For the `I`-adic topology on `O`, the Borel σ-algebra of `O^n` is generated by the adic cylinders.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Measure_AdicCylinders
import Theorems.Thm_TypeDensity_finite_coordinate_cylinder_basis
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.Topology.Algebra.MvPolynomial
set_option autoImplicit false

open TypeDensity
open MeasureTheory
open scoped BigOperators ENNReal

theorem TypeDensity.finite_coordinate_borel_cylinders (O : Type) [CommRing O] [TopologicalSpace O]
    [IsTopologicalRing O] (I : Ideal O) (htop : IsAdic I) (n : ℕ) [SecondCountableTopology O]
    [MeasurableSpace O] [BorelSpace O] :
    (inferInstance : MeasurableSpace (Fin n → O)) = adicCylinderSpace I (Fin n → O) := by sorry