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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 | import OAI.Analysis.FiniteFactor.Sampling.TestedFirstMoment
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
noncomputable section
/-- Squaring the constant-height slice leaves its reciprocal interval length. -/
theorem samplingWeight_square_slice (h α s : ℝ) (hh : 0 < h) (hh1 : h < 1)
(hα : 0 < α) (hs : s ∈ Set.Ico (0 : ℝ) 1) :
(∫ t, samplingWeight false h α s t ^ 2 ∂unitIntervalMeasure) =
if α ≤ samplingDistance s then 1/(h*samplingDistance s) else 0 := by
have he : (fun t => samplingWeight false h α s t ^ 2) =
fun t => (1/(h*samplingDistance s))*samplingWeight false h α s t := by
funext t
unfold samplingWeight
simp only [Bool.false_eq_true,ite_false]
split_ifs <;> ring
rw [he,integral_const_mul,samplingWeight_first_integral false h α s hh hh1 hα hs]
simp only [Bool.false_eq_true,ite_false]
split_ifs <;> simp
/-- Exact logarithmic integral of the cutoff reciprocal distance. -/
theorem samplingDistance_cutoff_integral (α : ℝ) (hα : 0 < α) (hα1 : α ≤ 1/2) :
(∫ s, (if α ≤ samplingDistance s then (samplingDistance s)⁻¹ else 0)
∂unitIntervalMeasure) = 2*Real.log (1/(2*α)) := by
have hab : α ≤ 1-α := by linarith
have hset : {s | α ≤ samplingDistance s} = Set.Icc α (1-α) := by
ext s
simp only [Set.mem_ofPred_eq,samplingDistance,le_min_iff,Set.mem_Icc]
constructor <;> intro hs <;> constructor <;> linarith [hs.1,hs.2]
have he : (fun s => if α ≤ samplingDistance s then (samplingDistance s)⁻¹ else 0) =
(Set.Icc α (1-α)).indicator (fun s => (samplingDistance s)⁻¹) := by
rw [← hset]
rfl
rw [he,integral_indicator measurableSet_Icc]
unfold unitIntervalMeasure
rw [Measure.restrict_restrict measurableSet_Icc,Set.inter_eq_left.mpr (show
Set.Icc α (1-α) ⊆ Set.Ico (0 : ℝ) 1 from fun s hs => ⟨hα.le.trans hs.1,by linarith [hs.2]⟩),
integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le hab]
have hc : ContinuousOn (fun s => (samplingDistance s)⁻¹) (Set.Icc α (1-α)) := by
apply ContinuousOn.inv₀ (by unfold samplingDistance; fun_prop)
intro s hs
have hd : 0 < samplingDistance s := lt_min (hα.trans_le hs.1) (by linarith [hs.2])
exact hd.ne'
have hi : IntervalIntegrable (fun s => (samplingDistance s)⁻¹) volume α (1-α) := hc.intervalIntegrable_of_Icc hab
have hleft : IntervalIntegrable (fun s => (samplingDistance s)⁻¹) volume α (1/2) :=
hi.mono_set (by rw [Set.uIcc_of_le hα1,Set.uIcc_of_le hab]; intro s hs; exact ⟨hs.1,by linarith [hs.2]⟩)
have hright : IntervalIntegrable (fun s => (samplingDistance s)⁻¹) volume (1/2) (1-α) :=
hi.mono_set (by rw [Set.uIcc_of_le (show (1/2 : ℝ) ≤ 1-α by linarith),Set.uIcc_of_le hab]; intro s hs; exact ⟨by linarith [hs.1],hs.2⟩)
rw [← intervalIntegral.integral_add_adjacent_intervals hleft hright]
have hl : (∫ s in α..(1/2 : ℝ), (samplingDistance s)⁻¹) = Real.log ((1/2)/α) := by
calc
_ = ∫ s in α..(1/2 : ℝ), s⁻¹ := by
apply intervalIntegral.integral_congr
rw [Set.uIcc_of_le hα1]
intro s hs
change (min s (1-s))⁻¹ = s⁻¹
rw [min_eq_left (by linarith [hs.2])]
_ = _ := integral_inv_of_pos hα (by norm_num)
have hr : (∫ s in (1/2 : ℝ)..(1-α), (samplingDistance s)⁻¹) = Real.log ((1/2)/α) := by
calc
_ = ∫ s in (1/2 : ℝ)..(1-α), (1-s)⁻¹ := by
apply intervalIntegral.integral_congr
rw [Set.uIcc_of_le (show (1/2 : ℝ) ≤ 1-α by linarith)]
intro s hs
change (min s (1-s))⁻¹ = (1-s)⁻¹
rw [min_eq_right (by linarith [hs.1])]
_ = ∫ s in α..(1/2 : ℝ), s⁻¹ := by
rw [intervalIntegral.integral_comp_sub_left (fun s : ℝ => s⁻¹) 1]
norm_num
_ = _ := integral_inv_of_pos hα (by norm_num)
rw [hl,hr,show (1/2 : ℝ)/α = 1/(2*α) by ring]
ring
theorem samplingWeight_square_integral (h α : ℝ) (hh : 0 < h) (hh1 : h < 1)
(hα : 0 < α) (hα1 : α ≤ 1/2) :
(∫ s, ∫ t, samplingWeight false h α s t ^ 2
∂unitIntervalMeasure ∂unitIntervalMeasure) = (2/h)*Real.log (1/(2*α)) := by
calc
_ = ∫ s, (1/h)*(if α ≤ samplingDistance s then (samplingDistance s)⁻¹ else 0)
∂unitIntervalMeasure := by
apply integral_congr_ae
filter_upwards [show ∀ᵐ s ∂unitIntervalMeasure, s ∈ Set.Ico (0 : ℝ) 1 from
ae_restrict_mem measurableSet_Ico] with s hs
rw [samplingWeight_square_slice h α s hh hh1 hα hs]
split_ifs <;> simp [div_eq_mul_inv,mul_comm]
_ = _ := by
rw [integral_const_mul,samplingDistance_cutoff_integral α hα hα1]
ring
end
end FiniteFactor
end OAI
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