∀ {F : Type u} {Γ₀ : Type v} [inst : Field F] [inst_1 : LinearOrderedCommGroupWithZero Γ₀] {v : Valuation F Γ₀}
  {O : Type w} [inst_2 : CommRing O] [inst_3 : Algebra O F] [IsPrincipalIdealRing O],
  v.Integers O → ¬DenselyOrdered ↑(Set.range ⇑v)