| import OAI.InformationTheory.SoftChannel.PairEntropy |
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| namespace OAI |
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| section |
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| noncomputable section |
| open Set MeasureTheory |
| open scoped Topology |
| namespace LeanBlast.CourtadeKumar |
| open SoftChannel204 |
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| lemma clipKernel_scaled (b v : ℝ) : |
| ell * clipKernel b (ell*v) = clipKernel (b/ell) v := by |
| have hL := ell_pos |
| have he : ell*v-b = ell*(v-b/ell) := by field_simp |
| have hm : max (ell*(v-b/ell)) 0 = ell*max (v-b/ell) 0 := by |
| simpa using (mul_max_of_nonneg (v-b/ell) 0 hL.le).symm |
| rw [clipKernel,clipKernel,he,hm] |
| simp only [mul_pow] |
| field_simp |
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| lemma clipKernel_integral_scaled (b q s : ℝ) : |
| (∫ h in q..s, clipKernel b h) = ∫ v in q/ell..s/ell, clipKernel (b/ell) v := by |
| have h := intervalIntegral.smul_integral_comp_mul_left (a := q/ell) (b := s/ell) |
| (clipKernel b) ell |
| simp only [smul_eq_mul, mul_div_cancel₀ _ ell_pos.ne'] at h |
| rw [← h, ← intervalIntegral.integral_const_mul] |
| apply intervalIntegral.integral_congr |
| intro v _ |
| exact clipKernel_scaled b v |
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| lemma entropy_interval_bound {x z q s : ℝ} (hx0 : 0 ≤ x) (hx1 : x < 1) |
| (hz0 : 0 ≤ z) (hzg : z < (1-x)/(1+x)) |
| (hql : ell*(1-x-(1+x)*z) ≤ q) (hqs : q ≤ s) |
| (hsL : s ≤ ell) (hJ : s-q ≤ ell*z) : |
| (∫ h in q..s, clipKernel (Real.log (1+x)) h) ≤ |
| -(3/2:ℝ)*Real.log (1-((1-x)/(1+x))*z) := by |
| have hL := ell_pos |
| have hp : 0 < 1+x := by linarith |
| have hlo : 0 < 1-x-(1+x)*z := by |
| have := (lt_div_iff₀ hp).mp hzg |
| nlinarith |
| have hq0 : 0 < q := (mul_pos hL hlo).trans_le hql |
| let u := min (q/ell) (1-z) |
| have hzl : 1-x-(1+x)*z ≤ 1-z := by nlinarith |
| have hul : 1-x-(1+x)*z ≤ u := le_min ((le_div_iff₀ hL).mpr (by simpa only [mul_comm] using hql)) hzl |
| have huh : u ≤ 1-z := min_le_right _ _ |
| have huq : u ≤ q/ell := min_le_left _ _ |
| have hss : s/ell ≤ u+z := by |
| dsimp [u] |
| rw [← min_add_add_right] |
| apply le_min |
| · apply (div_le_iff₀ hL).mpr |
| have : q/ell*ell = q := div_mul_cancel₀ q hL.ne' |
| nlinarith |
| · simpa using (div_le_one hL).mpr hsL |
| have hu0 : 0 < u := hlo.trans_le hul |
| have huz0 : 0 < u+z := by linarith |
| rw [clipKernel_integral_scaled] |
| calc |
| _ ≤ ∫ v in u..u+z, clipKernel (Real.log (1+x)/ell) v := |
| intervalIntegral.integral_mono_interval huq ((div_le_div_iff_of_pos_right hL).mpr hqs) |
| hss (Filter.Eventually.of_forall (fun v => clipKernel_nonneg _ v)) |
| (intervalIntegrable_clipKernel _ hu0 huz0) |
| _ ≤ _ := by |
| exact thinning_interval hx0 hx1 hz0 hzg hul huh |
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| lemma entropy_common_output {a k : ℝ} (ha : a ∈ Icc (-1) 1) (hk : k ∈ Icc (0:ℝ) 1) : |
| k*entropy a ≤ entropy (1-k+k*a) := by |
| have h := concaveOn_entropy.2 (by norm_num : (1:ℝ) ∈ Icc (-1) 1) ha |
| (by linarith [hk.2] : 0 ≤ 1-k) hk.1 (by ring : 1-k+k=1) |
| simpa only [smul_eq_mul,mul_one,entropy_one,mul_zero,zero_add] using h |
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| lemma pair_entropy_floor {x d : ℝ} (hx : 0 ≤ x) (hd : 0 ≤ d) (hxd : x+d < 1) : |
| (1-x)*entropy (d/(1-x)) ≤ (entropy (x+d)+entropy (x-d))/2 := by |
| have hs : 0 < 1-x := by linarith |
| have hr : d/(1-x) ∈ Icc (-1) 1 := by |
| constructor |
| · linarith [div_nonneg hd hs.le] |
| · exact (div_le_one hs).mpr (by linarith) |
| have hnr : -(d/(1-x)) ∈ Icc (-1) 1 := by constructor <;> linarith [hr.1,hr.2] |
| have hk : 1-x ∈ Icc (0:ℝ) 1 := ⟨hs.le,by linarith⟩ |
| have h₁ := entropy_common_output hr hk |
| have h₂ := entropy_common_output hnr hk |
| have he₁ : 1-(1-x)+(1-x)*(d/(1-x)) = x+d := by field_simp; ring |
| have he₂ : 1-(1-x)+(1-x)*(-(d/(1-x))) = x-d := by field_simp; ring |
| rw [he₁] at h₁ |
| rw [he₂,entropy_neg] at h₂ |
| linarith |
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| lemma thinning_window {x d e : ℝ} (hx : 0 ≤ x) (hd : 0 < d) (hxd : x+d < 1) |
| (hefloor : (1-x)*entropy (d/(1-x)) ≤ e) |
| (hehi : e ≤ (entropy (x+d)+entropy (x-d))/2) : |
| (∫ h in (e+psi x)..(e+ell-(entropy (x+d)+entropy (x-d))/2), |
| clipKernel (Real.log (1+x)) h) ≤ |
| -(3/2:ℝ)*Real.log (1-d^2/(1+x)^2) := by |
| have hs : 0 < 1-x := by linarith |
| have hp : 0 < 1+x := by linarith |
| have hh : |x|+|d| < 1 := by rw [abs_of_nonneg hx,abs_of_pos hd]; exact hxd |
| let z := d^2/((1-x)*(1+x)) |
| have hz : 0 ≤ z := by dsimp [z]; positivity |
| have hdz : z < (1-x)/(1+x) := by |
| dsimp [z] |
| apply (div_lt_div_iff₀ (mul_pos hs hp) hp).mpr |
| nlinarith [mul_pos (show 0 < 1-x-d by linarith) (show 0 < 1-x+d by linarith)] |
| have hratio : |d/(1-x)| ≤ 1 := by |
| rw [abs_of_pos (div_pos hd hs)] |
| exact (div_le_one hs).mpr (by linarith) |
| have hfloor := mul_le_mul_of_nonneg_left (ell_mul_one_sub_sq_le_entropy hratio) hs.le |
| have hfloorEq : (1-x)*(ell*(1-(d/(1-x))^2)) = ell*(1-x-(1+x)*z) := by |
| dsimp [z]; field_simp |
| rw [hfloorEq] at hfloor |
| have hql : ell*(1-x-(1+x)*z) ≤ e+psi x := by linarith [psi_nonneg x] |
| have hqs : e+psi x ≤ e+ell-(entropy (x+d)+entropy (x-d))/2 := by |
| have h := pairEntropyGap_nonneg x d hh |
| simp only [pairEntropyGap,entropy] at h ⊢ |
| linarith |
| have hJ : e+ell-(entropy (x+d)+entropy (x-d))/2-(e+psi x) ≤ ell*z := by |
| have h := pairEntropyGap_le_quadratic x d hh |
| have heq : 1-x^2 = (1-x)*(1+x) := by ring |
| rw [heq] at h |
| change pairEntropyGap x d ≤ ell*z at h |
| simp only [pairEntropyGap,entropy] at h ⊢ |
| linarith |
| have hzEq : ((1-x)/(1+x))*z = d^2/(1+x)^2 := by |
| dsimp [z]; field_simp |
| have H := entropy_interval_bound hx (by linarith) hz hdz hql hqs (by linarith) hJ |
| rwa [hzEq] at H |
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| end LeanBlast.CourtadeKumar |
| end |
| end |
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| end OAI |
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