| import OAI.Analysis.FiniteFactor.Sampling.SecondInterval |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical |
| noncomputable section |
|
|
| /-- The two closed intervals representing a positive superlevel of the |
| second-oriented tested reciprocal. Endpoint changes are dealt with separately. -/ |
| def samplingSecondLevelLeft (h α a c s r : ℝ) : Set ℝ := |
| Set.Icc (max 0 (max (s-(1-α)) (s-c))) |
| (min (min (min (samplingSecondLength h s) (min (s-α) (s-a))) (s-1/2)) (s-1+1/(h*r))) |
|
|
| def samplingSecondLevelRight (h α a c s r : ℝ) : Set ℝ := |
| Set.Icc (max (max (max 0 (max (s-(1-α)) (s-c))) (s-1/2)) (s-1/(h*r))) |
| (min (samplingSecondLength h s) (min (s-α) (s-a))) |
|
|
| /-- A reciprocal superlevel is the union of two endpoint intervals. -/ |
| theorem sampling_reciprocal_level (h r z : ℝ) (hh : 0 < h) (hr : 0 < r) |
| (hz : 0 < z) (hz1 : z < 1) : |
| r ≤ 1/(h*samplingDistance z) ↔ |
| (z ≤ 1/2 ∧ z ≤ 1/(h*r)) ∨ (1/2 ≤ z ∧ 1-z ≤ 1/(h*r)) := by |
| have hd : 0 < samplingDistance z := lt_min hz (by linarith) |
| have hd' : 0 < h*samplingDistance z := mul_pos hh hd |
| have hhr : 0 < h*r := mul_pos hh hr |
| rw [le_div_iff₀ hd'] |
| by_cases hc : z ≤ 1/2 |
| · have he : samplingDistance z = z := min_eq_left (by linarith) |
| rw [he] |
| constructor |
| · intro ha |
| exact Or.inl ⟨hc,(le_div_iff₀ hhr).mpr (by nlinarith)⟩ |
| · rintro (ha|ha) |
| · have hb := (le_div_iff₀ hhr).mp ha.2; nlinarith |
| · have hb := (le_div_iff₀ hhr).mp ha.2 |
| have heq : z = 1/2 := le_antisymm hc ha.1 |
| rw [heq] at * |
| nlinarith |
| · have he : samplingDistance z = 1-z := min_eq_right (by linarith) |
| rw [he] |
| constructor |
| · intro ha |
| exact Or.inr ⟨by linarith,(le_div_iff₀ hhr).mpr (by nlinarith)⟩ |
| · rintro (ha|ha) |
| · exact (hc ha.1).elim |
| · have hb := (le_div_iff₀ hhr).mp ha.2; nlinarith |
|
|
| /-- Away from three possible jumps, the exact level indicator is the sum of |
| two closed-interval indicators. No variation theorem is assumed. -/ |
| theorem samplingTestedSecond_level_intervals (h α : ℝ) (G : Set ℝ) (a c s r t : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) |
| (hs : s ∈ Set.Ico (0 : ℝ) 1) (ht : t ∈ Set.Ico (0 : ℝ) 1) |
| (hr : 0 < r) (hsG : s ∈ G) |
| (ht0 : t ≠ 0) (htL : t ≠ samplingSecondLength h s) (htmid : t ≠ s-1/2) : |
| (if r ≤ samplingTestedSecondWeight h α (Set.Icc a c) G s t then (1 : ℝ) else 0) = |
| (samplingSecondLevelLeft h α a c s r).indicator (fun _ => (1 : ℝ)) t + |
| (samplingSecondLevelRight h α a c s r).indicator (fun _ => (1 : ℝ)) t := by |
| rw [samplingTestedSecondWeight_interval h α G a c s t hh hh1 hα hs ht] |
| have hsupport : t ∈ Set.Ioo 0 (samplingSecondLength h s) ∩ |
| Set.Icc (max (s-(1-α)) (s-c)) (min (s-α) (s-a)) ↔ |
| 0 ≤ t ∧ t ≤ samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α ∧ |
| a ≤ s-t ∧ s-t ≤ c := by |
| simp only [Set.mem_inter_iff,Set.mem_Ioo,Set.mem_Icc,max_le_iff,le_min_iff] |
| constructor |
| · rintro ⟨h1,h2⟩; rcases h1 with ⟨h1,h1'⟩; rcases h2 with ⟨⟨h2,h3⟩,h4,h5⟩ |
| exact ⟨h1.le,h1'.le,by linarith,by linarith,by linarith,by linarith⟩ |
| · rintro ⟨h1,h2,h3,h4,h5,h6⟩ |
| exact ⟨⟨lt_of_le_of_ne h1 (Ne.symm ht0),lt_of_le_of_ne h2 htL⟩, |
| ⟨⟨by linarith,by linarith⟩,by linarith,by linarith⟩⟩ |
| by_cases hc : t ∈ Set.Ioo 0 (samplingSecondLength h s) ∩ |
| Set.Icc (max (s-(1-α)) (s-c)) (min (s-α) (s-a)) |
| · have hz := hsupport.mp hc |
| rw [Set.indicator_of_mem hc,ite_eq_left hsG] |
| simp only [sampling_reciprocal_level h r (s-t) hh hr (hα.trans_le hz.2.2.1) (by linarith [hz.2.2.2.1])] |
| simp only [samplingSecondLevelLeft,samplingSecondLevelRight,Set.indicator, |
| Set.mem_Icc,max_le_iff,le_min_iff] |
| have hm : s-t ≠ (1/2 : ℝ) := by intro he; apply htmid; linarith |
| split_ifs <;> simp_all <;> grind |
| · have hn := (not_congr hsupport).mp hc |
| rw [Set.indicator_of_notMem hc,ite_eq_right (by linarith : ¬ r ≤ (0 : ℝ))] |
| simp only [samplingSecondLevelLeft,samplingSecondLevelRight,Set.indicator, |
| Set.mem_Icc,max_le_iff,le_min_iff] |
| split_ifs <;> simp_all <;> grind |
|
|
| end |
| end FiniteFactor |
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| end OAI |
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