The coordinate remainder `(z, v) ↦ π • E (v ∑ zᵢ bᵢ)` raises `π`-adic order.

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.coordinate_remainder_raises_order (O R : Type) [CommRing O] [CommRing R] [Algebra O R]
    (n : ℕ) (pi : O) (b : Fin n → R) (E : R ≃ₗ[O] ((Fin n → O) × R)) :
    RaisesAdicOrder (Ideal.span {pi}) (coordinateRemainder pi b E) := by sorry