| 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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|
