isStronglyConnected_symmetrify — Mathlib

English

If mk a = mk b, then there exist paths a → b and b → a.

Русский

Если mk a = mk b, то существуют пути a → b и b → a.

LaTeX

$$$ (\\text{StronglyConnectedComponent.mk } a) = (\\text{StronglyConnectedComponent.mk } b) \\Rightarrow \\text{Nonempty}(\\text{Path}(a,b)) \\wedge \\text{Nonempty}(\\text{Path}(b,a)) $$$

Lean4

theorem isStronglyConnected_symmetrify (h : IsStronglyConnected V) : IsStronglyConnected (Symmetrify V) :=
  by
  intro a b
  obtain ⟨p⟩ := h a b
  induction p with
  | nil => exact ⟨Path.nil⟩
  | cons q e ih => exact ⟨ih.some.cons (Sum.inl e)⟩

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