Algebra types descend along an unramified extension `O → Oδ` of degree `δ`, scaled by `δ`.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Algebra_AlgebraType
import Definitions.Def_TypeDensity_Measure_AdicCylinders
import Definitions.Def_TypeDensity_Measure_HenselProducts
import Definitions.Def_TypeDensity_Unramified_Extension
import Definitions.Def_TypeDensity_Measure_PrimitiveLocus
import Definitions.Def_TypeDensity_Algebra_FactorInvariant
import Theorems.Thm_TypeDensity_closure_finite_descent
import Theorems.Thm_TypeDensity_unramified_generic_fibre_witness
import Theorems.Thm_TypeDensity_field_invariant_unramified_tower
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
import Mathlib.Topology.Algebra.MvPolynomial
import Mathlib.Algebra.Group.Pi.Units
import Mathlib.Order.CompletePartialOrder
import Mathlib.RingTheory.Polynomial.Resultant.Basic
import Mathlib.RingTheory.SimpleRing.Principal
import Mathlib.Algebra.GroupWithZero.Submonoid.CancelMulZero
import Mathlib.Algebra.Module.StablyFree.Basic
import Mathlib.NumberTheory.Padics.RingHoms
set_option autoImplicit false

open TypeDensity
open Polynomial

theorem TypeDensity.algebra_type_unramified_descent (O Oδ : Type) [CommRing O] [IsDomain O]
    [IsDiscreteValuationRing O] [CommRing Oδ] [IsDomain Oδ] [IsDiscreteValuationRing Oδ]
    [Algebra O Oδ] [Module.Finite O Oδ] [Module.Free O Oδ] (δ : ℕ) (hrank : Module.finrank O Oδ = δ)
    (pi : O) (hpi : Irreducible pi) (hpiδ : Irreducible (algebraMap O Oδ pi))
    [IsAdicComplete (IsLocalRing.maximalIdeal O) O] [Finite (IsLocalRing.ResidueField O)] (B : Type)
    [CommRing B] [Algebra O B] [Algebra Oδ B] [IsScalarTower O Oδ B] (σ : FactorType)
    (h : HasAlgebraType Oδ B σ) :
    HasAlgebraType O B (scaleType δ σ) := by sorry