exists_vert_degree_one_of_nontrivial

English

A nontrivial tree has a vertex of degree one.

Русский

Неденежный дерево имеет вершину степени единицы.

LaTeX

$$$\forall G: IsTree(G) \Rightarrow \exists v: V, G.degree(v) = 1$$$

Lean4

/-- A nontrivial tree has a vertex of degree one. -/
theorem exists_vert_degree_one_of_nontrivial [Fintype V] [Nontrivial V] [DecidableRel G.Adj] (h : G.IsTree) :
    ∃ v, G.degree v = 1 := by
  obtain ⟨v, hv⟩ := G.exists_minimal_degree_vertex
  use v
  rw [← hv]
  exact h.minDegree_eq_one_of_nontrivial

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