If two ideals of the same fixed tangent cell have equal strict transforms at the points of positive divisor multiplicity, and the first has the prescribed strict transforms at all datum points, then s

import Definitions.Def_TypeDensity_Basic
import Definitions.Def_TypeDensity_Formula_LawAlgebra
import Definitions.Def_TypeDensity_Measure_AdicCylinders
import Definitions.Def_TypeDensity_Measure_HenselProducts
import Definitions.Def_TypeDensity_Surfaces_Lengths
import Definitions.Def_TypeDensity_Surfaces_PowerSeries
import Definitions.Def_TypeDensity_Formula_FiniteFieldCounts
import Definitions.Def_TypeDensity_Formula_StateCombinatorics
import Definitions.Def_TypeDensity_Blowup_Directions
import Definitions.Def_TypeDensity_Unramified_Extension
import Definitions.Def_TypeDensity_Blowup_Charts
import Definitions.Def_TypeDensity_Algebra_AlgebraType
import Definitions.Def_TypeDensity_Measure_PrimitiveLocus
import Definitions.Def_TypeDensity_Algebra_FactorInvariant
import Definitions.Def_TypeDensity_Surfaces_TruncatedSeries
import Theorems.Thm_TypeDensity_directionPoint_properties
import Theorems.Thm_TypeDensity_count_normalizedFactors_divisorAffinePolynomial
import Theorems.Thm_TypeDensity_direction_point_overlap_maximal
import Theorems.Thm_TypeDensity_infinity_point_maximal
import Theorems.Thm_TypeDensity_reduced_form_eq_of_residue_coeff
import Theorems.Thm_TypeDensity_fixed_tangent_cell_generator_coefficients
import Theorems.Thm_TypeDensity_concrete_infinity_strict_transform_multiplicity
import Theorems.Thm_TypeDensity_concrete_strict_transform_generators
import Theorems.Thm_TypeDensity_concrete_strict_transform_multiplicity
import Theorems.Thm_TypeDensity_strict_transform_common_point_transport
import Theorems.Thm_TypeDensity_blowupChart_concrete_isNoetherianRing
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.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.SimpleRing.Principal
import Mathlib.FieldTheory.Finite.Extension
import Mathlib.RingTheory.AdicCompletion.Noetherian
import Mathlib.RingTheory.MvPowerSeries.Equiv
import Mathlib.RingTheory.PicardGroup
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.AdicCompletion.Completeness
import Mathlib.Algebra.GroupWithZero.Submonoid.CancelMulZero
import Mathlib.Algebra.Module.StablyFree.Basic
import Mathlib.NumberTheory.Padics.RingHoms
import Mathlib.Algebra.Azumaya.Basic
import Mathlib.RingTheory.AdicCompletion.RingHom
import Mathlib.RingTheory.TotallySplit
import Mathlib.RingTheory.AdicCompletion.LocalRing
import Mathlib.RingTheory.KrullDimension.NonZeroDivisors
import Mathlib.Algebra.Group.NatPowAssoc
set_option autoImplicit false

open TypeDensity
open scoped BigOperators ENNReal

theorem TypeDensity.cell_strict_transforms_agree_at_datum_points (O : Type) [CommRing O] [IsDomain O]
    [IsDiscreteValuationRing O] [IsAdicComplete (IsLocalRing.maximalIdeal O) O]
    [Finite (IsLocalRing.ResidueField O)] (pi : O) (hpi : Irreducible pi) (u : (TwoSeries O)ˣ)
    (a b : ℕ) (ha : 0 < a) [IsDomain (ConcreteSurface O pi u a b)]
    [IsRegularLocalRing (ConcreteSurface O pi u a b)]
    (S : SurfaceData O (ConcreteSurface O pi u a b))
    (hS : S.t = (surfaceT O pi u a b) ∧ S.x = (surfaceX O pi u a b) ∧ S.a = a ∧ S.b = b)
    (e : ((BlowupChart (ConcreteSurface O pi u a b) (surfaceT O pi u a b) (surfaceX O pi u a b)) ⧸
    chartExceptional (ConcreteSurface O pi u a b) (surfaceT O pi u a b) (surfaceX O pi u a b)) ≃+*
    Polynomial (IsLocalRing.ResidueField (ConcreteSurface O pi u a b)))
    (he : ∀ r : (ConcreteSurface O pi u a b), e
    (Ideal.Quotient.mk
    (chartExceptional (ConcreteSurface O pi u a b) (surfaceT O pi u a b) (surfaceX O pi u a b))
    ((chartBase (ConcreteSurface O pi u a b) (surfaceT O pi u a b) (surfaceX O pi u a b)) r)) =
    Polynomial.C (IsLocalRing.residue (ConcreteSurface O pi u a b) r))
    (heZ : e (Ideal.Quotient.mk
    (chartExceptional (ConcreteSurface O pi u a b) (surfaceT O pi u a b) (surfaceX O pi u a b))
    (chartSlope (ConcreteSurface O pi u a b) (surfaceT O pi u a b) (surfaceX O pi u a b))) =
    Polynomial.X) (Dv : ExceptionalDivisor (IsLocalRing.ResidueField (ConcreteSurface O pi u a b)))
    (hinf : (infinityPoint (ConcreteSurface O pi u a b) (surfaceT O pi u a b)
    (surfaceX O pi u a b)).IsPrime) (s : State) (d : ℕ) (J₁ J₂ : Ideal (ConcreteSurface O pi u a b))
    (h₁ : J₁ ∈ (fixedTangentCell O (ConcreteSurface O pi u a b) S s d Dv))
    (h₂ : J₂ ∈ (fixedTangentCell O (ConcreteSurface O pi u a b) S s d Dv))
    (heq : ∀ Q : divisorSupport Dv, divisorStrictTransform (ConcreteSurface O pi u a b)
    (surfaceT O pi u a b) (surfaceX O pi u a b) e Dv hinf d J₁ Q = divisorStrictTransform
    (ConcreteSurface O pi u a b) (surfaceT O pi u a b) (surfaceX O pi u a b) e Dv hinf d J₂ Q)
    (beta : BlowupRestriction (ConcreteSurface O pi u a b) (surfaceT O pi u a b)
    (surfaceX O pi u a b))
    (D : TwoChartCurveData (ConcreteSurface O pi u a b) (surfaceT O pi u a b) (surfaceX O pi u a b)
    beta) (hm : MatchesDivisor e D Dv)
    (hst₁ : ∀ s : D.ι ⊕ D.ι0, strictTransformAt (ConcreteSurface O pi u a b) (surfaceT O pi u a b)
    (surfaceX O pi u a b) (leftPoint D s) d J₁ = Ideal.span {leftData D s})
    (hst₁r : ∀ s : D.ι ⊕ D.ιinf, strictTransformAt (ConcreteSurface O pi u a b)
    (surfaceX O pi u a b) (surfaceT O pi u a b) (rightPoint D s) d J₁ = Ideal.span {rightData D s})
    :
    (∀ s : D.ι ⊕ D.ι0, strictTransformAt (ConcreteSurface O pi u a b) (surfaceT O pi u a b)
      (surfaceX O pi u a b) (leftPoint D s) d J₂ = Ideal.span {leftData D s}) ∧ ∀ s : D.ι ⊕ D.ιinf,
      strictTransformAt (ConcreteSurface O pi u a b) (surfaceX O pi u a b) (surfaceT O pi u a b)
      (rightPoint D s) d J₂ = Ideal.span {rightData D s} := by sorry