hasse_preconnected_of_succ — Mathlib

English

Componentwise characterization of CovBy in a product: CovBy x y decomposes in terms of x.fst, x.snd, y.fst, y.snd.

Русский

Покрывающее отношение CovBy в произведении распадается на компоненты порядка wc.

LaTeX

$$$\forall x y : Prod α β, \, CovBy x y = Or (And (CovBy x.fst y.fst) (Eq x.snd y.snd)) (And (CovBy x.snd y.snd) (Eq x.fst y.fst))$$$

Lean4

theorem hasse_preconnected_of_succ [SuccOrder α] [IsSuccArchimedean α] : (hasse α).Preconnected := fun a b =>
  by
  rw [reachable_iff_reflTransGen]
  exact
    reflTransGen_of_succ _ (fun c hc => Or.inl <| covBy_succ_of_not_isMax hc.2.not_isMax) fun c hc =>
      Or.inr <| covBy_succ_of_not_isMax hc.2.not_isMax

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