lean/OAI/Analysis/DirectCrouzeix/SupportNets.lean

import OAI.Analysis.DirectCrouzeix.ExteriorMapping

namespace OAI

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

namespace DirectCrouzeix.Geometry

theorem exists_axis_net {δ : ℝ} (hδ : 0 < δ) :
    ∃ D : Finset ℂ, (∀ ν ∈ D, ‖ν‖ = 1) ∧ 1 ∈ D ∧ -1 ∈ D ∧
      Complex.I ∈ D ∧ -Complex.I ∈ D ∧
      (∀ ν : ℂ, ‖ν‖ = 1 → ∃ μ ∈ D, ‖ν-μ‖ < δ) := by
  classical
  obtain ⟨S,hS⟩ := (isCompact_sphere (0:ℂ) 1).elim_finite_subcover
    (fun ν : sphere (0:ℂ) 1 => ball (ν:ℂ) δ) (fun _ => isOpen_ball) (by
      intro ν hν
      exact mem_iUnion.mpr ⟨⟨ν,hν⟩,mem_ball_self hδ⟩)
  let D := S.image Subtype.val ∪ {1,-1,Complex.I,-Complex.I}
  refine ⟨D,?_,?_,?_,?_,?_,?_⟩
  · intro ν hν
    rcases Finset.mem_union.mp hν with hν | hν
    · obtain ⟨μ,hμ,rfl⟩ := Finset.mem_image.mp hν
      simpa only [mem_sphere,dist_zero_right] using μ.property
    · simp only [Finset.mem_insert,Finset.mem_singleton] at hν
      rcases hν with rfl | rfl | rfl | rfl <;> simp
  · simp [D]
  · simp [D]
  · simp [D]
  · simp [D]
  · intro ν hν
    obtain ⟨μ,hμ,hνμ⟩ := mem_iUnion₂.mp (hS (by simpa only [mem_sphere,dist_zero_right] using hν))
    refine ⟨μ,Finset.mem_union_left _ (Finset.mem_image.mpr ⟨μ,hμ,rfl⟩),?_⟩
    simpa [dist_eq_norm] using hνμ

def convexSupport (K : Set ℂ) (ν : ℂ) : ℝ := sSup ((fun z => inner ℝ ν z) '' K)

theorem convexSupport_attained {K : Set ℂ} (hK : IsCompact K) (hne : K.Nonempty) (ν : ℂ) :
    ∃ w ∈ K, inner ℝ ν w = convexSupport K ν := by
  exact (hK.image ((innerSL ℝ ν).continuous)).sSup_mem (hne.image _)

theorem le_convexSupport {K : Set ℂ} (hK : IsCompact K) {z : ℂ} (hz : z ∈ K) (ν : ℂ) :
    inner ℝ ν z ≤ convexSupport K ν := by
  apply le_csSup (hK.image ((innerSL ℝ ν).continuous)).bddAbove
  exact mem_image_of_mem _ hz

theorem convexSupport_le {K : Set ℂ} (hne : K.Nonempty) {M : ℝ}
    (hM : ∀ z ∈ K, ‖z‖ ≤ M) {ν : ℂ} (hν : ‖ν‖ = 1) : convexSupport K ν ≤ M := by
  apply csSup_le (hne.image _)
  rintro x ⟨z,hz,rfl⟩
  exact (real_inner_le_norm ν z).trans (by simpa only [hν,one_mul] using hM z hz)

theorem direction_net_distance {K : Set ℂ} (hK : IsCompact K) (hne : K.Nonempty)
    (hconv : Convex ℝ K) {D : Finset ℂ} {δ M L : ℝ} (hδ : 0 < δ)
    (hM : ∀ w ∈ K, ‖w‖ ≤ M)
    (hnet : ∀ ν : ℂ, ‖ν‖ = 1 → ∃ μ ∈ D, ‖ν-μ‖ < δ)
    {z : ℂ} (hz : ‖z‖ ≤ L)
    (hs : ∀ μ ∈ D, inner ℝ μ z ≤ convexSupport K μ+δ) :
    ∃ w ∈ K, ‖z-w‖ ≤ δ*(1+L+M) := by
  obtain ⟨w,hw,hwmin⟩ := exists_norm_eq_iInf_of_complete_convex hne hK.isClosed.isComplete hconv z
  have hM0 : 0 ≤ M := (norm_nonneg w).trans (hM w hw)
  have hL0 : 0 ≤ L := (norm_nonneg z).trans hz
  refine ⟨w,hw,?_⟩
  by_cases he : z = w
  · rw [he,sub_self,norm_zero]
    positivity
  have hn : 0 < ‖z-w‖ := norm_pos_iff.mpr (sub_ne_zero.mpr he)
  let ν := (‖z-w‖⁻¹:ℝ) • (z-w)
  have hν : ‖ν‖ = 1 := by simp [ν,ne_of_gt hn]
  obtain ⟨μ,hμ,hνμ⟩ := hnet ν hν
  obtain ⟨u,hu,hsu⟩ := convexSupport_attained hK hne μ
  have hmin : inner ℝ ν (u-w) ≤ 0 := by
    have hh := (norm_eq_iInf_iff_real_inner_le_zero hconv hw).mp hwmin u hu
    dsimp only [ν]
    rw [real_inner_smul_left]
    exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt (inv_pos.mpr hn)) hh
  have hzw : inner ℝ ν (z-w) = ‖z-w‖ := by
    dsimp only [ν]
    rw [real_inner_smul_left,real_inner_self_eq_norm_sq]
    field_simp
  have hmain : ‖z-w‖ ≤ inner ℝ ν (z-u) := by
    have heq : z-w = (z-u)+(u-w) := by abel
    rw [← hzw,heq,inner_add_right]
    linarith
  have hμbound : inner ℝ μ (z-u) ≤ δ := by
    rw [inner_sub_right,hsu]
    linarith [hs μ hμ]
  have hnorm : ‖z-u‖ ≤ L+M := (norm_sub_le z u).trans (add_le_add hz (hM u hu))
  have herror : inner ℝ (ν-μ) (z-u) ≤ δ*(L+M) := by
    calc
      inner ℝ (ν-μ) (z-u) ≤ ‖ν-μ‖*‖z-u‖ := real_inner_le_norm _ _
      _ ≤ δ*(L+M) := mul_le_mul (le_of_lt hνμ) hnorm (norm_nonneg _) (le_of_lt hδ)
  have heq : inner ℝ ν (z-u) = inner ℝ μ (z-u)+inner ℝ (ν-μ) (z-u) := by
    rw [inner_sub_left]; ring
  rw [heq] at hmain
  nlinarith

theorem axis_support_bound {K : Set ℂ} (hne : K.Nonempty) {M δ : ℝ}
    (hM : ∀ w ∈ K, ‖w‖ ≤ M) {D : Finset ℂ}
    (h1 : 1 ∈ D) (hm1 : -1 ∈ D) (hI : Complex.I ∈ D) (hmI : -Complex.I ∈ D)
    {z : ℂ} (hs : ∀ μ ∈ D, inner ℝ μ z ≤ convexSupport K μ+δ) :
    ‖z‖ ≤ 2*(M+δ) := by
  have hb (μ : ℂ) (hμ : μ ∈ D) (hμN : ‖μ‖ = 1) : inner ℝ μ z ≤ M+δ :=
    (hs μ hμ).trans (add_le_add_left (convexSupport_le hne hM hμN) δ)
  have hr := hb 1 h1 (by simp)
  have hrr := hb (-1) hm1 (by simp)
  have hi := hb Complex.I hI (by simp)
  have hir := hb (-Complex.I) hmI (by simp)
  simp only [real_inner_eq_re_inner ℂ,RCLike.inner_apply] at hr hrr hi hir
  simp at hr hrr hi hir
  have ha : |z.re| ≤ M+δ := abs_le.mpr ⟨by linarith,hr⟩
  have hb : |z.im| ≤ M+δ := abs_le.mpr ⟨by linarith,hi⟩
  calc
    ‖z‖ ≤ |z.re|+|z.im| := Complex.norm_le_abs_re_add_abs_im z
    _ ≤ 2*(M+δ) := by linarith

end DirectCrouzeix.Geometry

noncomputable section

open Set

open scoped ComplexConjugate Matrix

namespace DirectCrouzeix.Geometry

end DirectCrouzeix.Geometry

end

end

end

end

end

end

end

end

end

end OAI