import OAI.Analysis.DirectCrouzeix.OuterApproximation

namespace OAI

universe u_144 u_145

noncomputable section

open scoped Matrix Matrix.Norms.L2Operator Kronecker

noncomputable section

open MeasureTheory Set Filter Metric

open scoped Topology Interval ENNReal NNReal ComplexConjugate

noncomputable section

open Filter Metric Set

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set

open scoped ComplexConjugate Matrix

namespace DirectCrouzeix

open Set Filter Metric

theorem expLevel_domain_bound {n m : ℕ} (hn : 0 < n) (hm : 0 < m)
    (A : Matrix (Fin n) (Fin n) ℂ) {ι : Type u_144} [Fintype ι]
    (v : ι → ℂ) (b : ι → ℝ) (hc : IsCompact {z | Geometry.expLevel v b z ≤ 1})
    (hW : ∀ z ∈ numericalRange A, Geometry.expLevel v b z < 1)
    (F : MatrixPolynomial m)
    (hF : ∀ z, Geometry.expLevel v b z ≤ 1 → ‖matrixPolynomialValue z F‖ ≤ 1) :
    ‖tensorPolynomial A F‖ ≤ 2 := by
  obtain ⟨a,ha⟩ := numericalRange_nonempty hn A
  obtain ⟨R,hR,c,g,hc0,hg,hinj,hder,hcircle,hout⟩ :=
    Geometry.exists_expLevel_exterior v b hc (hW a ha)
  have hge (ζ : ℂ) (hζ : 1 ≤ ‖ζ‖) : 1 ≤ Geometry.expLevel v b (Faber.exterior c g ζ) := by
    rcases eq_or_lt_of_le hζ with he|hl
    · exact le_of_eq (hcircle ζ he.symm).symm
    · exact le_of_lt (hout ζ hl)
  have hs (t : Faber.Angle) (w : ℂ) (hw : Geometry.expLevel v b w ≤ 1) :
      0 ≤ (conj (Faber.boundaryNormal c g t)*(Faber.boundary c g t-w)).re := by
    have hdt := Faber.hasDerivAt_exterior hg c (Faber.unitPoint_ne_zero t) (Faber.unitPoint_inv_mem hR t)
    have hh := Geometry.expLevel_holomorphic_support v b (hW a ha) (Faber.unitPoint_norm t)
      hdt.differentiableAt.hasDerivAt (hder _ (Faber.unitPoint_ne_zero t) (Faber.unitPoint_inv_mem hR t)) hcircle hge hw
    have he : Faber.unitPoint t*deriv (Faber.exterior c g) (Faber.unitPoint t) = Faber.boundaryNormal c g t := by
      rw [hdt.deriv]; rfl
    rw [he,real_inner_eq_re_inner ℂ,RCLike.inner_apply] at hh
    change 0 ≤ ((Faber.exterior c g (Faber.unitPoint t)-w)*conj (Faber.boundaryNormal c g t)).re at hh
    simpa only [Faber.boundary,mul_comm] using hh
  apply exterior_collar_bound hn hm A hR hg hc0 hinj hder
  · intro t s
    exact hs t _ (le_of_eq (hcircle _ (Faber.unitPoint_norm s)))
  · intro ζ hζ hz
    have h1 := hge ζ hζ
    have h2 := hW _ hz
    linarith
  · intro t w hw
    exact hs t w (le_of_lt (hW w hw))
  · intro t
    exact hF _ (le_of_eq (hcircle _ (Faber.unitPoint_norm t)))

theorem expLevel_domain_bound_scaled {n m : ℕ} (hn : 0 < n) (hm : 0 < m)
    (A : Matrix (Fin n) (Fin n) ℂ) {ι : Type u_145} [Fintype ι]
    (v : ι → ℂ) (b : ι → ℝ) (hc : IsCompact {z | Geometry.expLevel v b z ≤ 1})
    (hW : ∀ z ∈ numericalRange A, Geometry.expLevel v b z < 1)
    (F : MatrixPolynomial m) {ρ : ℝ} (hρ : 0 < ρ)
    (hF : ∀ z, Geometry.expLevel v b z ≤ 1 → ‖matrixPolynomialValue z F‖ ≤ ρ) :
    ‖tensorPolynomial A F‖ ≤ 2*ρ := by
  have hb := expLevel_domain_bound hn hm A v b hc hW (((ρ⁻¹:ℝ):ℂ) • F) (by
    intro z hz
    rw [matrixPolynomialValue_smul,norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hρ)]
    exact (mul_le_mul_of_nonneg_left (hF z hz) (le_of_lt (inv_pos.mpr hρ))).trans_eq (inv_mul_cancel₀ (ne_of_gt hρ)))
  rw [tensorPolynomial_smul,norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hρ)] at hb
  have hh := mul_le_mul_of_nonneg_left hb (le_of_lt hρ)
  rw [← mul_assoc,mul_inv_cancel₀ (ne_of_gt hρ),one_mul] at hh
  simpa only [mul_comm] using hh

end DirectCrouzeix

namespace DirectCrouzeix

open Set Metric

end DirectCrouzeix

end

end

end

end

end

end

end

end

end

end OAI