For a finite flat extension of local rings `O → B`, `e · f = [B : O]`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Theorems.Thm_TypeDensity_integral_local_extension_unique_prime
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheory.ArithmeticFunction.Moebius
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.RamificationInertia.Basic
import Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.RingTheory.RegularLocalRing.Defs
set_option autoImplicit false

open TypeDensity
open scoped BigOperators ENNReal

theorem TypeDensity.finite_flat_local_ramification_degree (O B : Type) [CommRing O] [IsDomain O]
    [IsLocalRing O] [CommRing B] [IsLocalRing B] [Algebra O B] [Module.Finite O B]
    [Module.Flat O B] :
    (IsLocalRing.maximalIdeal B).ramificationIdx O * (IsLocalRing.maximalIdeal B).inertiaDeg O =
      Module.finrank O B := by sorry