| import OAI.Analysis.FiniteFactor.Sampling.SecondLevels |
| import OAI.Analysis.FiniteFactor.Sampling.IntervalSums |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical |
| noncomputable section |
|
|
| / |
| simultaneously for every atomless law carried by [0,1). -/ |
| theorem samplingTestedSecond_level_exists (h α : ℝ) (G : Set ℝ) (a c s r : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) |
| (hs : s ∈ Set.Ico (0 : ℝ) 1) (hr : 0 < r) : |
| ∃ A B : Fin 2 → ℝ, ∀ (ν : Measure ℝ) [NullSingletonClass ν], |
| (∀ᵐ t ∂ν, t ∈ Set.Ico (0 : ℝ) 1) → |
| (fun t => if r ≤ samplingTestedSecondWeight h α (Set.Icc a c) G s t then (1 : ℝ) else 0) =ᵐ[ν] |
| (fun t => ∑ j, samplingClosedIndicator (A j) (B j) t) := by |
| by_cases hG : s ∈ G |
| · refine ⟨![max 0 (max (s-(1-α)) (s-c)), |
| max (max (max 0 (max (s-(1-α)) (s-c))) (s-1/2)) (s-1/(h*r))], |
| ![min (min (min (samplingSecondLength h s) (min (s-α) (s-a))) (s-1/2)) (s-1+1/(h*r)), |
| min (samplingSecondLength h s) (min (s-α) (s-a))],?_⟩ |
| intro ν _ hν |
| filter_upwards [hν, |
| ae_iff.mpr (show ν {t | ¬t ≠ 0} = 0 by simp), |
| ae_iff.mpr (show ν {t | ¬t ≠ samplingSecondLength h s} = 0 by simp), |
| ae_iff.mpr (show ν {t | ¬t ≠ s-1/2} = 0 by simp)] with t ht ht0 htL htm |
| simpa [Fin.sum_univ_two,samplingClosedIndicator,samplingSecondLevelLeft,samplingSecondLevelRight] |
| using samplingTestedSecond_level_intervals h α G a c s r t hh hh1 hα hs ht hr hG ht0 htL htm |
| · refine ⟨fun _ => 1,fun _ => 0,?_⟩ |
| intro ν _ _ |
| apply Eventually.of_forall |
| intro t |
| simp [samplingTestedSecondWeight,hG,not_le.mpr hr,samplingClosedIndicator] |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
