| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.GaussianStepCalculus |
| import OAI.Probability.SKBarriers.Gaussian.LocalMarginal |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| section LocalStepDerivative |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
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| theorem fderiv_local_gaussianStep {f : E × ℝ → ℝ} (hf : ContDiff ℝ 1 f) |
| (hg : GaussianLocallyDominated f) (hg₁ : GaussianLocallyDominated (fderiv ℝ f)) |
| (m : ℝ) (he : GaussianLocallyDominated (fun z => Real.exp (m*f z))) |
| (he₁ : GaussianLocallyDominated (fderiv ℝ (fun z => Real.exp (m*f z)))) (x : E) : |
| fderiv ℝ (gaussianStep m f) x = |
| ∫ y, leftRestrict (fderiv ℝ f (x,y)) ∂gaussianStepLaw m f x := by |
| have hd := hf.differentiable (by norm_num) |
| by_cases hm : m = 0 |
| · subst m |
| rw [gaussianStepLaw_zero] |
| have hfun : SK.Analytic.gaussianStep 0 f = (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) := by |
| funext x |
| simp only [SK.Analytic.gaussianStep,ite_true] |
| rw [hfun] |
| exact fderiv_local_gaussian_integral f hf hg hg₁ x |
| · let Z : E → ℝ := fun x => ∫ y, Real.exp (m*f (x,y)) ∂gaussianReal 0 1 |
| have hc : ContDiff ℝ 1 (fun z => Real.exp (m*f z)) := (contDiff_const.mul hf).exp |
| have hp : Z x ≠ 0 := (integral_exp_pos (he.integrable_section hc.continuous x)).ne' |
| have hi := hasFDerivAt_local_gaussian_integral (fun z => Real.exp (m*f z)) _ |
| hc.continuous (leftRestrict.continuous.comp |
| ((hc.fderiv_right (m := 0) (by norm_num)).continuous)) he (he₁.clm leftRestrict) |
| (leftRestrict_hasFDerivAt _ (hc.differentiable (by norm_num))) x |
| have hfun : gaussianStep m f = fun x => Real.log (Z x)/m := by |
| funext x |
| simp only [gaussianStep,ite_eq_right hm,positiveGaussianLogStep,Z] |
| rw [hfun] |
| have hder := (hi.log hp).mul_const m⁻¹ |
| simp only [div_eq_mul_inv] at ⊢ |
| rw [hder.fderiv] |
| rw [← integral_smul,← integral_smul,gaussianStepLaw,integral_tilted] |
| apply integral_congr_ae |
| filter_upwards [] with y |
| have hdexp := ((hd (x,y)).hasFDerivAt.const_mul m).exp.fderiv |
| simp only [Function.comp_apply] |
| rw [hdexp] |
| ext v |
| simp only [smul_apply,smul_eq_mul,leftRestrict, |
| ContinuousLinearMap.compL_apply,ContinuousLinearMap.flip_apply, |
| ContinuousLinearMap.comp_apply,ContinuousLinearMap.inl_apply] |
| dsimp only [Z] at hp ⊢ |
| field_simp [hm,hp] |
| end LocalStepDerivative |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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