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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 96 97 98 99 100 101 102 103 104 105 106 107 | import OAI.Analysis.FiniteFactor.Sampling.TestedSecondMoment
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical
noncomputable section
def samplingSecondLength (h s : ℝ) : ℝ := min (h*s/(1+h)) (h*(1-s)/(1-h))
theorem samplingSecondWeight_support_iff (h α s 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) :
samplingSecondWeight h α s t ≠ 0 ↔
0 < t ∧ t < samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α := by
have hplus : 0 < 1+h := by linarith
have hminus : 0 < 1-h := by linarith
constructor
· intro hw
have hf := (samplingWeight_second_no_wrap h α s t hh hh1 hα hs ht hw).2
have hc : α ≤ samplingDistance (s-t) ∧ 0 < t ∧ t < h*samplingDistance (s-t) := by
unfold samplingSecondWeight samplingWeight at hw
simp only [Bool.false_eq_true,ite_false,hf] at hw
split_ifs at hw with hc
· exact hc
· exact False.elim (hw rfl)
have hcut := le_min_iff.mp hc.1
have hleft : t < h*(s-t) := hc.2.2.trans_le
(mul_le_mul_of_nonneg_left (min_le_left _ _) hh.le)
have hright : t < h*(1-(s-t)) := hc.2.2.trans_le
(mul_le_mul_of_nonneg_left (min_le_right _ _) hh.le)
refine ⟨hc.2.1,?_,hcut.1,by linarith [hcut.2]⟩
unfold samplingSecondLength
rw [lt_min_iff,lt_div_iff₀ hplus,lt_div_iff₀ hminus]
constructor <;> nlinarith
· rintro ⟨ht0,htL,hcut,hcut'⟩
have hz : s-t ∈ Set.Ico (0 : ℝ) 1 := ⟨hα.le.trans hcut,by linarith⟩
have hf := Int.fract_eq_self.mpr hz
have hL := lt_min_iff.mp htL
have hl := (lt_div_iff₀ hplus).mp hL.1
have hr := (lt_div_iff₀ hminus).mp hL.2
have htl : t < h*(s-t) := by nlinarith
have htr : t < h*(1-(s-t)) := by nlinarith
have hd : α ≤ samplingDistance (s-t) := le_min hcut (by linarith)
have htd : t < h*samplingDistance (s-t) := by
unfold samplingDistance
rw [mul_min_of_nonneg _ _ hh.le]
exact lt_min htl htr
have hp : 0 < h*samplingDistance (s-t) := mul_pos hh (hα.trans_le hd)
simp [samplingSecondWeight,samplingWeight,hf,hd,ht0,htd,(hα.trans_le hd).ne',hh.ne']
/-- Exact reciprocal on the actual second-oriented support. -/
theorem samplingSecondWeight_interval (h α s 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) :
samplingSecondWeight h α s t =
if 0 < t ∧ t < samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α
then 1/(h*samplingDistance (s-t)) else 0 := by
by_cases hw : samplingSecondWeight h α s t ≠ 0
· have hc := (samplingSecondWeight_support_iff h α s t hh hh1 hα hs ht).mp hw
have hf := (samplingWeight_second_no_wrap h α s t hh hh1 hα hs ht hw).2
rw [ite_eq_left hc]
unfold samplingSecondWeight samplingWeight at *
simp only [Bool.false_eq_true,ite_false,hf] at *
split_ifs at * <;> simp_all
· rw [not_ne_iff] at hw
have hc : ¬ (0 < t ∧ t < samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α) := by
intro hc
exact ((samplingSecondWeight_support_iff h α s t hh hh1 hα hs ht).mpr hc) hw
rw [ite_eq_right hc,hw]
/-- The x-test intersects that support with a reflected interval, while the
y-test is constant in the phase variable. -/
theorem samplingTestedSecondWeight_interval (h α : ℝ) (G : Set ℝ) (a c s 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) :
samplingTestedSecondWeight h α (Set.Icc a c) G s t =
(Set.Ioo 0 (samplingSecondLength h s) ∩
Set.Icc (max (s-(1-α)) (s-c)) (min (s-α) (s-a))).indicator
(fun t => if s ∈ G then 1/(h*samplingDistance (s-t)) else 0) t := by
have heq : 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) ∧ s-t ∈ Set.Icc a c := by
simp only [Set.mem_inter_iff,Set.mem_Ioo,Set.mem_Icc,max_le_iff,le_min_iff]
constructor
· rintro ⟨ht0,hL⟩
exact ⟨⟨ht0.1,ht0.2,by linarith [hL.2.1],by linarith [hL.1.1]⟩,
⟨by linarith [hL.2.2],by linarith [hL.1.2]⟩⟩
· rintro ⟨⟨ht0,hL,hcut,hcut'⟩,ha,hc⟩
exact ⟨⟨ht0,hL⟩,⟨⟨by linarith,by linarith⟩,⟨by linarith,by linarith⟩⟩⟩
by_cases hw : samplingSecondWeight h α s t ≠ 0
· have hc := (samplingSecondWeight_support_iff h α s t hh hh1 hα hs ht).mp hw
have hf := (samplingWeight_second_no_wrap h α s t hh hh1 hα hs ht hw).2
rw [samplingTestedSecondWeight,samplingSecondWeight_interval h α s t hh hh1 hα hs ht,
ite_eq_left hc,hf]
simp only [Set.indicator,heq,hc,true_and]
split_ifs <;> simp_all
· have hc : ¬ (0 < t ∧ t < samplingSecondLength h s ∧ α ≤ s-t ∧ s-t ≤ 1-α) := by
intro hc
exact hw ((samplingSecondWeight_support_iff h α s t hh hh1 hα hs ht).mpr hc)
have hz := not_ne_iff.mp hw
simp only [samplingTestedSecondWeight,hz,ite_self,Set.indicator,heq,hc,false_and,ite_false]
end
end FiniteFactor
end OAI
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