{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-spanning-tree",
  "semantic_family_id": "sf-spanning-tree",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "生成樹"
  },
  "summary": {
    "zh-Hant": "生成樹是包含父圖全部頂點的連通無圈子圖。"
  },
  "statement": {
    "zh-Hant": "完整有限connected simple parent graph與選邊子圖。核選邊存在、全頂點、connected/acyclic；connected+n−1 edges或acyclic+n−1 edges可作exact certificate，n−1 alone不足。保留single vertex；不作weighted/MST claim。"
  },
  "formula": {
    "tex": "T(G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>T</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "T",
        "glyph": "T"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7b42c77fecff88395250442c10d152157040f0dbbedd086b17778e20b42fc190"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限無向無loop簡單父圖。"
    },
    {
      "id": "a2",
      "zh-Hant": "選邊以parent edge indices。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "spanning_tree",
      "role_zh": "生成樹",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "parent_graph",
      "role_zh": "父圖",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-tree",
      "title_zh": "樹",
      "reason_zh": "connected acyclic。"
    },
    {
      "id": "mko-graph",
      "title_zh": "有限圖",
      "reason_zh": "父圖與subgraph。"
    },
    {
      "id": "mko-connected-component",
      "title_zh": "連通分量",
      "reason_zh": "parent connectedness。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "包含全部parent vertices。",
      "選邊subgraph為tree。",
      "n−1 alone不足；single vertex合法；不宣稱minimum。"
    ],
    "common_misconception_zh": "只數n−1邊就叫tree，或把spanning tree當MST。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限unweighted graph。",
    "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minimum spanning tree。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-spanning-tree.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-11-04 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"
  }
}