Signed root matrix operator bound

import Mathlib.Analysis.CStarAlgebra.Matrix
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Permutation
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_DeterminantNormalization_Definitions
import Definitions.Def_D1Bound5_RootMatrixOperatorBound_Definitions
import Definitions.Def_D1Bound5_RootMatrixSigns_Definitions

section
open scoped Matrix.Norms.L2Operator

namespace D1Bound5




theorem _root_.D1Bound5.signed_root_matrix_operator_bound
    {R : Type*} [CommRing R] (f : R →+* ℂ)
    (a : Fin 4 → R) (u : Fin 4 → ℂ)
    (S : Matrix (Fin 4) (Fin 4) R)
    (hu : ∀ i, (u i)^2 = f (a i))
    (hdiag : ∀ i, S i i = a i)
    (hgood : ∀ i j, i ≠ j →
      SignedEntryGood f (a i) (a j) (u i) (u j) (S i j)) :
    ‖normalizedRootError (fun i => f (a i)) u
      (fun i j => f (S i j)) - halfOffDiagonal‖ <
      3 * (128/8649 : ℝ) := by sorry



end D1Bound5
end