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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 | 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
end OAI
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