File size: 3,115 Bytes
135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Gaussian.BoundedPrimitive
namespace OAI
section
noncomputable section
open scoped NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
attribute [local instance 2000] parameterNormedGroup parameterNormedSpace
theorem scalarHierarchyAverage_terminal_stability (n : ℕ) (m v : Fin n → ℝ)
(hm : ∀ i, m i ∈ Icc (0:ℝ) 1) (hmono : Monotone m)
{f g : ℝ → ℝ} (hf : BoundedDerivs f) (hg : BoundedDerivs g)
{ε : ℝ} (hε : 0 ≤ ε) (hfg : ∀ z, |f z-g z| ≤ ε)
{a b : ℝ → ℝ} (ha : Continuous a) (hb : Continuous b)
{C δ : ℝ} (hC : 0 ≤ C) (haC : ∀ z, |a z| ≤ C) (hbC : ∀ z, |b z| ≤ C)
(hab : ∀ z, |a z-b z| ≤ δ) (x : ℝ) :
|scalarHierarchyAverage n m v f a x-scalarHierarchyAverage n m v g b x| ≤
δ+2*C*(Real.exp (2*ε)-1) := by
rw [scalarHierarchyAverage_eq_integral n m v hf ha hC haC,
scalarHierarchyAverage_eq_integral n m v hg hb hC hbC]
exact hierarchyPathLaw_observable_stability n m hm hmono
((hf.translate x).compCLM (coordinateLinear n v))
((hg.translate x).compCLM (coordinateLinear n v)) hε
(fun z => hfg (x+coordinateLinear n v z))
(ha.comp (continuous_const.add (coordinateLinear n v).continuous))
(hb.comp (continuous_const.add (coordinateLinear n v).continuous))
hC (fun z => haC (x+coordinateLinear n v z)) (fun z => hbC (x+coordinateLinear n v z))
(fun z => hab (x+coordinateLinear n v z)) 0
theorem scalarCDFAverage_terminal_stability (β : ℝ) {α : ℝ → ℝ}
(hα : ∀ z, α z ∈ Icc (0:ℝ) 1) (hmono : Monotone α)
{f g a b da db : ℝ → ℝ} (hf : BoundedDerivs f) (hg : BoundedDerivs g)
{K L B C : ℝ≥0} (hfK : LipschitzWith K f) (hgL : LipschitzWith L g)
(ha : ∀ z, HasDerivAt a (da z) z) (hb : ∀ z, HasDerivAt b (db z) z)
(hda : Continuous da) (hdb : Continuous db)
(haB : ∀ z, |a z| ≤ B) (hbB : ∀ z, |b z| ≤ B)
(hdaC : ∀ z, |da z| ≤ C) (hdbC : ∀ z, |db z| ≤ C)
{ε δ : ℝ} (hε : 0 ≤ ε) (hfg : ∀ z, |f z-g z| ≤ ε)
(hab : ∀ z, |a z-b z| ≤ δ) (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x : ℝ) :
|scalarCDFAverage β α s t f a x-scalarCDFAverage β α s t g b x| ≤
δ+2*(B:ℝ)*(Real.exp (2*ε)-1) := by
have h₁ := dyadicScalarAverage_tendsto_C1 β hα hmono hf ha hda hfK haB hdaC s t ht x
have h₂ := dyadicScalarAverage_tendsto_C1 β hα hmono hg hb hdb hgL hbB hdbC s t ht x
apply le_of_tendsto (h₁.sub h₂).abs
apply Eventually.of_forall
intro n
apply scalarHierarchyAverage_terminal_stability _ _ _
(fun i => dyadicIntervals_mass_bounds hα n s t (List.get_mem _ _))
(fun i j hij => ?_)
hf hg hε hfg
(continuous_iff_continuousAt.mpr (fun z => (ha z).continuousAt))
(continuous_iff_continuousAt.mpr (fun z => (hb z).continuousAt))
B.coe_nonneg haB hbB hab x
rcases lt_or_eq_of_le hij with hij|rfl
· exact (List.pairwise_iff_get.mp (dyadicIntervals_pairwise hmono n s t)) i j hij
· exact le_rfl
end SK.Analytic
end
end
end OAI
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