| import OAI.InformationTheory.SoftChannel.Window |
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| namespace OAI |
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| section |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology |
| namespace LeanBlast.CourtadeKumar |
| open SoftChannel204 |
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| lemma continuousAt_rSecondDeriv {s : ℝ} (hs : s ∈ Ioo (0:ℝ) ell) : |
| ContinuousAt rSecondDeriv s := by |
| have hu := psiInv_mem_Ioo hs |
| have hi := continuousAt_psiInv hs |
| have ht := (hasDerivAt_artanh ⟨by linarith [hu.1],hu.2⟩).continuousAt.comp hi |
| have ht0 : Real.artanh (psiInv s) ≠ 0 := (Real.artanh_pos hu).ne' |
| have hq0 : 1-(psiInv s)^2 ≠ 0 := by nlinarith [hu.1,hu.2] |
| have hc := (((continuousAt_const (y := (1:ℝ))).add (hi.pow 2)).mul ht |>.sub hi).div |
| (((continuousAt_const.sub (hi.pow 2)).pow 2).mul (ht.pow 3)) |
| (mul_ne_zero (pow_ne_zero _ hq0) (pow_ne_zero _ ht0)) |
| apply hc.congr_of_eventuallyEq |
| filter_upwards [Ioi_mem_nhds hs.1] with x hx |
| simp [rSecondDeriv,ite_eq_right (not_le.mpr (show 0<x from hx))] |
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| lemma rate_window_point {u b : ℝ} (hu : u ∈ Ioo (0:ℝ) ell) : |
| -(2/3:ℝ)*clipKernel b (ell-u) ≤ (u-(ell-b))*rSecondDeriv u := by |
| have hsec := (rSecondDeriv_pos hu).le |
| have hs : 0 < ell-u := by linarith [hu.2] |
| by_cases h : ell-u ≤ b |
| · rw [clipKernel_of_le h,mul_zero] |
| exact mul_nonneg (by linarith) hsec |
| · have hb : b < ell-u := lt_of_not_ge h |
| rw [clipKernel_eq hb.le] |
| have hupper : rSecondDeriv u ≤ (2/3:ℝ)/(ell-u)^2 := by |
| exact (le_div_iff₀ (sq_pos_of_pos hs)).mpr (by |
| simpa only [mul_comm] using inverseEntropy_curvature_bound hu) |
| have hm := mul_le_mul_of_nonneg_left hupper (show 0 ≤ ell-u-b by linarith) |
| convert! neg_le_neg hm using 1 <;> ring |
| |
| lemma rate_window_euler {v I b k : ℝ} (hv : 0 < v) (hvI : v ≤ I) (hI : I < ell) |
| (hk : 0 ≤ k) |
| (hwindow : (∫ h in (ell-I)..(ell-v), clipKernel b h) ≤ 3*k) : |
| 0 ≤ ((I-(ell-b))*rDeriv I-r I)-((v-(ell-b))*rDeriv v-r v)+rDeriv v*k := by |
| have hdom : ∀ u ∈ uIcc v I, u ∈ Ioo (0:ℝ) ell := by |
| intro u hu |
| rw [uIcc_of_le hvI] at hu |
| exact ⟨hv.trans_le hu.1,hu.2.trans_lt hI⟩ |
| have hfc : ContinuousOn (fun u => (u-(ell-b))*rSecondDeriv u) (uIcc v I) := by |
| intro u hu |
| exact ((continuousAt_id.sub continuousAt_const).mul |
| (continuousAt_rSecondDeriv (hdom u hu))).continuousWithinAt |
| have hkc : ContinuousOn (fun u => -(2/3:ℝ)*clipKernel b (ell-u)) (uIcc v I) := by |
| intro u hu |
| exact (continuousAt_const.mul ((continuousAt_clipKernel b (by dsimp; linarith [(hdom u hu).2])).comp |
| (continuousAt_const.sub continuousAt_id))).continuousWithinAt |
| have hder : ∀ u ∈ uIcc v I, |
| HasDerivAt (fun z => (z-(ell-b))*rDeriv z-r z) ((u-(ell-b))*rSecondDeriv u) u := by |
| intro u hu |
| convert! (((hasDerivAt_id u).sub_const (ell-b)).mul (hasDerivAt_rDeriv (hdom u hu))).sub |
| (hasDerivAt_r (hdom u hu)) using 1; dsimp; ring |
| have heq := intervalIntegral.integral_eq_sub_of_hasDerivAt hder hfc.intervalIntegrable |
| have hcomp := intervalIntegral.integral_mono_on (μ := volume) hvI hkc.intervalIntegrable hfc.intervalIntegrable |
| (fun u hu => rate_window_point (hdom u (by rwa [uIcc_of_le hvI]))) |
| rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_comp_sub_left] at hcomp |
| rw [heq] at hcomp |
| have hsl := mul_le_mul_of_nonneg_right (rDeriv_ge_two hv.le (hvI.trans_lt hI)) hk |
| nlinarith |
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| end LeanBlast.CourtadeKumar |
| end |
| end |
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| end OAI |
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