edgeSet_replaceVertex_of_not_adj — Mathlib

English

With decidable equality, distance in the top graph is given by dist_top(u,v) = if u = v then 0 else 1.

Русский

При наличии разрешимости равенств расстояние в верхнем графе задаётся как dist_top(u,v) = if u = v then 0 иначе 1.

LaTeX

$$$ \operatorname{dist}_{\top}(u,v) = \operatorname{ite}(u = v, 0, 1) $$$

Lean4

theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) :
    (G.replaceVertex s t).edgeSet = G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) :=
  by
  ext e; refine e.inductionOn ?_
  simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
  intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]

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