The `n`-th coefficient of `P ∘ f` (for `f(0) = 0`) only involves the powers `f^d`, `d ≤ n`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Reciprocity_Defs
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.coeff_subst_eq_sum (n : ℕ) (P f : Ser) (hf : PowerSeries.constantCoeff f = 0) :
    PowerSeries.coeff n (P.subst f) =
      ∑ d ∈ Finset.range (n + 1), PowerSeries.coeff d P • PowerSeries.coeff n (f ^ d) := by sorry