| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.BoundedPrimitive |
| import OAI.Probability.SKBarriers.Hierarchy.TimeChainRepresentation |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def scalarTimeChainAverage (β : ℝ) (l : List (ℝ × ℝ≥0)) (f g : ℝ → ℝ) (x : ℝ) : ℝ := |
| scalarHierarchyAverage l.length (fun i => (l.get i).1) |
| (fun i => β*Real.sqrt ((l.get i).2:ℝ)) f g x |
|
|
| theorem scalarCDFAverage_eq_chainAverage {f g dg : ℝ → ℝ} |
| (hf : BoundedDerivs f) (hg : ∀ y, HasDerivAt g (dg y) y) (hdg : Continuous dg) |
| {K B C : ℝ≥0} (hfK : LipschitzWith K f) |
| (hB : ∀ y, |g y| ≤ B) (hC : ∀ y, |dg y| ≤ C) |
| (β : ℝ) {α : ℝ → ℝ} (hα : ∀ z, α z∈Icc (0:ℝ) 1) (hmono : Monotone α) |
| (l : List (ℝ × ℝ≥0)) (hm : ∀ p∈l, p.1∈Icc (0:ℝ) 1) |
| (s : ℝ) (t : ℝ≥0) (ht : t≤1) (hdur : chainDuration l=t) |
| (hmodel : TimeChainModels α s l) (x : ℝ) : |
| scalarCDFAverage β α s t f g x=scalarTimeChainAverage β l f g x := by |
| let G := boundedPrimitive g |
| have hc : Continuous g := continuous_iff_continuousAt.mpr (fun y => (hg y).continuousAt) |
| have hG : BoundedDerivs G := boundedPrimitive_regular hg hdg hB hC |
| have hGd : ∀ y, HasDerivAt G (g y) y := boundedPrimitive_hasDerivAt hc |
| have hGB : LipschitzWith B G := boundedPrimitive_lipschitz hc hB |
| have he : (fun a => scalarCDFOperator β α s t (primitivePerturbation f G a) x)= |
| fun a => scalarTimeChain β l (primitivePerturbation f G a) x := by |
| funext a |
| exact scalarCDFOperator_eq_chain (primitivePerturbation_regular hf hG a) |
| (primitivePerturbation_lipschitz hfK hGB a) β hα hmono l hm s t ht hdur hmodel x |
| rw [scalarCDFAverage_eq_terminalDerivative β α s t hf hG hGd] |
| have hd := (scalarCDFOperator_terminal_hasDerivAt β hα hmono hf hG hGd hg hfK hGB hB hC s t ht x 0).deriv |
| rw [he] at hd |
| rw [← hd] |
| simp only [scalarTimeChain_eq_hierarchy] |
| rw [scalarHierarchy_primitive_derivative _ _ _ hf hG hGd, |
| primitivePerturbation_zero] |
| simp only [add_zero,scalarTimeChainAverage] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
