;all c=[0, 1, 2, 3, 4, 5]",
      "after": "left values=[0, 0, 0, 0, 0, 0];right values=[0, 0, 0, 0, 0, 0];all equal=是",
      "relation": "implies",
      "rule_id": "ring_complete_axiom_third_element",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個有序(a,b)列出所有c的兩側，使用父表真正的加乘運算，不把元素標籤當整數。即使某條失敗，也繼續核其他責任。",
        "en": "For each ordered(a,b),list both sides for every c using actual parent-table operations,not integer arithmetic on labels.Continue auditing other obligations even after a failure."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S179",
      "uses": [
        "G001",
        "S178"
      ],
      "before": "additive associativity first failure=null;additive commute=是;zero=0;inverses=[0, 4, 3, 2, 1, 5];multiplication first failure=null;one=1;left distributivity first failure=null;right distributivity first failure=null",
      "after": "additive Abelian=是;multiplication associative=是;left distributive=是;right distributive=是;distinct identities=是;qualified=是;priority failure=null",
      "relation": "implies",
      "rule_id": "ring_all_obligations_and_failure_priority",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留所有個別旗標，再按加法、乘法結合、左分配、右分配、單位元相異的宣告優先序報首錯；一個通過不能代替其他責任。",
        "en": "Retain every individual flag,then report the first failure by the declared priority:additive structure,multiplicative associativity,left distributivity,right distributivity,distinct identities.One passing obligation cannot replace another."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      0,
      1,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 1, 是)",
    "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-20261014-advance-20261027-r01-finite-unital-ring-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261027-r01-finite-unital-ring-foundations/questions.jsonl",
      "batch_sha256": "893d9fa40c38bc7b418db0fb5e98c39f9d8e500e86654b1884dd6c0077d8375a",
      "original_line": 80,
      "original_line_sha256": "e13234245cc22397a94d0ac5e3e57ed6244575110dc93af7403d7e538078947d",
      "question_compact_sha256": "0c4f032e6fdb87aad7ccfb88fbef64d4e96aeab0616a9bf9ee150529347391ef",
      "english_row_compact_sha256": "7a1ea9797113364e2e8e46b7e2814ca9d13c82b3d1e97097a72801f948af60cf",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261027-r01-finite-unital-ring-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261027-r01-finite-unital-ring-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-unital-ring": {
        "sha256": "e930e72ae14b44b56feb441e48526b614a1bfb2f281a1518ec6c7b19b1de48bf",
        "english_sha256": "0e72ca06145d9acc6b850e3f01c4fc86eec243a747e18cc1dcc09f6abfc706f6",
        "original_path": "public/data/mko/finite-unital-ring.json",
        "english_path": "public/data/i18n/editions/mko-finite-unital-ring.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a02aa2dccf790869cf864b821d0e60119686214561153daba232d921e07e7924",
    "original_candidate_sha256": "b71288cadcdad379e929ea386240adcc4e710c2d5f87e25e0ca0ebaf721e6889",
    "original_candidate_line_sha256": "b71288cadcdad379e929ea386240adcc4e710c2d5f87e25e0ca0ebaf721e6889"
  },
  "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-20261026-007-p26",
    "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": "0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5",
      "question_first_witness_sha256": "ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 179,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "11456bcc269eb16633a072db0e5930c8adffdc06bf71427b90d5ba6b422af715",
      "arithmetic_steps": 0
    },
    "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."
  }
}