| import OAI.Probability.GaussianPropeller.ConeReduction |
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| namespace OAI |
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| universe uE |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open scoped RealInnerProductSpace |
| open scoped RealInnerProductSpace |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal RealInnerProductSpace |
| open Filter |
| open scoped Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology |
| open scoped RealInnerProductSpace |
| open Set Filter |
| open scoped Topology RealInnerProductSpace |
| open scoped NNReal |
| open Set Filter |
| open scoped Topology RealInnerProductSpace NNReal |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology RealInnerProductSpace |
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|
| namespace GaussianPropeller.OneCell |
|
|
| lemma nonneg_of_deriv_nonneg (f f' : ℝ → ℝ) (hf : Continuous f) |
| (hd : ∀ x, HasDerivAt f (f' x) x) (hpos : ∀ x, 0 ≤ x → 0 ≤ f' x) |
| (h0 : 0 ≤ f 0) {x : ℝ} (hx : 0 ≤ x) : 0 ≤ f x := by |
| have hm : MonotoneOn f (Ici 0) := monotoneOn_of_hasDerivWithinAt_nonneg (convex_Ici 0) |
| hf.continuousOn (fun x _ => (hd x).hasDerivWithinAt) |
| (fun x hx => hpos x (interior_subset hx)) |
| exact h0.trans (hm (by simp) hx hx) |
|
|
| lemma exp_neg_le_quadratic {x : ℝ} (hx : 0 ≤ x) : |
| Real.exp (-x) ≤ 1-x+x^2/2 := by |
| have h := nonneg_of_deriv_nonneg |
| (fun x => (1-x+x^2/2)-Real.exp (-x)) (fun x => -1+x+Real.exp (-x)) |
| (by fun_prop) (fun x => by convert |
| (((hasDerivAt_const x 1).sub (hasDerivAt_id x)).add |
| ((hasDerivAt_pow 2 x).div_const 2)).sub ((hasDerivAt_id x).neg.exp) using 1 <;> first | rfl | (solve | ring) | (dsimp; ring)) |
| (fun x _ => by have hh := Real.add_one_le_exp (-x); linarith) |
| (by simp) hx |
| linarith |
|
|
| lemma exp_neg_ge_cubic {x : ℝ} (hx : 0 ≤ x) : |
| 1-x+x^2/2-x^3/6 ≤ Real.exp (-x) := by |
| have h := nonneg_of_deriv_nonneg |
| (fun x => Real.exp (-x)-(1-x+x^2/2-x^3/6)) |
| (fun x => -Real.exp (-x)+1-x+x^2/2) |
| (by fun_prop) (fun x => by convert |
| (((hasDerivAt_id x).neg.exp).sub |
| ((((hasDerivAt_const x 1).sub (hasDerivAt_id x)).add |
| ((hasDerivAt_pow 2 x).div_const 2)).sub ((hasDerivAt_pow 3 x).div_const 6))) using 1 <;> first | rfl | (solve | ring) | (dsimp; ring)) |
| (fun x hx => by have hh := exp_neg_le_quadratic hx; linarith) |
| (by simp) hx |
| linarith |
|
|
| lemma exp_neg_le_quartic {x : ℝ} (hx : 0 ≤ x) : |
| Real.exp (-x) ≤ 1-x+x^2/2-x^3/6+x^4/24 := by |
| have h := nonneg_of_deriv_nonneg |
| (fun x => (1-x+x^2/2-x^3/6+x^4/24)-Real.exp (-x)) |
| (fun x => -1+x-x^2/2+x^3/6+Real.exp (-x)) |
| (by fun_prop) (fun x => by convert |
| (((((hasDerivAt_const x 1).sub (hasDerivAt_id x)).add |
| ((hasDerivAt_pow 2 x).div_const 2)).sub ((hasDerivAt_pow 3 x).div_const 6)).add |
| ((hasDerivAt_pow 4 x).div_const 24)).sub ((hasDerivAt_id x).neg.exp) using 1 <;> first | rfl | (solve | ring) | (dsimp; ring)) |
| (fun x hx => by have hh := exp_neg_ge_cubic hx; linarith) |
| (by simp) hx |
| linarith |
|
|
| lemma exp_neg_sq_upper (x : ℝ) : |
| Real.exp (-x^2/2) ≤ 1-x^2/2+x^4/8-x^6/48+x^8/384 := by |
| convert exp_neg_le_quartic (show (0:ℝ) ≤ x^2/2 by positivity) using 1 <;> ring_nf |
|
|
| noncomputable def Q (x : ℝ) : ℝ := x+x^3/2-x^5/24+x^7/240-x^9/2688+x^11/3456 |
| noncomputable def P (x : ℝ) : ℝ := (398943/1000000:ℝ)*Q x-x^2/2 |
| noncomputable def P' (x : ℝ) : ℝ := (398943/1000000:ℝ)* |
| (1+3*x^2/2-5*x^4/24+7*x^6/240-9*x^8/2688+11*x^10/3456)-x |
| noncomputable def P'' (x : ℝ) : ℝ := (398943/1000000:ℝ)* |
| (3*x-5*x^3/6+7*x^5/40-3*x^7/112+55*x^9/1728)-1 |
|
|
| lemma hasDerivAt_P (x : ℝ) : HasDerivAt P (P' x) x := by |
| unfold P P' Q |
| convert (((((((hasDerivAt_id x).add ((hasDerivAt_pow 3 x).div_const 2)).sub |
| ((hasDerivAt_pow 5 x).div_const 24)).add ((hasDerivAt_pow 7 x).div_const 240)).sub |
| ((hasDerivAt_pow 9 x).div_const 2688)).add ((hasDerivAt_pow 11 x).div_const 3456)).const_mul |
| (398943/1000000:ℝ)).sub ((hasDerivAt_pow 2 x).div_const 2) using 1 <;> first | rfl | (solve | ring) |
|
|
| lemma hasDerivAt_P' (x : ℝ) : HasDerivAt P' (P'' x) x := by |
| unfold P' P'' |
| convert (((((((hasDerivAt_const x 1).add (((hasDerivAt_pow 2 x).const_mul 3).div_const 2)).sub |
| (((hasDerivAt_pow 4 x).const_mul 5).div_const 24)).add (((hasDerivAt_pow 6 x).const_mul 7).div_const 240)).sub |
| (((hasDerivAt_pow 8 x).const_mul 9).div_const 2688)).add (((hasDerivAt_pow 10 x).const_mul 11).div_const 3456)).const_mul |
| (398943/1000000:ℝ)).sub (hasDerivAt_id x) using 1 <;> first | rfl | (solve | ring) |
|
|
| lemma P''_nonpos {x : ℝ} (hx : x ∈ Icc (0:ℝ) 1) : P'' x ≤ 0 := by |
| have hx2 : x^2 ≤ 1 := pow_le_one₀ hx.1 hx.2 |
| have hx5 : x^5 ≤ 1 := pow_le_one₀ hx.1 hx.2 |
| have hx7 : 0 ≤ x^7 := pow_nonneg hx.1 _ |
| have hx9 : x^9 ≤ 1 := pow_le_one₀ hx.1 hx.2 |
| have hf : 0 ≤ (1-x)*(13-5*x-5*x^2) := mul_nonneg (by linarith [hx.2]) (by linarith only [hx.2,hx2]) |
| have ha : 3*x-5*x^3/6 ≤ 13/6 := by nlinarith only [hf] |
| unfold P'' |
| linarith only [ha,hx5,hx7,hx9] |
|
|
| lemma concaveOn_P : ConcaveOn ℝ (Icc (0:ℝ) 1) P := |
| concaveOn_of_hasDerivWithinAt2_nonpos (convex_Icc 0 1) |
| (by unfold P Q; fun_prop) (fun x _ => (hasDerivAt_P x).hasDerivWithinAt) |
| (fun x _ => (hasDerivAt_P' x).hasDerivWithinAt) |
| (fun x hx => P''_nonpos (interior_subset hx)) |
|
|
| lemma P_global_upper {x : ℝ} (hx : x ∈ Icc (0:ℝ) 1) : P x ≤ 81/800 := by |
| have hq : (153/250:ℝ) ∈ Icc (0:ℝ) 1 := by norm_num |
| have hpq : P (153/250) ≤ 10123/100000 := by norm_num [P,Q] |
| have hdq : P' (153/250) ∈ Icc (0:ℝ) (11/1000000) := by norm_num [P'] |
| rcases lt_trichotomy x (153/250) with hlt | heq | hgt |
| · have hh := concaveOn_P.le_slope_of_hasDerivAt hx hq hlt (hasDerivAt_P (153/250)) |
| rw [slope_def_field] at hh |
| have hh' := (le_div_iff₀ (sub_pos.mpr hlt)).mp hh |
| have hg : 0 ≤ P' (153/250)*(153/250-x) := mul_nonneg hdq.1 (sub_nonneg.mpr hlt.le) |
| linarith only [hpq,hh',hg] |
| · rw [heq] |
| linarith only [hpq] |
| · have hh := concaveOn_P.slope_le_of_hasDerivAt hq hx hgt (hasDerivAt_P (153/250)) |
| rw [slope_def_field] at hh |
| have hh' := (div_le_iff₀ (sub_pos.mpr hgt)).mp hh |
| have hg : P' (153/250)*(x-153/250) ≤ 11/1000000 := by |
| calc |
| _ ≤ (11/1000000)*(x-153/250) := mul_le_mul_of_nonneg_right hdq.2 (sub_nonneg.mpr hgt.le) |
| _ ≤ _ := by linarith only [hx.2] |
| linarith only [hpq,hh',hg] |
|
|
| noncomputable def density (x : ℝ) : ℝ := Real.exp (-x^2/2) |
| noncomputable def gaussianConst : ℝ := (Real.sqrt (2*Real.pi))⁻¹ |
| noncomputable def tail (a : ℝ) : ℝ := ∫ x in Ioi a, density x |
|
|
| lemma density_integrable : Integrable density := by |
| change Integrable (fun x : ℝ => Real.exp (-x^2/2)) |
| simpa only [ show ∀ x : ℝ, -x^2/2 = -(1/2:ℝ)*x^2 by intro x; ring] |
| using integrable_exp_neg_mul_sq (by norm_num : (0:ℝ) < 1/2) |
|
|
| lemma mul_density_integrable : Integrable (fun x : ℝ => x*density x) := by |
| simpa only [density, show ∀ x : ℝ, -x^2/2 = -(1/2:ℝ)*x^2 by intro x; ring] |
| using integrable_mul_exp_neg_mul_sq (by norm_num : (0:ℝ) < 1/2) |
|
|
| lemma sq_density_integrable : Integrable (fun x : ℝ => x^2*density x) := by |
| have h := integrable_rpow_mul_exp_neg_mul_sq (by norm_num : (0:ℝ) < 1/2) |
| (by norm_num : (-1:ℝ) < 2) |
| simpa only [density, Real.rpow_two, show ∀ x : ℝ, -x^2/2 = -(1/2:ℝ)*x^2 by intro x; ring] |
| using h |
|
|
| lemma hasDerivAt_density (x : ℝ) : HasDerivAt density (-x*density x) x := by |
| unfold density |
| convert (((hasDerivAt_pow 2 x).neg.div_const 2).exp) using 1 |
| first | rfl | (solve | ring) | (dsimp; ring) |
|
|
| lemma tendsto_density : Tendsto density atTop (𝓝 0) := by |
| apply Real.tendsto_exp_atBot.comp |
| convert (tendsto_pow_atTop two_ne_zero).const_mul_atTop_of_neg (by norm_num : (-(1/2:ℝ)) < 0) using 1 |
| ext x |
| ring |
|
|
| lemma tail_first_moment (a : ℝ) : ∫ x in Ioi a, x*density x = density a := by |
| have hd (x : ℝ) : HasDerivAt (fun x => -density x) (x*density x) x := by |
| convert (hasDerivAt_density x).neg using 1 |
| ring |
| have ht : Tendsto (fun x => -density x) atTop (𝓝 (-0)) := tendsto_density.neg |
| simpa only [neg_zero, zero_sub, neg_neg] using |
| integral_Ioi_of_hasDerivAt_of_tendsto' (fun x _ => hd x) |
| mul_density_integrable.integrableOn ht |
|
|
| lemma tail_zero : gaussianConst*tail 0 = 1/2 := by |
| have h : tail 0 = Real.sqrt (2*Real.pi)/2 := by |
| simpa only [tail, density, show ∀ x : ℝ, -x^2/2 = -(1/2:ℝ)*x^2 by intro x; ring, |
| show Real.pi/(1/2:ℝ) = 2*Real.pi by ring] using integral_gaussian_Ioi (1/2) |
| rw [h, gaussianConst] |
| field_simp |
|
|
| lemma tail_sub {a : ℝ} (ha : 0 ≤ a) : tail a = tail 0 - ∫ x in (0:ℝ)..a, density x := by |
| have h := intervalIntegral.integral_Ioi_sub_Ioi density_integrable.integrableOn ha |
| change tail 0-tail a = _ at h |
| linarith |
|
|
| lemma gaussianConst_bounds : (39/100:ℝ) ≤ gaussianConst ∧ gaussianConst ≤ 398943/1000000 := by |
| have hp := Real.pi_pos |
| have hs := Real.sq_sqrt (show 0 ≤ 2*Real.pi by positivity) |
| have hspos : 0 < Real.sqrt (2*Real.pi) := by positivity |
| constructor |
| · rw [gaussianConst, inv_eq_one_div, le_div_iff₀ hspos] |
| have hh := Real.pi_lt_d2 |
| nlinarith |
| · rw [gaussianConst, inv_eq_one_div, div_le_iff₀ hspos] |
| have hh := Real.pi_gt_d6 |
| nlinarith |
|
|
| lemma tendsto_mul_density : Tendsto (fun x : ℝ => x*density x) atTop (𝓝 0) := by |
| have ht := (tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact |
| (by norm_num : (0:ℝ) < 1/2) 1).mono_left atTop_le_cocompact |
| apply ht.congr' |
| filter_upwards [eventually_ge_atTop (0:ℝ)] with x hx |
| simp only [Real.rpow_one, abs_of_nonneg hx, density] |
| rw [show -(1/2:ℝ)*x^2 = -x^2/2 by ring] |
|
|
| lemma tail_second_moment (a : ℝ) : ∫ x in Ioi a, x^2*density x = a*density a+tail a := by |
| have hd (x : ℝ) : HasDerivAt (fun x => -(x*density x)) |
| (x^2*density x-density x) x := by |
| convert ((hasDerivAt_id x).mul (hasDerivAt_density x)).neg using 1 <;> first | rfl | (dsimp; ring) |
| have ht : Tendsto (fun x : ℝ => -(x*density x)) atTop (𝓝 (-0)) := tendsto_mul_density.neg |
| have hh := integral_Ioi_of_hasDerivAt_of_tendsto' (a := a) (fun x _ => hd x) |
| (sq_density_integrable.sub density_integrable).integrableOn ht |
| rw [integral_sub sq_density_integrable.integrableOn density_integrable.integrableOn] at hh |
| change (∫ x in Ioi a, x^2*density x)-tail a = -0- -(a*density a) at hh |
| linarith |
|
|
| noncomputable def densityUpper (x : ℝ) : ℝ := 1-x^2/2+x^4/8-x^6/48+x^8/384 |
| noncomputable def densityLower (x : ℝ) : ℝ := 1-x^2/2+x^4/8-x^6/48 |
| noncomputable def densityPrimitive (x : ℝ) : ℝ := x-x^3/6+x^5/40-x^7/336+x^9/3456 |
| noncomputable def densityLowerPrimitive (x : ℝ) : ℝ := x-x^3/6+x^5/40-x^7/336 |
|
|
| lemma integral_density_upper {a : ℝ} (ha : 0 ≤ a) : |
| ∫ x in (0:ℝ)..a, density x ≤ densityPrimitive a := by |
| have hd (x : ℝ) : HasDerivAt densityPrimitive (densityUpper x) x := by |
| unfold densityPrimitive densityUpper |
| convert (((((hasDerivAt_id x).sub ((hasDerivAt_pow 3 x).div_const 6)).add |
| ((hasDerivAt_pow 5 x).div_const 40)).sub ((hasDerivAt_pow 7 x).div_const 336)).add |
| ((hasDerivAt_pow 9 x).div_const 3456)) using 1 <;> first | rfl | ring |
| have hh : ∫ x in (0:ℝ)..a, densityUpper x = densityPrimitive a := by |
| simpa only [densityPrimitive, zero_pow (by decide : 3≠0), zero_pow (by decide : 5≠0), |
| zero_pow (by decide : 7≠0), zero_pow (by decide : 9≠0), zero_div, sub_zero, add_zero] |
| using intervalIntegral.integral_eq_sub_of_hasDerivAt (a := (0:ℝ)) (b := a) (fun x _ => hd x) |
| (by unfold densityUpper; apply Continuous.intervalIntegrable; fun_prop) |
| rw [← hh] |
| exact intervalIntegral.integral_mono_on ha density_integrable.intervalIntegrable |
| (by unfold densityUpper; apply Continuous.intervalIntegrable; fun_prop) |
| (fun x _ => exp_neg_sq_upper x) |
|
|
| lemma integral_density_lower : (479/560:ℝ) ≤ ∫ x in (0:ℝ)..1, density x := by |
| have hd (x : ℝ) : HasDerivAt densityLowerPrimitive (densityLower x) x := by |
| unfold densityLowerPrimitive densityLower |
| convert ((((hasDerivAt_id x).sub ((hasDerivAt_pow 3 x).div_const 6)).add |
| ((hasDerivAt_pow 5 x).div_const 40)).sub ((hasDerivAt_pow 7 x).div_const 336)) using 1 <;> |
| first | rfl | ring |
| have hh : ∫ x in (0:ℝ)..1, densityLower x = 479/560 := by |
| have h := intervalIntegral.integral_eq_sub_of_hasDerivAt (a := (0:ℝ)) (b := 1) |
| (fun x _ => hd x) (by unfold densityLower; apply Continuous.intervalIntegrable; fun_prop) |
| norm_num [densityLowerPrimitive] at h ⊢ |
| linarith |
| rw [← hh] |
| apply intervalIntegral.integral_mono_on (by norm_num) |
| (by unfold densityLower; apply Continuous.intervalIntegrable; fun_prop) |
| density_integrable.intervalIntegrable |
| intro x _ |
| have h := exp_neg_ge_cubic (show (0:ℝ) ≤ x^2/2 by positivity) |
| convert h using 1 <;> first | rfl | (unfold densityLower; ring) | (unfold density; congr 1; ring) |
|
|
| lemma tail_second_bound : gaussianConst*(∫ x in Ioi (1:ℝ), x^2*density x)/4 ≤ 81/800 := by |
| rw [tail_second_moment, one_mul, tail_sub (by norm_num : (0:ℝ) ≤ 1)] |
| have hc := gaussianConst_bounds |
| have h2 := tail_zero |
| have hlow := integral_density_lower |
| have he : density 1 ≤ 233/384 := by |
| have h := exp_neg_sq_upper 1 |
| norm_num at h |
| simpa only [density, one_pow, neg_div] using h |
| have hneg : density 1-(∫ x in (0:ℝ)..1, density x) ≤ -(31/125:ℝ) := by |
| linarith |
| have hh := mul_le_mul_of_nonneg_left hneg (le_trans (by norm_num : (0:ℝ) ≤ 39/100) hc.1) |
| have hh' := mul_le_mul_of_nonpos_right hc.1 (by norm_num : -(31/125:ℝ) ≤ 0) |
| nlinarith only [hh,hh',h2] |
|
|
| lemma normalized_excess_small {a : ℝ} (ha : a ∈ Icc (0:ℝ) 1) : |
| a*gaussianConst*(density a-a*tail a) ≤ 81/800 := by |
| have hp : 0 ≤ gaussianConst := gaussianConst_bounds.1.trans' (by norm_num) |
| have hi : 0 ≤ ∫ x in (0:ℝ)..a, density x := |
| intervalIntegral.integral_nonneg ha.1 (fun x _ => (Real.exp_pos _).le) |
| have hd : 0 ≤ density a := (Real.exp_pos _).le |
| have hu : a*density a+a^2*(∫ x in (0:ℝ)..a, density x) ≤ Q a := by |
| have h1 := mul_le_mul_of_nonneg_left (exp_neg_sq_upper a) ha.1 |
| have h2 := mul_le_mul_of_nonneg_left (integral_density_upper ha.1) (sq_nonneg a) |
| unfold Q densityPrimitive at * |
| change a*Real.exp (-a^2/2)+a^2*(∫ x in (0:ℝ)..a, density x) ≤ _ |
| nlinarith only [h1,h2] |
| have hq : 0 ≤ Q a := (add_nonneg (mul_nonneg ha.1 hd) (mul_nonneg (sq_nonneg a) hi)).trans hu |
| have h1 := mul_le_mul_of_nonneg_left hu hp |
| have h2 := mul_le_mul_of_nonneg_right gaussianConst_bounds.2 hq |
| have h3 := P_global_upper ha |
| have h0 := tail_zero |
| rw [tail_sub ha.1] |
| unfold P at h3 |
| nlinarith only [h1,h2,h3,h0] |
|
|
| lemma normalized_excess_large {a : ℝ} (ha : 1 ≤ a) : |
| a*gaussianConst*(density a-a*tail a) ≤ 81/800 := by |
| have he : ∫ x in Ioi a, a*(x-a)*density x = a*(density a-a*tail a) := by |
| have hf : (fun x : ℝ => a*(x-a)*density x) = |
| (fun x => a*(x*density x-a*density x)) := by ext x; ring |
| rw [hf, integral_const_mul, integral_sub mul_density_integrable.integrableOn |
| (density_integrable.const_mul a).integrableOn, integral_const_mul, tail_first_moment] |
| rfl |
| have hu : ∫ x in Ioi a, a*(x-a)*density x ≤ (∫ x in Ioi a, x^2*density x)/4 := by |
| rw [← integral_div] |
| apply setIntegral_mono_on |
| (by convert (mul_density_integrable.sub (density_integrable.const_mul a)).const_mul a |>.integrableOn using 1; ext x; dsimp; ring) |
| (sq_density_integrable.div_const 4).integrableOn measurableSet_Ioi |
| intro x _ |
| have h : a*(x-a) ≤ x^2/4 := by nlinarith only [sq_nonneg (x-2*a)] |
| have hd : 0 ≤ density x := (Real.exp_pos _).le |
| exact (mul_le_mul_of_nonneg_right h hd).trans_eq (by ring) |
| have hm : (∫ x in Ioi a, x^2*density x) ≤ ∫ x in Ioi (1:ℝ), x^2*density x := by |
| exact setIntegral_mono_set sq_density_integrable.integrableOn |
| (ae_of_all _ (fun x => mul_nonneg (sq_nonneg x) (Real.exp_pos _).le)) |
| (ae_of_all _ (fun _ hx => Ioi_subset_Ioi ha hx)) |
| rw [he] at hu |
| have hc : 0 ≤ gaussianConst := gaussianConst_bounds.1.trans' (by norm_num) |
| have hu' := mul_le_mul_of_nonneg_left hu hc |
| have hm' := mul_le_mul_of_nonneg_left hm hc |
| have hb := tail_second_bound |
| nlinarith only [hu',hm',hb] |
|
|
| lemma normalized_excess_bound {a : ℝ} (ha : 0 ≤ a) : |
| a*gaussianConst*(density a-a*tail a) ≤ 81/800 := by |
| rcases le_total a 1 with h | h |
| · exact normalized_excess_small ⟨ha,h⟩ |
| · exact normalized_excess_large h |
|
|
| lemma integral_gaussianReal_std (f : ℝ → ℝ) : |
| ∫ x, f x ∂gaussianReal 0 1 = gaussianConst * ∫ x, f x*density x := by |
| rw [integral_gaussianReal_eq_integral_smul (by norm_num : (1:NNReal) ≠ 0), ← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards [] with x |
| simp only [gaussianPDFReal, NNReal.coe_one, sub_zero, mul_one, smul_eq_mul, gaussianConst, density] |
| ring |
|
|
| lemma integral_excess_gaussian (a : ℝ) : ∫ x : ℝ, max (x-a) 0 ∂gaussianReal 0 1 = |
| gaussianConst*(density a-a*tail a) := by |
| rw [integral_gaussianReal_std] |
| congr 1 |
| have hh : ∫ x : ℝ, max (x-a) 0*density x = ∫ x in Ioi a, (x-a)*density x := by |
| rw [← setIntegral_eq_integral_of_forall_compl_eq_zero (s := Ioi a)] |
| · exact setIntegral_congr_fun measurableSet_Ioi (fun x hx => by rw [max_eq_left (sub_nonneg.mpr hx.le)]) |
| · intro x hx |
| simp only [mem_Ioi, not_lt] at hx |
| rw [max_eq_right (sub_nonpos.mpr hx), zero_mul] |
| rw [hh] |
| simp_rw [sub_mul] |
| rw [integral_sub mul_density_integrable.integrableOn (density_integrable.const_mul a).integrableOn, |
| integral_const_mul, tail_first_moment] |
| rfl |
|
|
| lemma gaussian_excess_bound {a : ℝ} (ha : 0 ≤ a) : |
| a*(∫ x : ℝ, max (x-a) 0 ∂gaussianReal 0 1) ≤ 81/800 := by |
| rw [integral_excess_gaussian] |
| simpa only [mul_assoc] using normalized_excess_bound ha |
|
|
| section GeneralCell |
| variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| lemma map_unit_inner_gaussian (e : E) (he : ‖e‖=1) : |
| (stdGaussian E).map (fun x => ⟪e,x⟫) = gaussianReal 0 1 := by |
| change (stdGaussian E).map (InnerProductSpace.toDual ℝ E e) = _ |
| rw [IsGaussian.map_eq_gaussianReal, integral_strongDual_stdGaussian, |
| variance_dual_stdGaussian, LinearIsometryEquiv.norm_map, he] |
| norm_num |
|
|
| lemma projection_rearrangement {A : Set E} (hA : MeasurableSet A) |
| (e : E) (he : ‖e‖=1) (a : ℝ) : |
| ∫ x in A, ⟪e,x⟫ ∂stdGaussian E ≤ a*(stdGaussian E).real A+ |
| ∫ x : ℝ, max (x-a) 0 ∂gaussianReal 0 1 := by |
| let f : E → ℝ := fun x => ⟪e,x⟫ |
| have hi : Integrable f (stdGaussian E) := IsGaussian.integrable_id.const_inner e |
| have hconst : Integrable (fun _ : E => a) (stdGaussian E) := integrable_const a |
| have hpos : Integrable (fun x : E => max (f x-a) 0) (stdGaussian E) := |
| (hi.sub hconst).pos_part |
| have hm : ∫ x, max (f x-a) 0 ∂stdGaussian E = |
| ∫ x : ℝ, max (x-a) 0 ∂gaussianReal 0 1 := by |
| rw [← integral_map (by fun_prop : AEMeasurable f (stdGaussian E)) |
| (by fun_prop : AEStronglyMeasurable (fun x : ℝ => max (x-a) 0) ((stdGaussian E).map f)), |
| map_unit_inner_gaussian e he] |
| have hp (x : E) : A.indicator f x ≤ A.indicator (fun _ => a) x+max (f x-a) 0 := by |
| by_cases hx : x∈A |
| · rw [Set.indicator_of_mem hx, Set.indicator_of_mem hx] |
| linarith only [le_max_left (f x-a) 0] |
| · rw [Set.indicator_of_notMem hx, Set.indicator_of_notMem hx, zero_add] |
| exact le_max_right _ _ |
| have hh := integral_mono_ae (hi.indicator hA) ((hconst.indicator hA).add hpos) |
| (ae_of_all _ hp) |
| simp only [Pi.add_apply] at hh |
| rw [integral_indicator hA, integral_add (hconst.indicator hA) hpos, |
| integral_indicator hA, setIntegral_const, smul_eq_mul, hm, mul_comm] at hh |
| exact hh |
|
|
| lemma cell_norm_sq_le {A : Set E} (hA : MeasurableSet A) : |
| ‖∫ x in A, x ∂stdGaussian E‖^2 ≤ (81/200:ℝ)*(stdGaussian E).real A := by |
| let z : E := ∫ x in A, x ∂stdGaussian E |
| change ‖z‖^2 ≤ _ |
| by_cases hz : z=0 |
| · rw [hz, norm_zero, zero_pow (by decide : 2≠0)] |
| positivity |
| have hn : 0 < ‖z‖ := norm_pos_iff.mpr hz |
| have hP : 0 < (stdGaussian E).real A := by |
| apply lt_of_le_of_ne measureReal_nonneg |
| intro h |
| have hzero : stdGaussian E A = 0 := (measureReal_eq_zero_iff (by finiteness)).mp h.symm |
| apply hz |
| exact setIntegral_measure_zero _ hzero |
| let e : E := ‖z‖⁻¹ • z |
| have he : ‖e‖=1 := by |
| rw [norm_smul, norm_inv, Real.norm_eq_abs, abs_of_pos hn, inv_mul_cancel₀ hn.ne'] |
| have hm : ∫ x in A, ⟪e,x⟫ ∂stdGaussian E = ‖z‖ := by |
| have hi : Integrable (fun x : E => x) (stdGaussian E) := IsGaussian.integrable_id |
| rw [integral_inner hi.restrict] |
| change ⟪‖z‖⁻¹ • z,z⟫ = ‖z‖ |
| rw [real_inner_smul_left, real_inner_self_eq_norm_sq] |
| field_simp |
| let a : ℝ := ‖z‖/(2*(stdGaussian E).real A) |
| have ha : 0 ≤ a := by dsimp [a]; positivity |
| have hr := projection_rearrangement hA e he a |
| rw [hm] at hr |
| have hr' := mul_le_mul_of_nonneg_left hr ha |
| have hb := gaussian_excess_bound ha |
| have hh : a*‖z‖ ≤ a^2*(stdGaussian E).real A+81/800 := by |
| nlinarith only [hr',hb] |
| dsimp [a] at hh |
| have hden : (stdGaussian E).real A ≠ 0 := hP.ne' |
| field_simp at hh |
| nlinarith only [hh,hP] |
|
|
| lemma normalized_cell_probability {A : Set E} (hA : MeasurableSet A) |
| {C : ℝ} (hC : 9/(8*Real.pi) < C) : |
| (884/1000:ℝ)*(‖∫ x in A, x ∂stdGaussian E‖/Real.sqrt C)^2 ≤ |
| (stdGaussian E).real A := by |
| have hp : 0 < Real.pi := Real.pi_pos |
| have hCp : 0 < C := (by positivity : (0:ℝ) < 9/(8*Real.pi)).trans hC |
| have hnum : (81/200:ℝ)*(884/1000) < 9/(8*Real.pi) := by |
| apply (lt_div_iff₀ (by positivity)).mpr |
| nlinarith only [Real.pi_lt_d4] |
| have hCc : (81/200:ℝ)*(884/1000) < C := hnum.trans hC |
| have hz := cell_norm_sq_le hA |
| rw [div_pow, Real.sq_sqrt hCp.le, ← mul_div_assoc] |
| apply (div_le_iff₀ hCp).mpr |
| have hm := mul_le_mul_of_nonneg_left hz (by norm_num : (0:ℝ) ≤ 884/1000) |
| have hP : 0 ≤ (stdGaussian E).real A := measureReal_nonneg |
| have hc := mul_le_mul_of_nonneg_right hCc.le hP |
| nlinarith only [hm,hc] |
|
|
| end GeneralCell |
| end GaussianPropeller.OneCell |
|
|
| end OAI |
|
|
