The inverse dilation commutes with the plethystic powers `P_w` (for `g(0) = 1`).

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Reciprocity_Defs
import Definitions.Def_TypeDensity_Reciprocity_Quantum
import Theorems.Thm_TypeDensity_qpow_zero
import Theorems.Thm_TypeDensity_coeff_codilate
import Theorems.Thm_TypeDensity_codilate_dilate
import Theorems.Thm_TypeDensity_dilate_codilate
import Theorems.Thm_TypeDensity_dilate_pPow
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.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.PowerSeries.Expand
import Mathlib.RingTheory.PowerSeries.Log
set_option autoImplicit false

open TypeDensity

theorem TypeDensity.codilate_pPow (w : RatFunc ℚ) (g : Ser) (hg : PowerSeries.constantCoeff g = 1) :
    codilate (pPow w g) = pPow w (codilate g) := by sorry