Structural facts on the coefficient space `O^n`: its Borel σ-algebra is generated by `π`-adic cylinders, the coefficient measure is a Haar probability measure, its `π`-adic quotients are finite, and s

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Blowup_Directions
import Definitions.Def_TypeDensity_Measure_AdicCylinders
import Definitions.Def_TypeDensity_Measure_HenselProducts
import Definitions.Def_TypeDensity_Surfaces_Lengths
import Definitions.Def_TypeDensity_Unramified_Extension
import Definitions.Def_TypeDensity_Blowup_Charts
import Theorems.Thm_TypeDensity_finite_coordinate_borel_cylinders
import Theorems.Thm_TypeDensity_finite_coordinate_linear_continuous
import Theorems.Thm_TypeDensity_residue_quotient_card
import Theorems.Thm_TypeDensity_coefficient_measure_probability
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.Topology.Algebra.MvPolynomial
import Mathlib.Algebra.Group.Pi.Units
import Mathlib.Order.CompletePartialOrder
import Mathlib.RingTheory.Henselian
import Mathlib.RingTheory.Polynomial.Resultant.Basic
import Mathlib.RingTheory.RegularLocalRing.Defs
import Mathlib.RingTheory.SimpleRing.Principal
import Mathlib.FieldTheory.Finite.Extension
import Mathlib.RingTheory.AdicCompletion.Noetherian
import Mathlib.RingTheory.MvPowerSeries.Equiv
import Mathlib.RingTheory.PicardGroup
set_option autoImplicit false

open TypeDensity

theorem TypeDensity.coefficient_space_strict_instances (O : Type) [CommRing O] [IsDomain O]
    [IsDiscreteValuationRing O] [IsAdicComplete (IsLocalRing.maximalIdeal O) O]
    [Finite (IsLocalRing.ResidueField O)] [TopologicalSpace O] [IsTopologicalRing O]
    [CompactSpace O] [T2Space O] [SecondCountableTopology O] [MeasurableSpace O] [BorelSpace O]
    (htop : IsAdic (IsLocalRing.maximalIdeal O)) (pi : O) (hpi : Irreducible pi) (n : ℕ) :
    (inferInstance : MeasurableSpace (Fin n → O)) = adicCylinderSpace (Ideal.span {pi}) (Fin n → O)
        ∧ MeasureTheory.Measure.IsAddHaarMeasure (coefficientMeasure O n) ∧
        MeasureTheory.IsProbabilityMeasure (coefficientMeasure O n) ∧
        (∀ j, Finite (AdicQuotient (Ideal.span {pi}) (Fin n → O) j)) ∧ ∀ (a : ℕ)
        (L : (Fin n → O) →ₗ[O] (Fin n → O)), Continuous (fun z : Fin n → O => (pi^a) • L z) := by sorry