| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.BoundedPrimitive |
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| namespace OAI |
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| section |
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| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
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| 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 |
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| 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 |
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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