{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-eulerian-trail",
  "semantic_family_id": "sf-eulerian-trail",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Euler trail與circuit"
  },
  "summary": {
    "zh-Hant": "Euler trail恰用每條邊一次；非零邊支撐連通且奇度數為0或2時分別有circuit或open trail。"
  },
  "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。"
  },
  "formula": {
    "tex": "deg(v)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>deg</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "deg",
        "glyph": "deg"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "v",
          "glyph": "v"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "56b2ef5aba51c832934a32b608f9d39a5992245bb3487bdce9c081ac98209d70"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限無loop簡單無向圖及明示edge list。"
    },
    {
      "id": "a2",
      "zh-Hant": "零邊convention固定為非空頂點集的length-zero circuit。"
    }
  ],
  "symbols": [
    {
      "token": "\\deg",
      "role": "vertex_degree",
      "role_zh": "頂點度數",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "euler_trail",
      "role_zh": "Euler trail",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph-walk-simple-path",
      "title_zh": "圖walk與simple path",
      "reason_zh": "trail與path的區別。"
    },
    {
      "id": "mko-graph",
      "title_zh": "有限簡單無向圖",
      "reason_zh": "完整頂點與邊。"
    },
    {
      "id": "mko-connected-component",
      "title_zh": "連通分量",
      "reason_zh": "非孤立支撐的連通性。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "trail只禁止邊重複，不禁止頂點重訪。",
      "0個奇度頂點對應circuit，2個對應open trail。",
      "判準作用於非孤立支撐，不推出Hamilton path或cycle。"
    ],
    "common_misconception_zh": "把trail當simple path，或讓孤立頂點錯誤否決Euler trail。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限簡單無向圖；有向、多重圖與Hamilton問題另列。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入Hierholzer構造與有向Euler判準。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-eulerian-trail.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-11-05 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}