∀ {K : Type u_1} {Γ₀ : Type u_2} [inst : Field K] [inst_1 : LinearOrderedCommGroupWithZero Γ₀] [inst_2 : Valued K Γ₀]
  [hc : CompactSpace ↥(Valued.integer K)], IsPrincipalIdealRing ↥(Valued.integer K)