For separable `F` the route-F and original polynomial types agree: every factor is separable, so its closure is finite by `closure_finite_of_separable`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Algebra_AlgebraType
import Theorems.Thm_TypeDensity_closure_finite_of_separable
import Theorems.Thm_TypeDensity_HasPolynomialTypeF_to_original
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
import Mathlib.RingTheory.Henselian
set_option autoImplicit false

open TypeDensity
open scoped TensorProduct
open Polynomial IsLocalRing

theorem TypeDensity.hasPolynomialTypeF_iff_of_separable (O : Type) [CommRing O] [IsDomain O]
    [IsDiscreteValuationRing O] {F : (BaseField O)[X]} {σ : FactorType} (hs : F.Separable) :
    HasPolynomialTypeF O F σ ↔ HasPolynomialType O F σ := by sorry