Bennett exists integral root ratio

import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Operations
import Mathlib.Analysis.Analytic.Binomial
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Ring.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Combinatorics.Enumerative.Catalan.Basic
import Mathlib.Data.Finset.Insert
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Card
import Mathlib.LinearAlgebra.Lagrange
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
import Mathlib.RingTheory.PowerSeries.Binomial
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring

section
namespace D1Bound5






theorem _root_.D1Bound5.bennett_exists_integral_root_ratio
    {R : Type*} [CommRing R] (f : R →+* ℂ)
    (a c d z r v : R) (θ : ℂ)
    (ha : f a ≠ 0) (hc : f c ≠ 0) (hz : f z ≠ 0) (hθ0 : θ ≠ 0)
    (hr : r ^ 2 = a * c + 1) (hv : v ^ 2 = a * d + 1)
    (hzsq : z ^ 2 = c * d + 1) (hθ : θ ^ 2 = 1 + 1 / (f a * f c)) :
    ∃ s : R, (s = r * v ∨ s = -(r * v)) ∧
      0 ≤ ((f s / (f a * f z)) / θ).re ∧
      ‖θ - f s / (f a * f z)‖ ≤
        ‖θ‖ * ‖f c - f a‖ / (‖f a‖ * ‖f z‖ ^ 2) := by sorry





end D1Bound5
end