| 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 |
|
|
