∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizationMonoid α]
  (h : ∀ (a b : α), ∃ c, ∀ (d : α), d ∣ a ∧ d ∣ b ↔ d ∣ c) (x x_1 : α),
  normalize (normalize (Classical.choose ⋯)) = normalize (Classical.choose ⋯)