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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.Analysis.FiniteFactor.Sampling.Smoothing
namespace OAI
namespace FiniteFactor
open MeasureTheory
noncomputable section
/-- Interval membership cannot change under a displacement smaller than
delta when the starting point is farther than delta from both endpoints. -/
theorem interval_membership_stable (a b δ t u : ℝ) (_hδ : 0 ≤ δ)
(ha : δ < |t-a|) (hb : δ < |t-b|) (htu : |t-u| < δ) :
u ∈ Set.Icc a b ↔ t ∈ Set.Icc a b := by
have hd := abs_lt.mp htu
constructor
· intro hu
constructor
· by_contra ht
have hh : t-a < 0 := by linarith
rw [abs_of_neg hh] at ha
linarith [hu.1]
· by_contra ht
have hh : 0 < t-b := by linarith
rw [abs_of_pos hh] at hb
linarith [hu.2]
· intro ht
rw [abs_of_nonneg (sub_nonneg.mpr ht.1)] at ha
rw [abs_of_nonpos (sub_nonpos.mpr ht.2)] at hb
constructor <;> linarith
/-- A separated pair of representatives, with the first away from the cut,
has a fractional displacement outside the kernel's exceptional arc. -/
theorem fract_displacement_middle (δ t u : ℝ) (hδ : 0 < δ)
(ht : δ ≤ t) (ht' : t ≤ 1-δ) (hu : u ∈ Set.Ico (0 : ℝ) 1)
(hsep : δ ≤ |t-u|) : Int.fract (t-u) ∈ Set.Icc δ (1-δ) := by
by_cases hpos : 0 ≤ t-u
· rw [abs_of_nonneg hpos] at hsep
rw [Int.fract_eq_self.mpr ⟨hpos, by linarith [hu.1]⟩]
constructor <;> linarith [hu.1]
· have hneg : t-u < 0 := lt_of_not_ge hpos
rw [abs_of_neg hneg] at hsep
have hf : ⌊t-u⌋ = (-1 : ℤ) := Int.floor_eq_iff.mpr ⟨by norm_num; linarith [hu.2], by norm_num; linarith⟩
rw [Int.fract, hf]
norm_num
constructor <;> linarith [hu.2]
/-- The actual interval convolution differs from its indicator only by
the proved kernel tail, away from interval endpoints and the circle cut. -/
theorem samplingSmooth_interval_error (H : ℕ) (hH : 0 < H) (a b δ t : ℝ)
(hδ : 0 < δ) (ht : δ ≤ t) (ht' : t ≤ 1-δ)
(ha : δ < |t-a|) (hb : δ < |t-b|) :
|samplingSmooth H (Set.Icc a b) t - (Set.Icc a b).indicator (fun _ => (1 : ℝ)) t| ≤
1/(4*(H : ℝ)*δ^2) := by
let E := Set.Icc a b
let K := fun u => samplingKernel H (t-u)
let C := 1/(4*(H : ℝ)*δ^2)
have hC : 0 ≤ C := by dsimp [C]; positivity
have hki : Integrable K unitIntervalMeasure :=
((samplingKernel_continuous H).comp (continuous_const.sub continuous_id)).integrableOn_Icc.mono_set
Set.Ico_subset_Icc_self
have hcut (B : Set ℝ) (hBm : MeasurableSet B)
(hB : ∀ u ∈ B, ¬(u ∈ E ↔ t ∈ E)) :
(∫ u in B, K u ∂unitIntervalMeasure) ≤ C := by
rw [← integral_indicator hBm]
have hle : ∀ᵐ u ∂unitIntervalMeasure, B.indicator K u ≤ C := by
filter_upwards [ae_restrict_mem measurableSet_Ico] with u hu
by_cases hub : u ∈ B
· rw [Set.indicator_of_mem hub]
have hsep : δ ≤ |t-u| := by
by_contra hh
exact hB u hub (interval_membership_stable a b δ t u hδ.le ha hb (lt_of_not_ge hh))
have hf := fract_displacement_middle δ t u hδ ht ht' hu hsep
dsimp [K,C]
rw [← samplingKernel_fract H (t-u)]
exact samplingKernel_tail_bound H hH δ _ hδ hf.1 hf.2
· rw [Set.indicator_of_notMem hub]
exact hC
have hh := integral_mono_ae (hki.indicator hBm) (integrable_const C) hle
simpa using hh
have hbounds := samplingSmooth_bounds H hH E t
by_cases hte : t ∈ E
· rw [Set.indicator_of_mem hte, abs_of_nonpos (by linarith [hbounds.2])]
have hh := hcut Eᶜ measurableSet_Icc.compl (fun u hu he => hu (he.mpr hte))
have hsplit := integral_add_compl (s := E) measurableSet_Icc hki
have hone : (∫ u, K u ∂unitIntervalMeasure) = 1 := samplingKernel_shift_integral H hH t
change (∫ u in E, K u ∂unitIntervalMeasure) + (∫ u in Eᶜ, K u ∂unitIntervalMeasure) = _ at hsplit
change -(samplingSmooth H E t-1) ≤ C
have heq : (∫ u in E, K u ∂unitIntervalMeasure) = samplingSmooth H E t := rfl
rw [hone,heq] at hsplit
linarith
· rw [Set.indicator_of_notMem hte, sub_zero, abs_of_nonneg hbounds.1]
exact hcut E measurableSet_Icc (fun u hu he => hte (he.mp hu))
end
end FiniteFactor
end OAI
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