| 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 |
| |
