Bennett Cleared Matrix det formula

import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.CharZero.Infinite
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.Algebra.Polynomial.Roots
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.PowerSeries.Binomial
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_BennettDeterminantConstant_Definitions
import Definitions.Def_D1Bound5_BennettDeterminantFormula_Definitions

section
open Finset

namespace D1Bound5

variable {F : Type*} [Field F] [CharZero F]












theorem _root_.D1Bound5.bennettClearedMatrix_det_formula
    (n : ℕ) (a : Fin (n + 1) → F) (ha : Function.Injective a)
    (k : ℕ) (hk : 1 ≤ k) (ν : F) (h0 : a 0 = 0) :
    (bennettClearedMatrix a k ν).det =
      Polynomial.C (bennettBinomialProduct (n + 1) k ν * bennettDirectedProduct a k) := by sorry



end D1Bound5
end