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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 | 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
end
end
end OAI
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