English
Let v be a vertex inside a cycle path p; for any w, p.toSubgraph.Adj v w holds if and only if G.Adj v w.
Русский
Пусть v belongs к вершинам цикла; тогда для любого w, p.toSubgraph.Adj v w эквивалентно G.Adj v w.
LaTeX
$$∀ w, p.toSubgraph.Adj(v,w) ⇔ G.Adj(v,w)$$
Lean4
theorem adj_toSubgraph_iff_of_isCycles [LocallyFinite G] {u} {p : G.Walk u u} (hp : p.IsCycle) (hcyc : G.IsCycles)
(hv : v ∈ p.toSubgraph.verts) : ∀ w, p.toSubgraph.Adj v w ↔ G.Adj v w :=
by
refine fun w ↦ Subgraph.adj_iff_of_neighborSet_equiv (?_ : Inhabited _).default (Set.toFinite _)
apply Classical.inhabited_of_nonempty
rw [← Cardinal.eq, ← Set.cast_ncard (Set.toFinite _), ← Set.cast_ncard (finite_neighborSet_toSubgraph p),
hcyc
(Set.Nonempty.mono (p.toSubgraph.neighborSet_subset v) <|
Set.nonempty_of_ncard_ne_zero <| by simp [hp.ncard_neighborSet_toSubgraph_eq_two (by aesop)]),
hp.ncard_neighborSet_toSubgraph_eq_two (by simp_all)]
