| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.SmoothedObservable |
|
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| namespace OAI |
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| section |
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| noncomputable section |
| open scoped NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
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|
| 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] |
|
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| 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 |
|
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| end |
| end |
|
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| end OAI |
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