{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-directed-graph",
  "semantic_family_id": "sf-finite-directed-graph",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限有向圖"
  },
  "summary": {
    "zh-Hant": "有限loop-free simple digraph由有限vertices與不含loop、無平行同向重邊的ordered-pair edge set組成。"
  },
  "statement": {
    "zh-Hant": "固定loop-free simple convention：每edge=(u,v)、u!=v，(u,v)與(v,u)不同，E為set故同向parallel duplicates禁止。out-degree與in-degree分開，兩者全圖總和皆等於|E|。walk每一步必須符合edge orientation；loops由negative boundary拒絕。"
  },
  "formula": {
    "tex": "D(V,E)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        },
        {
          "type": "symbol",
          "name": "E",
          "glyph": "E"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "0da39141f4e77dca16a885a18163c68468be0342696954b5ee6be336f9efe8a4"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "vertices標為0至n-1且n>=1。"
    },
    {
      "id": "a2",
      "zh-Hant": "edges canonical lexicographic、unique、loop-free。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "finite_directed_graph",
      "role_zh": "有限有向圖",
      "scope": "representative_formula"
    },
    {
      "token": "E",
      "role": "ordered_edge_set",
      "role_zh": "有向邊集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-relation",
      "title_zh": "關係",
      "reason_zh": "ordered pairs。"
    },
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "vertex與edge sets。"
    },
    {
      "id": "mko-cardinality-comparison",
      "title_zh": "集合基數比較",
      "reason_zh": "degree counts。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "edge orientation是語義的一部分。",
      "in/out degrees各自總和為edge count。",
      "loop與multiedge需要不同模型。"
    ],
    "common_misconception_zh": "把(u,v)與(v,u)視為同edge，或沿用undirected degree。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不含loops、parallel edges或undirected connectivity conventions。",
    "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topological ordering與flow networks。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-directed-graph.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-11-08 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"
  }
}