For the `I`-adic topology on `O`, the adic cylinders form a basis of the product topology of `O^n`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Measure_AdicCylinders
import Theorems.Thm_TypeDensity_finite_coordinate_adic_congruence
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_cylinder_basis (O : Type) [CommRing O] [TopologicalSpace O]
    [IsTopologicalRing O] (I : Ideal O) (htop : IsAdic I) (n : ℕ) :
    TopologicalSpace.IsTopologicalBasis (adicCylinders I (Fin n → O)) := by sorry