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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 | import Mathlib
import OAI.Probability.SKBarriers.Parisi.CDFTerminalStability
import OAI.Probability.SKBarriers.Gaussian.LipschitzObservable
namespace OAI
section
noncomputable section
open scoped NNReal Topology
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem scalarCDFAverage_terminal_stability_lipschitz (β : ℝ) {α : ℝ → ℝ}
(hα : ∀ z, α z∈Icc (0:ℝ) 1) (hmono : Monotone α)
{f g a b : ℝ → ℝ} (hf : BoundedDerivs f) (hg : BoundedDerivs g)
{K L B A C : ℝ≥0} (hfK : LipschitzWith K f) (hgL : LipschitzWith L g)
(haA : LipschitzWith A a) (hbC : LipschitzWith C b)
(haB : ∀ z, |a z| ≤ B) (hbB : ∀ z, |b z| ≤ B)
{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
apply le_of_tendsto ((dyadicScalarAverage_tendsto_lipschitz β hα hmono hf hfK haA haB s t ht x).sub
(dyadicScalarAverage_tendsto_lipschitz β hα hmono hg hgL hbC hbB s t ht x)).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 haA.continuous hbC.continuous 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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