Congruence modulo `I^j M` is an equivalence relation, coarser for smaller `j`, and trivial for `j = 0`.

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 scoped ENNReal

theorem TypeDensity.adic_congruence_rules (R M : Type) [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
    :
    (∀ j (x : M), adicCongr I j x x) ∧ (∀ j (x y : M), adicCongr I j x y → adicCongr I j y x) ∧
        (∀ j (x y z : M), adicCongr I j x y → adicCongr I j y z → adicCongr I j x z) ∧
        (∀ i j (x y : M), i ≤ j → adicCongr I j x y → adicCongr I i x y) ∧
        (∀ x y : M, adicCongr I 0 x y) := by sorry