Full ring scaled Root Matrix det norm lower

import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.Complex.Norm
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Int.GCD
import Mathlib.Data.ZMod.Basic
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SchurComplement
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_DeterminantNormalization_Definitions

section
open scoped NumberField BigOperators
open Module

namespace D1Bound5






theorem _root_.D1Bound5.full_ring_scaledRootMatrix_det_norm_lower
    {K n : Type*} [Field K] [NumberField K]
    [NumberField.IsTotallyComplex K] [Fintype n] [DecidableEq n]
    (hK : finrank ℚ K = 2) (phi : K →+* ℂ)
    (hn : Even (Fintype.card n))
    (a : n → 𝓞 K) (M : Matrix n n (𝓞 K))
    (hd : ∀ i, M i i = a i)
    (hs : ∀ i j, i ≠ j → (M i j)^2 = a i*a j+1)
    (u : n → ℂ) (hu : ∀ i, (u i)^2 = phi (a i : K)) :
    (∏ i, ‖phi (a i : K)‖) ≤
      ‖(scaledRootMatrix u
        (M.map (phi.comp (algebraMap (𝓞 K) K)))).det‖ := by sorry

end D1Bound5
end