The type of a finite product of algebras is the sum of their types.

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Algebra_AlgebraType
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.algebra_type_pi (O : Type) [CommRing O] [IsDomain O] (n : ℕ) (B : Fin n → Type)
    [∀ j, CommRing (B j)] [∀ j, Algebra O (B j)] (σ : Fin n → FactorType)
    (h : ∀ j, HasAlgebraType O (B j) (σ j)) : HasAlgebraType O (∀ j, B j) (∑ j, σ j) := by sorry