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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 42 43 44 45 46 47 48 49 50 51 52 | 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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