| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.DyadicCDFStability |
| import OAI.Probability.SKBarriers.Gaussian.BoundedPrimitive |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped Topology NNReal |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem scalarCDFValue_lipschitz (β : ℝ) {α : ℝ → ℝ} |
| (hα : ∀ s, α s ∈ Icc (0:ℝ) 1) (hmono : Monotone α) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) : LipschitzWith 1 (scalarCDFValue β α s t) := by |
| apply lipschitzWith_of_nnnorm_deriv_le |
| (fun x => (scalarCDFValue_hasDerivAt β hα hmono s t ht x).differentiableAt) |
| intro x |
| rw [(scalarCDFValue_hasDerivAt β hα hmono s t ht x).deriv] |
| exact_mod_cast (scalarCDF_derivative_bounds β hα hmono s t ht x).1 |
|
|
| theorem scalarCDFHessian_abs_le_one (β : ℝ) {α : ℝ → ℝ} |
| (hα : ∀ s, α s ∈ Icc (0:ℝ) 1) (hmono : Monotone α) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x : ℝ) : |
| |scalarCDFHessian β α s t x| ≤ 1 := by |
| have H := scalarCDF_derivative_bounds β hα hmono s t ht x |
| rw [abs_of_nonneg H.2.1] |
| linarith [H.2.2,sq_nonneg (scalarCDFGradient β α s t x)] |
|
|
| theorem scalarCDFGradient_lipschitz (β : ℝ) {α : ℝ → ℝ} |
| (hα : ∀ s, α s ∈ Icc (0:ℝ) 1) (hmono : Monotone α) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) : LipschitzWith 1 (scalarCDFGradient β α s t) := by |
| apply lipschitzWith_of_nnnorm_deriv_le |
| (fun x => (scalarCDFGradient_hasDerivAt β hα hmono s t ht x).differentiableAt) |
| intro x |
| rw [(scalarCDFGradient_hasDerivAt β hα hmono s t ht x).deriv] |
| exact_mod_cast scalarCDFHessian_abs_le_one β hα hmono s t ht x |
|
|
| theorem scalarCDFValue_regular (β : ℝ) {α : ℝ → ℝ} |
| (hα : ∀ s, α s ∈ Icc (0:ℝ) 1) (hmono : Monotone α) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) : BoundedDerivs (scalarCDFValue β α s t) := |
| boundedDerivs_of_derivatives (scalarCDFValue_hasDerivAt β hα hmono s t ht) |
| (scalarCDFGradient_hasDerivAt β hα hmono s t ht) |
| (scalarCDFHessian_lipschitz β hα hmono s t ht).continuous |
| (B:=1) (C:=1) (fun x => (scalarCDF_derivative_bounds β hα hmono s t ht x).1) |
| (scalarCDFHessian_abs_le_one β hα hmono s t ht) |
|
|
| theorem scalarCDFGradient_cdf_stability (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) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x h : ℝ) (hh : 0<h) : |
| |scalarCDFGradient β α s t x-scalarCDFGradient β γ s t x| ≤ |
| 2*(scalarTimeMassConstant β*(∫ z in s..s+t, |α z-γ z|))/h+2*h := by |
| have H := derivative_bound_of_uniform_value_and_lipschitz _ _ (1+1) _ |
| (fun z => (scalarCDFValue_hasDerivAt β hα hαm s t ht z).sub |
| (scalarCDFValue_hasDerivAt β hγ hγm s t ht z)) |
| ((scalarCDFGradient_lipschitz β hα hαm s t ht).sub |
| (scalarCDFGradient_lipschitz β hγ hγm s t ht)) |
| (scalarCDFValue_cdf_lipschitz β hα hαm hγ hγm s t ht) x h hh |
| norm_num only [NNReal.coe_add,NNReal.coe_one,one_add_one_eq_two] at H |
| exact H |
|
|
| theorem scalarCDFHessian_cdf_stability (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) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x h k : ℝ) (hh : 0<h) (hk : 0<k) : |
| |scalarCDFHessian β α s t x-scalarCDFHessian β γ s t x| ≤ |
| 2*(2*(scalarTimeMassConstant β*(∫ z in s..s+t, |α z-γ z|))/h+2*h)/k+ |
| 2*(susceptibilityLipschitzConstant:ℝ)*k := by |
| have H := derivative_bound_of_uniform_value_and_lipschitz _ _ |
| (susceptibilityLipschitzConstant+susceptibilityLipschitzConstant) _ |
| (fun z => (scalarCDFGradient_hasDerivAt β hα hαm s t ht z).sub |
| (scalarCDFGradient_hasDerivAt β hγ hγm s t ht z)) |
| ((scalarCDFHessian_lipschitz β hα hαm s t ht).sub |
| (scalarCDFHessian_lipschitz β hγ hγm s t ht)) |
| (fun z => scalarCDFGradient_cdf_stability β hα hαm hγ hγm s t ht z h hh) x k hk |
| simpa only [NNReal.coe_add,two_mul] using H |
|
|
| theorem scalarCDFGradient_sq_regular (β : ℝ) {α : ℝ → ℝ} |
| (hα : ∀ s, α s ∈ Icc (0:ℝ) 1) (hmono : Monotone α) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) : |
| (∀ x, HasDerivAt (fun y => (scalarCDFGradient β α s t y)^2) |
| (2*scalarCDFGradient β α s t x*scalarCDFHessian β α s t x) x) ∧ |
| Continuous (fun x => 2*scalarCDFGradient β α s t x*scalarCDFHessian β α s t x) ∧ |
| (∀ x, |(scalarCDFGradient β α s t x)^2| ≤ (1:ℝ)) ∧ |
| (∀ x, |2*scalarCDFGradient β α s t x*scalarCDFHessian β α s t x| ≤ (2:ℝ)) := by |
| refine ⟨?_,continuous_const.mul (scalarCDFGradient_lipschitz β hα hmono s t ht).continuous |>.mul |
| (scalarCDFHessian_lipschitz β hα hmono s t ht).continuous,?_,?_⟩ |
| · intro x |
| convert (scalarCDFGradient_hasDerivAt β hα hmono s t ht x).pow 2 using 1 |
| norm_num |
| · intro x |
| rw [abs_pow] |
| exact pow_le_one₀ (abs_nonneg _) (scalarCDF_derivative_bounds β hα hmono s t ht x).1 |
| · intro x |
| calc |
| _ = 2*|scalarCDFGradient β α s t x| * |scalarCDFHessian β α s t x| := by simp only [abs_mul,abs_of_pos (by norm_num : (0:ℝ)<2)] |
| _ ≤ 2*1*1 := mul_le_mul (mul_le_mul_of_nonneg_left |
| (scalarCDF_derivative_bounds β hα hmono s t ht x).1 (by norm_num)) |
| (scalarCDFHessian_abs_le_one β hα hmono s t ht x) (abs_nonneg _) (by norm_num) |
| _ = 2 := by norm_num |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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