{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R355",
  "content_hash": null,
  "slug": "gtc-formalization-graceful-def",
  "type": "formalization",
  "title": "Draft Lean definition of a graceful labeling",
  "summary": "A proposed Lean definition of gracefulness for a finite simple graph, specialized to trees by the edge count.",
  "relevance": "For Graceful tree conjecture, record gtc-formalization-graceful-def (“Draft Lean definition of a graceful labeling”) states a machine-checkable theorem or proof obligation. The record states: A proposed Lean definition of gracefulness for a finite simple graph, specialized to trees by the edge count.",
  "relevance_source": "recorded",
  "body": "Written against Mathlib's SimpleGraph. The labeling is a function to Fin (m+1), with injectivity and the edge-label condition stated separately, so the statement stays close to the informal definition. Not compiled; the deliverable is the statement.",
  "status": "draft",
  "evidence_grade": "unverified_formalization",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "formalization",
    "citation": {
      "url": "https://arxiv.org/abs/1403.1564",
      "locator": "draft for this project"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": "def IsGraceful {V : Type*} [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj] (l : V → ℕ) : Prop :=\n  Function.Injective l ∧\n  (∀ v, l v ≤ G.edgeFinset.card) ∧\n  (G.edgeFinset.image (fun e => Sym2.lift ⟨fun a b => max (l a) (l b) - min (l a) (l b), by intros; simp [max_comm, min_comm]⟩ e))\n    = Finset.Icc 1 G.edgeFinset.card",
  "source": {
    "url": "https://arxiv.org/abs/1403.1564",
    "locator": "draft for this project"
  },
  "models": [],
  "relations": [
    {
      "slug": "graceful-tree-conjecture",
      "title": "Graceful tree conjecture",
      "object_type": "problem",
      "relation": "formalizes",
      "direction": "outgoing"
    },
    {
      "slug": "R356",
      "title": "Draft Lean statement of the graceful tree conjecture",
      "object_type": "formalization",
      "relation": "depends_on",
      "direction": "incoming"
    },
    {
      "slug": "graceful-tree-conjecture",
      "title": "graceful tree conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}