The multiplicity of an affine direction `φ` in `c · divisorAffinePolynomial Dv` is `Dv φ`;

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Blowup_Directions
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
set_option autoImplicit false

open TypeDensity
open scoped Classical

theorem TypeDensity.count_normalizedFactors_divisorAffinePolynomial (k : Type) [Field k]
    (Dv : ExceptionalDivisor k) (lc : k) (hlc : lc ≠ 0) (φ : AffineDirection k) :
    Multiset.count φ.val (UniqueFactorizationMonoid.normalizedFactors
        (Polynomial.C lc * divisorAffinePolynomial k Dv)) = Dv (some φ) ∧
        (φ.val ∣ divisorAffinePolynomial k Dv ↔ 0 < Dv (some φ)) := by sorry