{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-eulerian-trail-01",
  "mko_id": "mko-eulerian-trail",
  "title": {
    "zh-Hant": "含孤立頂點的Euler circuit",
    "en": "An Euler circuit with an isolated vertex"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-eulerian-trail-01",
    "source": {
      "mko_id": "mko-eulerian-trail",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "bd4c7accbd4091e00d7a37abe72e796dc663a93d6cdb230cfaa24a0ec125ad41",
      "english_sha256": "e03f13cd5f3980be232f919922b722c842b0d6d7a7dca8b3ba1e5ea56bf7f340"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "完整核每條邊恰一次與非孤立支撐條件。",
      "en": "Audit every edge exactly once and the nonisolated support condition."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={0,1,2,3}，E={01,12,02}，3孤立；walk W=(0,1,2,0)。",
          "en": "V={0,1,2,3}, E={01,12,02};3 is isolated; walk W=(0,1,2,0)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "E=\\{01,12,02\\}",
        "after_tex": "\\deg=(2,2,2,0)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐頂點算degree，三個非孤立點皆偶度。",
          "en": "Compute every degree; all three nonisolated vertices have even degree."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "無向每邊對degree總和貢獻二。",
            "en": "Each undirected edge contributes two to the degree sum."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "\\{v:\\deg(v)>0\\}=\\{0,1,2\\}",
        "after_tex": "0-1-2-0\\text{ connects the entire edge support}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "非孤立支撐connected，孤立3不破壞它。",
          "en": "The nonisolated support is connected; isolated 3 does not break it."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不要求3出現在用邊circuit。",
            "en": "The edge-using circuit need not visit3."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "W_0,W_1=(0,1)",
        "after_tex": "\\{0,1\\}=01\\in E",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第一walk步使用合法原邊。",
          "en": "The first walk step uses a legal original edge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "walk相鄰點必有邊。",
            "en": "Consecutive walk vertices require an edge."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "W_1,W_2=(1,2)",
        "after_tex": "\\{1,2\\}=12\\in E",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二步使用不同邊12。",
          "en": "The second step uses distinct edge 12."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不是重新使用01。",
            "en": "It does not reuse01."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "W_2,W_3=(2,0)",
        "after_tex": "\\{2,0\\}=02\\in E",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第三步回0使用02，方向不影響無向邊。",
          "en": "The third step returns to 0 using02; orientation does not change an undirected edge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "02與20是同一unordered edge。",
            "en": "02 and 20 denote the same unordered edge."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "\\text{edge sequence }(01,12,02)",
        "after_tex": "\\text{distinct edges and edge set exactly }E",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三邊無重複且全E皆用，得到Euler trail。",
          "en": "The three edges are distinct and exhaust E, giving an Euler trail."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "頂點0重訪合法，trail禁止的是重邊。",
            "en": "Revisiting0 is allowed; a trail forbids repeated edges."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "W_0=W_3=0",
        "after_tex": "W\\text{ is an Euler circuit}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "起終點相同使trail閉合。",
          "en": "Equal start and end close the trail into a circuit."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "Euler使用全部邊，不等同Hamilton訪每點一次。",
            "en": "Euler concerns all edges, not Hamilton's once-per-vertex requirement."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "E=\\varnothing,\\quad V\\ne\\varnothing",
        "after_tex": "W=(v)\\text{ is a length-zero Euler circuit for }v\\in V",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "明示零邊convention，空頂點圖不在本scope。",
          "en": "State the zero-edge convention; empty-vertex graphs are outside scope."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "單點walk無邊可重複且覆蓋全部空E。",
            "en": "The singleton walk repeats no edge and covers empty E."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "W=(0,1,2,0)完整使用三邊一次，是Euler circuit；孤立3不妨礙。",
      "en": "W=(0,1,2,0) uses all three edges once and is an Euler circuit; isolated 3 is harmless."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "孤立3使Euler circuit不可能。",
          "en": "Isolated 3 makes an Euler circuit impossible."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "明示W已使用全部邊。",
          "en": "The supplied W already uses every edge."
        }
      },
      {
        "claim": {
          "zh-Hant": "trail不可重訪0。",
          "en": "A trail cannot revisit0."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "trail禁止重邊而非重頂點。",
          "en": "A trail forbids repeated edges, not vertices."
        }
      },
      {
        "claim": {
          "zh-Hant": "(0,1,0,2)是此圖Euler trail。",
          "en": "(0,1,0,2) is an Euler trail here."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "01被使用兩次且12未用。",
          "en": "It uses01 twice and omits12."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "有限簡單無向圖中 trail 可重複頂點但不可重複邊。存在判準中的連通性責任只要求所有非孤立頂點位於同一分量；孤立點不破壞 edge support。非空頂點零邊圖採長度零 circuit，空頂點圖不在本輪範圍。",
          "en": "A trail in a finite simple undirected graph may repeat vertices but not edges. The connectivity part of the existence criterion requires only that all nonisolated vertices lie in one component; isolated vertices do not break the edge support. A nonempty vertex set with no edges uses the length-zero-circuit convention; the empty-vertex graph is out of scope."
        },
        "source": {
          "mko_id": "mko-eulerian-trail",
          "sha256": "bd4c7accbd4091e00d7a37abe72e796dc663a93d6cdb230cfaa24a0ec125ad41",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限無向簡單圖的邊為無loop、無重複unordered pairs；圖性質須對完整明示V、E逐一檢查。 walk每步沿原邊，trail不重複邊；Euler trail使用全部邊恰一次；circuit為起終點相同的trail；degree計incident邊數。",
          "en": "A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. Each walk step uses an original edge; a trail repeats no edge; an Euler trail uses every edge exactly once; a circuit is a closed trail; degree counts incident edges."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "2+2+2+0",
        "rhs": "2*3"
      },
      {
        "id": "N02",
        "lhs": "3+1",
        "rhs": "4"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "有限簡單無向；零邊採length-zero circuit。",
      "en": "Finite simple undirected; zero edges use the length-zero-circuit convention."
    },
    "md_symbols": [
      [
        "V,E",
        "完整頂點與邊集合",
        "complete vertex and edge sets"
      ],
      [
        "W",
        "頂點walk序列",
        "vertex walk sequence"
      ]
    ],
    "md_error_controls": [
      {
        "title": "unordered edges / Unordered edges",
        "zh": "把2→0正規化為02再比較。",
        "en": "Normalize2-to0 to02 before comparing edges."
      },
      {
        "title": "支撐 / Support",
        "zh": "連通條件作用於degree>0的頂點。",
        "en": "Apply connectivity to vertices with positive degree."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "cd1f3d71a3456a1a2c90e91c98b3c616a6c81cfb80bc766effb371a9418f4eed",
    "original_source_sha256": "e58be6b3dd89120f7fd12c07583f1e4197ebffd5d8058f2882ca58fc9b81553c",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S086",
      "BR142-S085",
      "BR142-P029",
      "BR142-P028",
      "BR142-S081",
      "BR142-S082",
      "BR142-S083",
      "BR142-S084"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}