Bennett Polynomial Residue residual tail has Sum

import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Operations
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.PartialFractions
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Analysis.Analytic.Binomial
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.PowerSeries.Binomial
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_BennettLocalResidues_Definitions
import Definitions.Def_D1Bound5_BennettResiduePolynomials_Definitions

section
open Finset

namespace D1Bound5





section Nodes

variable {I : Type*} [Fintype I] [DecidableEq I]






theorem _root_.D1Bound5.bennettPolynomialResidue_residual_tail_hasSum
    (a : I → ℂ) (ha : Function.Injective a) (k : ℕ) (hk : 1 ≤ k)
    (ν x : ℂ) (i : I) (hx : ∀ j, ‖a j * x‖ < 1) :
    HasSum (fun t : ℕ =>
      (Ring.choose ((k : ℂ) + ν) (Fintype.card I * k + t) *
        ∑ j, bennettLocalResidue a
          (fun l => k + (if i = l then 1 else 0) - 1)
          (Polynomial.X ^ (Fintype.card I * k + t)) j) *
            x ^ (Fintype.card I * k + t))
      (∑ j, (bennettPolynomialResidue univ a k ν i j).eval x *
        (1 + a j * x) ^ ν) := by sorry

end Nodes

end D1Bound5
end