Exists cleared Pole decomposition zero residue sum

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]













end CommRing

section Field

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




theorem _root_.D1Bound5.exists_clearedPole_decomposition_zero_residue_sum
    (a : ι → F) (ha : Function.Injective a) (e : ι → ℕ) (f : F[X])
    (hdeg : f.degree < ((∑ i, (e i + 1)) - 1 : ℕ)) :
    ∃ c : (i : ι) → Fin (e i + 1) → F,
      (f = ∑ i, ∑ j : Fin (e i + 1), C (c i j) * clearedPoleNumerator a e i j) ∧
      ∑ i, c i (Fin.last (e i)) = 0 := by sorry

end Field

end D1Bound5
end