| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.GaussianSecondStep |
| import OAI.Probability.SKBarriers.Calculus.ParameterComposition |
| import OAI.Probability.SKBarriers.Calculus.ParameterIntegral |
| import OAI.Probability.SKBarriers.Calculus.ParameterLinear |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| section ParameterSmooth |
| variable {P E : Type} [NormedAddCommGroup P] [NormedSpace ℝ P] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| def ParamRegular (f : P × E → ℝ) : Prop := |
| ContDiff ℝ 2 f ∧ ParamLinearGrowth f ∧ ParamExpGrowth (fderiv ℝ f) ∧ |
| ParamExpGrowth (fderiv ℝ (fderiv ℝ f)) |
|
|
| theorem ParamRegular.const_mul {f : P × E → ℝ} (hf : ParamRegular f) (c : ℝ) : |
| ParamRegular (fun z => c*f z) := by |
| refine ⟨contDiff_const.mul hf.1,hf.2.1.const_mul c,?_,?_⟩ |
| · apply hf.2.2.1.of_norm_le (abs_nonneg c) |
| intro z |
| rw [((hf.1.differentiable (by norm_num) z).hasFDerivAt.const_mul c).fderiv] |
| simp only [norm_smul,Real.norm_eq_abs] |
| exact le_rfl |
| · apply hf.2.2.2.of_norm_le (abs_nonneg c) |
| intro z |
| rw [fderiv_fderiv_const_mul f hf.1 c] |
| simpa only [Real.norm_eq_abs] using ContinuousLinearMap.opNorm_smul_le c (fderiv ℝ (fderiv ℝ f) z) |
|
|
| theorem param_gaussian_smooth (f : P × (E × ℝ) → ℝ) (hf : ContDiff ℝ 2 f) |
| (hg : ParamExpGrowth f) (hg₁ : ParamExpGrowth (fderiv ℝ f)) |
| (hg₂ : ParamExpGrowth (fderiv ℝ (fderiv ℝ f))) : |
| let Z := fun z : P × E => ∫ y, f (z.1,(z.2,y)) ∂gaussianReal 0 1 |
| ContDiff ℝ 2 Z ∧ ParamExpGrowth (fderiv ℝ Z) ∧ |
| ParamExpGrowth (fderiv ℝ (fderiv ℝ Z)) := by |
| let L : (P × E) × ℝ →L[ℝ] P × (E × ℝ) := |
| (ContinuousLinearEquiv.prodAssoc ℝ P E ℝ).toContinuousLinearMap |
| let ι : P × (E × ℝ) → (P × E) × ℝ := fun z => ((z.1,z.2.1),z.2.2) |
| let g := fun z => f (L z) |
| have hc : ContDiff ℝ 2 g := hf.comp L.contDiff |
| have hg' : ParamExpGrowth (fun z => g (ι z)) := hg |
| have h₁ : ParamExpGrowth (fun z => fderiv ℝ g (ι z)) := |
| param_fderiv_compCLM_growth f L ι (hf.differentiable (by norm_num)) hg₁ |
| have h₂ : ParamExpGrowth (fun z => fderiv ℝ (fderiv ℝ g) (ι z)) := |
| param_second_compCLM_growth f L ι hf hg₂ |
| exact ⟨param_contDiff_gaussian_integral g hc hg' h₁ h₂, |
| param_fderiv_gaussian_integral_growth g (hc.of_le (by norm_num)) hg' h₁, |
| param_second_gaussian_integral_growth g hc hg' h₁ h₂⟩ |
|
|
| theorem ParamRegular.gaussian_integral {f : P × (E × ℝ) → ℝ} (hf : ParamRegular f) : |
| ParamRegular (fun z : P × E => ∫ y, f (z.1,(z.2,y)) ∂gaussianReal 0 1) := by |
| have h := param_gaussian_smooth f hf.1 hf.2.1.expGrowth hf.2.2.1 hf.2.2.2 |
| exact ⟨h.1,hf.2.1.gaussian_integral,h.2⟩ |
| end ParameterSmooth |
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
