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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import OAI.Probability.GaussianPropeller.Polar
namespace OAI
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
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology RealInnerProductSpace
open MeasureTheory Set Filter
open scoped Topology BigOperators
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology ENNReal
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology ENNReal
open Metric
namespace GaussianPropeller.Cap
open OneCell ProbabilityBounds Polar
open scoped Topology
lemma p_eq_integral_indicator (a : ℝ) :
p a = ∫ x : ℝ, (Ioi a).indicator (fun _ => (1:ℝ)) x ∂gaussianReal 0 1 := by
rw [integral_gaussianReal_std]
have he : (fun x : ℝ => (Ioi a).indicator (fun _ => (1:ℝ)) x*density x) =
(Ioi a).indicator density := by
funext x
by_cases hx : x∈Ioi a <;> simp [hx]
rw [he,integral_indicator measurableSet_Ioi]
rfl
lemma p_le_one (a : ℝ) : p a ≤ 1 := by
rw [p_eq_integral_indicator, integral_indicator measurableSet_Ioi, setIntegral_const,
smul_eq_mul, mul_one]
exact measureReal_le_one
lemma density_le_one (a : ℝ) : density a ≤ 1 := by
unfold density
rw [Real.exp_le_one_iff]
linarith [sq_nonneg a]
variable {n : ℕ}
abbrev CE (n : ℕ) := EuclideanSpace ℝ (Fin n)
noncomputable def capP (n : ℕ) (a : ℝ) : ℝ := ∫ y : CE n, p (a*‖y‖) ∂stdGaussian (CE n)
noncomputable def capM (n : ℕ) (a : ℝ) : ℝ := gaussianConst * ∫ y : CE n, density (a*‖y‖) ∂stdGaussian (CE n)
lemma integrable_p_norm (a : ℝ) : Integrable (fun y : CE n => p (a*‖y‖)) (stdGaussian (CE n)) := by
apply Integrable.mono' (integrable_const (1:ℝ))
(((continuous_iff_continuousAt.mpr (fun x => (hasDerivAt_p x).continuousAt)).comp (by fun_prop)).aestronglyMeasurable)
filter_upwards [] with y
dsimp only [Function.comp_def]
rw [Real.norm_eq_abs,abs_of_pos (p_pos _)]
exact p_le_one _
lemma capP_zero (n : ℕ) : capP n 0 = 1/2 := by
simp [capP,p,tail_zero]
lemma hasDerivAt_capP (a : ℝ) : HasDerivAt (capP n)
(-gaussianConst * ∫ y : CE n, ‖y‖*density (a*‖y‖) ∂stdGaussian (CE n)) a := by
have hc : 0 < gaussianConst := by dsimp [gaussianConst]; positivity
have hd := hasDerivAt_integral_of_dominated_loc_of_deriv_le (s:=univ)
(bound:= fun y : CE n => gaussianConst*‖y‖)
(F:= fun s y => p (s*‖y‖))
(F':=fun s y => -gaussianConst*(‖y‖*density (s*‖y‖)))
(by simp : univ ∈ 𝓝 a)
(Filter.Eventually.of_forall (fun s => (integrable_p_norm s).aestronglyMeasurable))
(integrable_p_norm a)
(by unfold density; fun_prop : AEStronglyMeasurable (fun y : CE n => -gaussianConst*(‖y‖*density (a*‖y‖))) (stdGaussian (CE n)))
(by
filter_upwards [] with y
intro s _
simp only [norm_mul,norm_neg,Real.norm_eq_abs,abs_of_pos hc,abs_of_nonneg (norm_nonneg y),abs_of_pos (show 0 < density (s*‖y‖) from Real.exp_pos _)]
exact mul_le_mul_of_nonneg_left (by nlinarith only [density_le_one (s*‖y‖),norm_nonneg y] :
‖y‖*density (s*‖y‖) ≤ ‖y‖) hc.le)
(IsGaussian.integrable_id.norm.const_mul gaussianConst)
(by
filter_upwards [] with y
intro s _
convert (hasDerivAt_p (s*‖y‖)).comp s ((hasDerivAt_id s).mul_const ‖y‖) using 1 <;> first | rfl | ring)
have hd2 := hd.2
rw [integral_const_mul] at hd2
exact hd2
lemma capM_eq (hn : 0<n) (a : ℝ) :
capM n a = gaussianConst * (Real.sqrt (1+a^2))⁻¹^n := by
let : NeZero n := ⟨by omega⟩
have hh := integral_norm_pow_exp (ι:=Fin n) 0 (a:=a^2/2) (by positivity)
have hs := radial_scale (n-1) (b:=a^2/2+1/2) (by positivity)
simp only [Fintype.card_fin, add_zero, pow_zero, one_mul] at hh
unfold capM
have he : (fun y : CE n => density (a*‖y‖)) = fun y => Real.exp (-(a^2/2)*‖y‖^2) := by
funext y; unfold density; congr 1; ring
rw [he,hh,hs,show 2*(a^2/2+1/2)=1+a^2 by ring,
show n-1+1=n by omega, mul_div_cancel_right₀ _ (radial_pos _ (by norm_num)).ne']
lemma capP_deriv_two (a : ℝ) : HasDerivAt (capP 2)
(-(1/2:ℝ)*(Real.sqrt (1+a^2))⁻¹^3) a := by
have hh := integral_norm_pow_exp (ι:=Fin 2) 1 (a:=a^2/2) (by positivity)
have hs := radial_scale 2 (b:=a^2/2+1/2) (by positivity)
norm_num only [Fintype.card_fin, Nat.reduceSub, Nat.reduceAdd, pow_one, radial_half_one] at hh
have hd := hasDerivAt_capP (n:=2) a
have he : (fun y : CE 2 => ‖y‖*density (a*‖y‖)) = fun y => ‖y‖*Real.exp (-(a^2/2)*‖y‖^2) := by
funext y; unfold density; congr 2; ring
rw [he,hh,div_one,hs,show 2*(a^2/2+1/2)=1+a^2 by ring,radial_half_two] at hd
convert hd using 1
dsimp [gaussianConst]
field_simp
lemma capP_two (a : ℝ) : capP 2 a = (1-a/Real.sqrt (1+a^2))/2 := by
have hp (x : ℝ) : 0 < 1+x^2 := by positivity
have hd (x : ℝ) : HasDerivAt (fun x : ℝ => (1-x/Real.sqrt (1+x^2))/2)
(-(1/2:ℝ)*(Real.sqrt (1+x^2))⁻¹^3) x := by
convert ((hasDerivAt_const x 1).sub ((hasDerivAt_id x).div
(((hasDerivAt_pow 2 x).const_add 1).sqrt (hp x).ne') (Real.sqrt_pos.mpr (hp x)).ne')).div_const 2 using 1 <;> first | rfl | (dsimp; field_simp; nlinarith only [Real.sq_sqrt (hp x).le])
have he := is_const_of_deriv_eq_zero (fun x => ((hasDerivAt_capP (n:=2) x).sub (hd x)).differentiableAt)
have he' : ∀ x, deriv (fun x => capP 2 x-(1-x/Real.sqrt (1+x^2))/2) x=0 := by
intro x
have hx := ((capP_deriv_two x).fun_sub (hd x)).deriv
rw [sub_self] at hx
exact hx
have hg := he he' a 0
simp only [Pi.sub_apply, capP_zero,zero_div,sub_zero,one_div] at hg
linarith
end GaussianPropeller.Cap
end OAI
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