lean/OAI/Analysis/C0Absorption/Grid.lean

import OAI.Analysis.C0Absorption.Detection

namespace OAI

open Set Filter Topology
open scoped NNReal BigOperators ZeroAtInfty
open NormedSpace

namespace C0Absorption
noncomputable section
open Set Filter Topology
open scoped NNReal BigOperators ZeroAtInfty

abbrev BandGrid (j : ℕ) := Fin j → GridIndex (2^j)
abbrev PrefixCube (j : ℕ) := Fin j → Icc (-1 : ℝ) 1

def bandPoint (j : ℕ) (ξ : BandGrid j) : PrefixCube j :=
  fun k => gridPoint (2^j) (by positivity) (ξ k)

def prefixRestrict (j : ℕ) (s : C0Ball) : PrefixCube j := fun k =>
  ⟨s.val k, abs_le.mp ((c0_norm_apply_le s.val k).trans (by
    have hh := s.property
    simpa only [Metric.mem_closedBall,dist_zero_right] using hh))⟩

theorem prefixRestrict_lipschitz (j : ℕ) : LipschitzWith 1 (prefixRestrict j) := by
  apply LipschitzWith.of_dist_le_mul
  intro x y
  rw [NNReal.coe_one,one_mul]
  apply (dist_pi_le_iff dist_nonneg).mpr
  intro k
  exact BoundedContinuousFunction.dist_coe_le_dist (f := x.val.toBCF) (g := y.val.toBCF) k

def bandBump (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : ℝ :=
  max 0 (1-dist s (bandPoint j ξ)/dyadic j)

def bandVector (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : ℝ :=
  min 1 (max 0 (2-dist s (bandPoint j ξ)/dyadic j))

def bandDenominator (j : ℕ) (s : PrefixCube j) : ℝ := ∑ ξ : BandGrid j, bandBump j ξ s

def bandRow (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) : ℝ :=
  bandBump j ξ s / bandDenominator j s

theorem bandBump_nonneg (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) :
    0 ≤ bandBump j ξ s := le_max_left _ _

theorem bandVector_nonneg (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) :
    0 ≤ bandVector j ξ s := le_min zero_le_one (le_max_left _ _)

theorem bandVector_le_one (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) :
    bandVector j ξ s ≤ 1 := min_le_left _ _

theorem bandVector_eq_one {j : ℕ} {ξ : BandGrid j} {s : PrefixCube j}
    (h : bandBump j ξ s ≠ 0) : bandVector j ξ s=1 := by
  have hh : 0 < 1-dist s (bandPoint j ξ)/dyadic j := by
    by_contra hn
    exact h (max_eq_left (by linarith))
  unfold bandVector
  rw [min_eq_left]
  exact le_max_of_le_right (by linarith)

theorem bandVector_support {j : ℕ} {ξ : BandGrid j} {s : PrefixCube j}
    (h : bandVector j ξ s ≠ 0) : dist s (bandPoint j ξ) < 2*dyadic j := by
  have hmax : max 0 (2-dist s (bandPoint j ξ)/dyadic j) ≠ 0 := by
    intro hh
    exact h (by simp only [bandVector,hh,min_eq_right zero_le_one])
  have hh : 0 < 2-dist s (bandPoint j ξ)/dyadic j := by
    by_contra hn
    exact hmax (max_eq_left (by linarith))
  exact (div_lt_iff₀ (dyadic_pos j)).mp (by linarith)

theorem const_dist_div_lipschitz {X : Type*} [PseudoMetricSpace X] (q : X) (a : ℝ) (j : ℕ) :
    LipschitzWith (2^j) (fun s => a-dist s q/dyadic j) := by
  apply LipschitzWith.of_dist_le_mul
  intro s t
  rw [Real.dist_eq,sub_sub_sub_cancel_left,← sub_div,abs_div,abs_of_pos (dyadic_pos j)]
  calc
    _ ≤ dist t s/dyadic j := div_le_div_of_nonneg_right (abs_dist_sub_le t s q) (dyadic_pos j).le
    _ = (2^j : ℝ≥0)*dist s t := by simp only [dyadic,div_inv_eq_mul,NNReal.coe_pow,NNReal.coe_ofNat,dist_comm]; ring

theorem bandBump_lipschitz (j : ℕ) (ξ : BandGrid j) :
    LipschitzWith (2^j) (bandBump j ξ) :=
  (const_dist_div_lipschitz (bandPoint j ξ) 1 j).const_max 0

theorem bandVector_lipschitz (j : ℕ) (ξ : BandGrid j) :
    LipschitzWith (2^j) (bandVector j ξ) :=
  ((const_dist_div_lipschitz (bandPoint j ξ) 2 j).const_max 0).const_min 1

theorem bandDenominator_ge (j : ℕ) (s : PrefixCube j) : 1/2 ≤ bandDenominator j s := by
  classical
  let ξ : BandGrid j := fun k => nearestGridIndex (2^j) (by positivity) (s k)
  have hd : dist s (bandPoint j ξ) ≤ dyadic j/2 := by
    apply (dist_pi_le_iff (div_nonneg (dyadic_pos j).le (by norm_num) : 0 ≤ dyadic j/2)).mpr
    intro k
    have hh := nearestGrid_dist (2^j) (by positivity) (s k)
    change dist (s k) (gridPoint _ _ (nearestGridIndex _ _ (s k))) ≤ _
    refine hh.trans_eq ?_
    push_cast
    unfold dyadic
    ring
  have hb : 1/2 ≤ bandBump j ξ s := by
    apply le_max_of_le_right
    have hh : dist s (bandPoint j ξ)/dyadic j ≤ 1/2 := (div_le_iff₀ (dyadic_pos j)).mpr (by linarith)
    linarith
  exact hb.trans (Finset.single_le_sum (fun a _ => bandBump_nonneg j a s) (Finset.mem_univ ξ))

theorem bandDenominator_pos (j : ℕ) (s : PrefixCube j) : 0 < bandDenominator j s :=
  lt_of_lt_of_le (by norm_num) (bandDenominator_ge j s)

theorem bandRow_nonneg (j : ℕ) (ξ : BandGrid j) (s : PrefixCube j) :
    0 ≤ bandRow j ξ s := div_nonneg (bandBump_nonneg j ξ s) (bandDenominator_pos j s).le

theorem bandRow_sum (j : ℕ) (s : PrefixCube j) : ∑ ξ : BandGrid j, bandRow j ξ s=1 := by
  simp only [bandRow,← Finset.sum_div]
  change bandDenominator j s/bandDenominator j s=1
  exact div_self (bandDenominator_pos j s).ne'

theorem bandRow_vector (j : ℕ) (s : PrefixCube j) :
    ∑ ξ : BandGrid j, bandRow j ξ s*bandVector j ξ s=1 := by
  rw [← bandRow_sum j s]
  apply Finset.sum_congr rfl
  intro ξ _
  by_cases h : bandBump j ξ s=0
  · simp only [bandRow,h,zero_div,zero_mul]
  · rw [bandVector_eq_one h,mul_one]

theorem normalized_sum_distance {A : Type*} [Fintype A] (b c : A → ℝ)
    (_hb : ∀ i, 0 ≤ b i) (hc : ∀ i, 0 ≤ c i)
    (hB : 1/2 ≤ ∑ i,b i) (hC : 1/2 ≤ ∑ i,c i) :
    (∑ i, |b i/(∑ i,b i)-c i/(∑ i,c i)|) ≤ 4*∑ i,|b i-c i| := by
  let B := ∑ i,b i
  let C := ∑ i,c i
  have hB' : 0 < B := lt_of_lt_of_le (by norm_num) hB
  have hC' : 0 < C := lt_of_lt_of_le (by norm_num) hC
  have hdist : |C-B| ≤ ∑ i,|b i-c i| := by
    rw [abs_sub_comm]
    change |(∑ i,b i)-(∑ i,c i)| ≤ _
    rw [← Finset.sum_sub_distrib]
    exact Finset.abs_sum_le_sum_abs _ _
  have hrow : ∑ i,c i/C=1 := by rw [← Finset.sum_div]; exact div_self hC'.ne'
  have he (i : A) : b i/B-c i/C=(b i-c i)/B+(c i/C)*((C-B)/B) := by
    field_simp
    ; ring
  calc
    _ ≤ ∑ i, (|b i-c i|/B+(c i/C)*(|C-B|/B)) := by
      apply Finset.sum_le_sum
      intro i _
      rw [he]
      refine (abs_add_le _ _).trans_eq ?_
      simp only [abs_mul,abs_div,abs_of_pos hB',abs_of_nonneg (hc i),abs_of_pos hC']
    _ = (∑ i,|b i-c i|)/B+|C-B|/B := by
      rw [Finset.sum_add_distrib,← Finset.sum_div,← Finset.sum_mul,hrow,one_mul]
    _ ≤ (∑ i,|b i-c i|)/B+(∑ i,|b i-c i|)/B :=
      add_le_add_right ((div_le_div_iff_of_pos_right hB').mpr hdist) _
    _ ≤ 4*∑ i,|b i-c i| := by
      have hh : (∑ i,|b i-c i|)/B ≤ 2*∑ i,|b i-c i| := by
        apply (div_le_iff₀ hB').mpr
        have ha : 0 ≤ ∑ i,|b i-c i| := Finset.sum_nonneg (fun i _ => abs_nonneg _)
        nlinarith
      linarith

def bandL (j : ℕ) : ℝ≥0 := 4*Fintype.card (BandGrid j)*2^j

theorem bandRow_distance (j : ℕ) (s t : PrefixCube j) :
    (∑ ξ : BandGrid j,|bandRow j ξ s-bandRow j ξ t|) ≤ bandL j*dist s t := by
  refine (normalized_sum_distance (fun ξ => bandBump j ξ s) (fun ξ => bandBump j ξ t)
    (fun ξ => bandBump_nonneg j ξ s) (fun ξ => bandBump_nonneg j ξ t)
    (bandDenominator_ge j s) (bandDenominator_ge j t)).trans ?_
  calc
    _ ≤ 4*∑ ξ : BandGrid j, (2 : ℝ)^j*dist s t := by
      gcongr with ξ
      simpa only [Real.dist_eq,NNReal.coe_pow,NNReal.coe_ofNat] using
        (bandBump_lipschitz j ξ).dist_le_mul s t
    _ = bandL j*dist s t := by simp [bandL]; ring

end
end C0Absorption

end OAI