∀ {R : Type u_1} {S : Type u_2} [inst : CommSemiring R] [inst_1 : IsLocalRing R] [inst_2 : CommSemiring S]
  [inst_3 : IsLocalRing S] (f : R →+* S),
  [IsLocalHom f, ⇑f '' ↑(IsLocalRing.maximalIdeal R) ⊆ ↑(IsLocalRing.maximalIdeal S),
      Ideal.map f (IsLocalRing.maximalIdeal R) ≤ IsLocalRing.maximalIdeal S,
      IsLocalRing.maximalIdeal R ≤ Ideal.comap f (IsLocalRing.maximalIdeal S),
      Ideal.comap f (IsLocalRing.maximalIdeal S) = IsLocalRing.maximalIdeal R].TFAE