{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-graph-walk-simple-path",
  "semantic_family_id": "sf-graph-walk-simple-path",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "圖walk與simple path"
  },
  "summary": {
    "zh-Hant": "有限無向簡單圖的walk要求每對連續頂點相鄰；simple path另要求所有頂點互異。"
  },
  "statement": {
    "zh-Hant": "本輪固定有限無向無loop簡單圖。頂點序列(v0,...,vk)為walk，若每個連續無序對都是邊；其edge length為k。simple path是頂點不重複的walk。walk可重複頂點或邊，cycle與trail另有額外責任。"
  },
  "formula": {
    "tex": "W(v)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>W</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "W",
        "glyph": "W"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "v",
          "glyph": "v"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "a96ec6de20531dafd572a1a9806e31453ff565563aafca7178b4aa9f1b7151cb"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "頂點標籤為0到n-1，完整無向簡單邊集合與頂點序列明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "空序列不作walk；單頂點序列是長度0的simple path。"
    }
  ],
  "symbols": [
    {
      "token": "G",
      "role": "simple_graph",
      "role_zh": "有限無向簡單圖",
      "scope": "representative_formula"
    },
    {
      "token": "v",
      "role": "vertex_sequence",
      "role_zh": "頂點序列",
      "scope": "representative_formula"
    },
    {
      "token": "k",
      "role": "edge_length",
      "role_zh": "邊長度",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph",
      "title_zh": "有限圖",
      "reason_zh": "鄰接由完整邊集合給出。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "walk以有序頂點序列表示。"
    },
    {
      "id": "mko-relation",
      "title_zh": "二元關係",
      "reason_zh": "無向鄰接視為對稱關係。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先逐步核每個連續頂點對是否相鄰。",
      "長度是邊數，等於頂點序列長度減一。",
      "simple path禁止重複頂點；一般walk允許重複。"
    ],
    "common_misconception_zh": "把任何頂點清單當walk、把頂點數當長度，或把無重複邊的trail與simple path混同。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理明示有限無向簡單圖；不宣稱directed walk、trail或cycle資格。",
    "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": "加入連通分量。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-graph-walk-simple-path.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-10-27 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"
  }
}