| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.CDFTiltedStability |
| import OAI.Probability.SKBarriers.Gaussian.LipschitzChainRepresentation |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem scalarCDFAverage_cdf_stability_lipschitz (β : ℝ) {α γ : ℝ → ℝ} |
| (hα : ∀ z, α z∈Icc (0:ℝ) 1) (hαm : Monotone α) |
| (hγ : ∀ z, γ z∈Icc (0:ℝ) 1) (hγm : Monotone γ) |
| {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 h : ℝ) (hh : 0<h) : |
| |scalarCDFAverage β α s t f g x-scalarCDFAverage β γ s t f g x| ≤ |
| 2*(scalarTimeMassConstantK β (K+B+B)*(∫ z in s..s+t, |α z-γ z|))/h+ |
| 2*((L:ℝ)+2*(B:ℝ)^2)*h := by |
| apply le_of_forall_pos_le_add |
| intro ε hε |
| let d := ε/(2*((L:ℝ)+1)) |
| have hL : 0<(L:ℝ)+1 := by positivity |
| have hd : 0<d := div_pos hε (mul_pos (by norm_num) hL) |
| have hsmall : (L:ℝ)*d ≤ ε/2 := by |
| calc |
| _ ≤ ((L:ℝ)+1)*d := mul_le_mul_of_nonneg_right (by linarith) hd.le |
| _ = ε/2 := by dsimp [d]; field_simp |
| have hga (z) : |g z-scalarWindow g d z| ≤ (L:ℝ)*d := by |
| rw [abs_sub_comm] |
| exact scalarWindow_approx hgL hd z |
| have Hα := scalarCDFAverage_observable_distance β hα hαm hf hfK hgL |
| (scalarWindow_lipschitz hgL hd) hB (scalarWindow_bound hgL.continuous hB hd) |
| (mul_nonneg L.coe_nonneg hd.le) hga s t ht x |
| have Hγ := scalarCDFAverage_observable_distance β hγ hγm hf hfK hgL |
| (scalarWindow_lipschitz hgL hd) hB (scalarWindow_bound hgL.continuous hB hd) |
| (mul_nonneg L.coe_nonneg hd.le) hga s t ht x |
| have Hc := scalarCDFAverage_cdf_stability_C1 β hα hαm hγ hγm hf |
| (scalarWindow_hasDerivAt hgL.continuous d) (scalarWindowDerivative_continuous hgL.continuous d) |
| hfK (scalarWindow_bound hgL.continuous hB hd) (scalarWindowDerivative_lipschitz_bound hgL hd) |
| s t ht x h hh |
| have H₁ := abs_sub_le (scalarCDFAverage β α s t f g x) |
| (scalarCDFAverage β α s t f (scalarWindow g d) x) (scalarCDFAverage β γ s t f g x) |
| have H₂ := abs_sub_le (scalarCDFAverage β α s t f (scalarWindow g d) x) |
| (scalarCDFAverage β γ s t f (scalarWindow g d) x) (scalarCDFAverage β γ s t f g x) |
| rw [abs_sub_comm (scalarCDFAverage β γ s t f (scalarWindow g d) x)] at H₂ |
| linarith |
|
|
| end SK.Analytic |
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|
| end |
| end |
|
|
| end OAI |
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|
