∀ {R : Type u} [inst : CommRing R] {S : Type v} [inst_1 : CommRing S] {f : R →+* S} {p : Ideal R} {P : Ideal S}
  [inst_2 : IsDedekindDomain S] [inst_3 : DecidableEq (Ideal S)],
  Ideal.map f p ≠ ⊥ →
    P.IsPrime →
      P ≠ ⊥ → Ideal.ramificationIdx f p P = Multiset.count P (UniqueFactorizationMonoid.factors (Ideal.map f p))