Coefficient formula for a product of two laws: a convolution over the types of `L`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Theorems.Thm_TypeDensity_lawInterp_lawMul
import Theorems.Thm_TypeDensity_lawInterp_coeff
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.Finite.GaloisField
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.RingTheory.RegularLocalRing.Defs
set_option autoImplicit false

open TypeDensity
open scoped BigOperators ENNReal

theorem TypeDensity.law_coeff_mul (L M : Law) (σ : FactorType) (q : ℚ) :
    (lawCoeff (lawMul L M) σ).eval q = ∑ τ ∈ (L.map Prod.fst).toFinset,
      (if τ ≤ σ then (lawCoeff L τ).eval q * (lawCoeff M (σ - τ)).eval q else 0) := by sorry