| 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 |
|
|
