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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 | import OAI.Analysis.FiniteFactor.OrbitSampling.RationalTests
import Mathlib.MeasureTheory.Integral.DominatedConvergence
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
open scoped Classical ENNReal Topology
noncomputable section
theorem sourceSamplingWidth_tendsto : Tendsto sourceSamplingWidth atTop (𝓝 0) := by
have hh := (tendsto_rpow_neg_atTop (by norm_num : (0 : ℝ) < 3/5)).comp
(tendsto_natCast_atTop_atTop.comp sourceBlockSize_tendsto)
convert hh using 1
funext stage
simp only [sourceSamplingWidth,Function.comp_apply]
congr 1; norm_num
theorem sourceSamplingCutoff_tendsto : Tendsto sourceSamplingCutoff atTop (𝓝 0) := by
have hh := (tendsto_rpow_neg_atTop (by norm_num : (0 : ℝ) < 2)).comp
(tendsto_natCast_atTop_atTop.comp sourceBlockSize_tendsto)
exact hh
theorem sampling_interval_small_displacement (s a c t : ℝ)
(_ha : s ≠ a) (hc : s ≠ c) (ht : 0 ≤ t)
(hta : t < |s-a|) (htc : t < |s-c|) :
s+t ∈ Set.Icc a c ↔ s ∈ Set.Icc a c := by
have hleft : a ≤ s+t ↔ a ≤ s := by
by_cases h : a ≤ s
· constructor
· exact fun _ => h
· intro _; linarith
· rw [abs_of_neg (by linarith : s-a < 0)] at hta
constructor <;> intro hh <;> linarith
have hright : s+t ≤ c ↔ s ≤ c := by
by_cases h : s ≤ c
· have hsc : s < c := lt_of_le_of_ne h hc
rw [abs_of_neg (by linarith : s-c < 0)] at htc
constructor <;> intro hh <;> linarith
· constructor <;> intro hh <;> linarith
exact and_congr hleft hright
/-- Away from the interval endpoints, a sufficiently short retained diagonal
step has exactly the same interval test as its starting coordinate. -/
theorem samplingTestedFirstDensity_diagonal_exact (h α s a c : ℝ) (E : Set ℝ)
(hh : 0 < h) (hh1 : h < 1) (hα : 0 < α)
(hs : s ∈ Set.Ico (0 : ℝ) 1) (hcut : α ≤ samplingDistance s)
(ha : s ≠ a) (hc : s ≠ c) (hha : h < |s-a|) (hhc : h < |s-c|) :
samplingTestedFirstDensity false h α E (Set.Icc a c) s =
if s ∈ E ∧ s ∈ Set.Icc a c then 1 else 0 := by
have he (t : ℝ) : samplingTestedFirstWeight false h α E (Set.Icc a c) s t =
if s ∈ E ∧ s ∈ Set.Icc a c then samplingWeight false h α s t else 0 := by
by_cases hw : samplingWeight false h α s t = 0
· simp [samplingTestedFirstWeight,hw]
· have ho := samplingWeight_orders false h α s t hh hh1 hα hs hw
have hsp : 0 < t ∧ t < h*samplingDistance s := by
unfold samplingWeight at hw
simp only [Bool.false_eq_true,ite_false] at hw
split_ifs at hw with hcond
· exact hcond.2
· exact False.elim (hw rfl)
have hd : samplingDistance s ≤ 1 := (min_le_right s (1-s)).trans (by linarith [hs.1])
have hth : t < h := hsp.2.trans_le (by nlinarith)
have hmem := sampling_interval_small_displacement s a c t ha hc hsp.1.le
(hth.trans hha) (hth.trans hhc)
simp only [samplingTestedFirstWeight,ho.1,hmem]
unfold samplingTestedFirstDensity
simp_rw [he]
by_cases htest : s ∈ E ∧ s ∈ Set.Icc a c
· simp only [ite_eq_left htest]
simpa only [Bool.false_eq_true,ite_false,ite_eq_left hcut] using
samplingWeight_first_integral false h α s hh hh1 hα hs
· simp only [ite_eq_right htest,integral_zero]
/-- At each interior starting point away from the two tested endpoints, the
actual diagonal density eventually equals the uniform diagonal test. -/
theorem sourceSampling_diagonal_eventually (s a c : ℝ) (E : Set ℝ)
(hs : s ∈ Set.Ioo (0 : ℝ) 1) (ha : s ≠ a) (hc : s ≠ c) :
∀ᶠ stage in atTop,
samplingTestedFirstDensity false (sourceSamplingWidth stage) (sourceSamplingCutoff stage)
E (Set.Icc a c) s = if s ∈ E ∧ s ∈ Set.Icc a c then 1 else 0 := by
have hd : 0 < samplingDistance s := lt_min hs.1 (by linarith [hs.2])
have hcut := sourceSamplingCutoff_tendsto.eventually (gt_mem_nhds hd)
have hwa := sourceSamplingWidth_tendsto.eventually (gt_mem_nhds (abs_pos.mpr (sub_ne_zero.mpr ha)))
have hwc := sourceSamplingWidth_tendsto.eventually (gt_mem_nhds (abs_pos.mpr (sub_ne_zero.mpr hc)))
filter_upwards [hcut,hwa,hwc] with stage hcut hwa hwc
have hp := sourceSampling_parameters stage
exact samplingTestedFirstDensity_diagonal_exact _ _ s a c E hp.1 hp.2.1 hp.2.2
⟨hs.1.le,hs.2⟩ hcut.le ha hc hwa hwc
end
end FiniteFactor
end OAI
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