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a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Variational.BackwardScore
import OAI.Probability.Perceptron.Variational.FieldRounding
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
def cdfIndicator (a t : Time) : ℝ := if a≤t then 1 else 0
lemma cdfIndicator_integrable (a : Time) : Integrable (cdfIndicator a) timeLaw := by
apply Integrable.mono' (integrable_const (1:ℝ))
((measurable_const.ite (measurableSet_le measurable_const measurable_id) measurable_const).aestronglyMeasurable)
exact ae_of_all _ fun _ => by split_ifs <;> norm_num
lemma cdfIndicator_integral (a : Time) : (∫ t, cdfIndicator a t ∂timeLaw)=1-(a:ℝ) := by
have he := timeTail_atom_integral a 0 1
rw [show Ici (0:Time)=univ from Ici_bot,Measure.restrict_univ] at he
change (∫ s, (if a ≤ s then (1:ℝ) else 0) ∂timeLaw)=1*(1-max (a:ℝ) 0) at he
simpa only [cdfIndicator,one_mul,max_eq_left a.2.1] using he
lemma cdfIndicator_order {a b : Time} (hab : a≤b) (t : Time) : cdfIndicator b t≤cdfIndicator a t := by
unfold cdfIndicator
by_cases hb : b≤t
· rw [ite_eq_left hb,ite_eq_left (hab.trans hb)]
· rw [ite_eq_right hb]
split_ifs <;> norm_num
lemma cdfIndicator_abs_integral (a b : Time) :
(∫ t, |cdfIndicator a t-cdfIndicator b t| ∂timeLaw)=|(a:ℝ)-b| := by
have hle {a b : Time} (hab : a≤b) :
(∫ t, |cdfIndicator a t-cdfIndicator b t| ∂timeLaw)=(b:ℝ)-a := by
simp_rw [abs_of_nonneg (sub_nonneg.mpr (cdfIndicator_order hab _))]
rw [integral_sub (cdfIndicator_integrable a) (cdfIndicator_integrable b),
cdfIndicator_integral,cdfIndicator_integral]
ring
rcases le_total a b with hab | hba
· rw [hle hab,abs_sub_comm,abs_of_nonneg (sub_nonneg.mpr (show (a:ℝ)≤b from hab))]
· simp_rw [abs_sub_comm (cdfIndicator a _) (cdfIndicator b _)]
rw [hle hba,abs_of_nonneg (sub_nonneg.mpr (show (b:ℝ)≤a from hba))]
lemma quantileTrial_L1_le (q r : Time→Time) (hq : Measurable q) (hr : Measurable r) :
(∫ t, |quantileTrial q t-quantileTrial r t| ∂timeLaw)≤
∫ u, |(q u:ℝ)-r u| ∂timeLaw := by
let F := fun p : Time×Time => |cdfIndicator (q p.2) p.1-cdfIndicator (r p.2) p.1|
have hmq : Measurable (fun p : Time×Time => cdfIndicator (q p.2) p.1) :=
measurable_const.ite (measurableSet_le (hq.comp measurable_snd) measurable_fst) measurable_const
have hmr : Measurable (fun p : Time×Time => cdfIndicator (r p.2) p.1) :=
measurable_const.ite (measurableSet_le (hr.comp measurable_snd) measurable_fst) measurable_const
have hF : Integrable F (timeLaw.prod timeLaw) := by
apply Integrable.mono' (integrable_const (1:ℝ)) ((hmq.sub hmr).abs.aestronglyMeasurable)
exact ae_of_all _ fun p => by dsimp [F,cdfIndicator]; split_ifs <;> norm_num
have hpoint (t : Time) : |quantileTrial q t-quantileTrial r t|≤∫ u, F (t,u) ∂timeLaw := by
rw [quantileTrial_eq_integral q hq,quantileTrial_eq_integral r hr]
have hi (v : Time→Time) (hv : Measurable v) :
Integrable (fun u => if v u≤t then (1:ℝ) else 0) timeLaw := by
apply Integrable.mono' (integrable_const (1:ℝ))
((measurable_const.ite (measurableSet_le hv measurable_const) measurable_const).aestronglyMeasurable)
exact ae_of_all _ fun _ => by split_ifs <;> norm_num
rw [← integral_sub (hi q hq) (hi r hr)]
simpa only [Real.norm_eq_abs,F,cdfIndicator] using (norm_integral_le_integral_norm
(fun u : Time => (if q u≤t then (1:ℝ) else 0)-(if r u≤t then (1:ℝ) else 0)))
calc
_ ≤ ∫ t, ∫ u, F (t,u) ∂timeLaw ∂timeLaw :=
integral_mono ((trial_integrable (quantileTrial q)).sub (trial_integrable (quantileTrial r))).abs
hF.integral_prod_left hpoint
_ = ∫ u, |(q u:ℝ)-r u| ∂timeLaw := by
rw [integral_integral_swap hF]
exact integral_congr_ae (ae_of_all _ fun u => cdfIndicator_abs_integral (q u) (r u))
theorem controlValue_quantile_abs_sub_le (P : Measure BrownianPath) [IsProbabilityMeasure P]
(g : Jet3) (q r : Time→Time) (hq : Measurable q) (hr : Measurable r) :
|controlValue P g.f (quantileTrial q)-controlValue P g.f (quantileTrial r)|≤
(3/2:ℝ)*(‖g.d1‖₊:ℝ)^2*∫ u, |(q u:ℝ)-r u| ∂timeLaw := by
apply (controlValue_trial_abs_sub_le P g.f _ _ ‖g.d1‖₊ g.lipschitz).trans
exact mul_le_mul_of_nonneg_left (quantileTrial_L1_le q r hq hr) (by positivity)
end SphericalPerceptronFreeEnergy
end
end OAI
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