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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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import Mathlib
import OAI.Probability.SKBarriers.Gaussian.SmoothedObservable
namespace OAI
section
noncomputable section
open scoped NNReal Topology
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem dyadicScalarAverage_tendsto_lipschitz (β : ℝ) {α : ℝ → ℝ}
(hα : ∀ z, α z ∈ Icc (0:ℝ) 1) (hmono : Monotone α)
{f g : ℝ → ℝ} (hf : BoundedDerivs f)
{K B L : ℝ≥0} (hfK : LipschitzWith K f) (hgL : LipschitzWith L g)
(hB : ∀ z, |g z| ≤ B) (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x : ℝ) :
Tendsto (fun n => dyadicScalarAverage β α n s t f g x) atTop
(𝓝 (scalarCDFAverage β α s t f g x)) := by
apply CauchySeq.tendsto_limUnder
rw [Metric.cauchySeq_iff]
intro ε hε
let h := ε/(4*((L:ℝ)+1))
have hL : 0<(L:ℝ)+1 := by positivity
have hh : 0<h := div_pos hε (mul_pos (by norm_num) hL)
have hsmall : (L:ℝ)*h ≤ ε/4 := by
calc
_ ≤ ((L:ℝ)+1)*h := mul_le_mul_of_nonneg_right (by linarith) hh.le
_ = ε/4 := by dsimp [h]; field_simp
let C : ℝ≥0 := ⟨2*(B:ℝ)/h,div_nonneg (by positivity) hh.le⟩
have hW := dyadicScalarAverage_tendsto_C1 β hα hmono hf
(scalarWindow_hasDerivAt hgL.continuous h) (scalarWindowDerivative_continuous hgL.continuous h)
hfK (scalarWindow_bound hgL.continuous hB hh) (C:=C)
(scalarWindowDerivative_bound hB hh) s t ht x
have hc := Metric.cauchySeq_iff.mp hW.cauchySeq (ε/4) (by positivity)
obtain ⟨N,hN⟩ := hc
refine ⟨N,fun m hm n hn => ?_⟩
have he (j : ℕ) : |dyadicScalarAverage β α j s t f g x-
dyadicScalarAverage β α j s t f (scalarWindow g h) x| ≤ ε/4 := by
apply (scalarHierarchyAverage_observable_distance hf
⟨hgL.continuous,B,B.coe_nonneg,hB⟩
⟨continuous_iff_continuousAt.mpr (fun z => (scalarWindow_hasDerivAt hgL.continuous h z).continuousAt),
B,B.coe_nonneg,scalarWindow_bound hgL.continuous hB hh⟩
(mul_nonneg L.coe_nonneg hh.le) (fun z => ?_) _ _ _ x).trans hsmall
rw [abs_sub_comm]
exact scalarWindow_approx hgL hh z
have hd := hN m hm n hn
simp only [Real.dist_eq] at hd ⊢
have H₁ := abs_sub_le (dyadicScalarAverage β α m s t f g x)
(dyadicScalarAverage β α m s t f (scalarWindow g h) x)
(dyadicScalarAverage β α n s t f g x)
have H₂ := abs_sub_le (dyadicScalarAverage β α m s t f (scalarWindow g h) x)
(dyadicScalarAverage β α n s t f (scalarWindow g h) x)
(dyadicScalarAverage β α n s t f g x)
rw [abs_sub_comm (dyadicScalarAverage β α n s t f (scalarWindow g h) x)] at H₂
linarith [he m,he n]
theorem scalarCDFAverage_observable_distance (β : ℝ) {α : ℝ → ℝ}
(hα : ∀ z, α z∈Icc (0:ℝ) 1) (hmono : Monotone α)
{f g a : ℝ → ℝ} (hf : BoundedDerivs f)
{K B C L M : ℝ≥0} (hfK : LipschitzWith K f)
(hgL : LipschitzWith L g) (haM : LipschitzWith M a)
(hB : ∀ z, |g z| ≤ B) (hC : ∀ z, |a z| ≤ C)
{δ : ℝ} (hδ : 0≤δ) (hga : ∀ z, |g z-a z| ≤ δ)
(s : ℝ) (t : ℝ≥0) (ht : t≤1) (x : ℝ) :
|scalarCDFAverage β α s t f g x-scalarCDFAverage β α s t f a x| ≤ δ := by
apply le_of_tendsto ((dyadicScalarAverage_tendsto_lipschitz β hα hmono hf hfK hgL hB s t ht x).sub
(dyadicScalarAverage_tendsto_lipschitz β hα hmono hf hfK haM hC s t ht x)).abs
exact Eventually.of_forall (fun n => scalarHierarchyAverage_observable_distance hf
⟨hgL.continuous,B,B.coe_nonneg,hB⟩ ⟨haM.continuous,C,C.coe_nonneg,hC⟩ hδ hga _ _ _ x)
end SK.Analytic
end
end
end OAI
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