Cleared Pole Numerator degree lt total

import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.PartialFractions
import Mathlib.Algebra.Polynomial.RingDivision
import Definitions.Def_D1Bound5_BennettResidueCoefficient_Definitions

section
open Finset Polynomial

namespace D1Bound5

/-!
This module proves existence of the exact cleared partial-fraction decomposition
used by the residue-sum argument. The decomposition is a conclusion, not an
assumption. Identifying its coefficients with the manuscript's explicit finite
binomial sums remains a separate obligation.
-/

section CommRing

variable {R ι : Type*} [CommRing R] [Nontrivial R]
variable [Fintype ι] [DecidableEq ι]










theorem _root_.D1Bound5.clearedPoleNumerator_degree_lt_total (a : ι → R) (e : ι → ℕ)
    (i : ι) (j : Fin (e i + 1)) :
    (clearedPoleNumerator a e i j).degree < (∑ l, (e l + 1) : ℕ) := by sorry



end CommRing

section Field

variable {F ι : Type*} [Field F] [Fintype ι] [DecidableEq ι]





end Field

end D1Bound5
end