Adic congruences in an `O`-algebra are compatible with multiplication.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Measure_AdicCylinders
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.algebra_product_congruence (O R : Type) [CommRing O] [CommRing R] [Algebra O R]
    (I : Ideal O) (j : ℕ) (x x2 y y2 : R) (hx : adicCongr I j x x2) (hy : adicCongr I j y y2) :
    adicCongr I j (x*y) (x2*y2) := by sorry