| import Mathlib |
| import OAI.Probability.SKBarriers.SpinGlass.Basic |
| import OAI.Probability.SKBarriers.Gaussian.GaussianDomination |
| import OAI.Probability.SKBarriers.Scalar.RawDerivatives |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology Interval |
| noncomputable section |
| open MeasureTheory Set |
| open scoped Interval |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped Topology |
| namespace SK.Analytic |
| section MarginalDerivatives |
| variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
|
|
| def leftRestrict : ((E × ℝ) →L[ℝ] F) →L[ℝ] E →L[ℝ] F := |
| (ContinuousLinearMap.compL ℝ E (E × ℝ) F).flip (ContinuousLinearMap.inl ℝ E ℝ) |
|
|
| theorem leftRestrict_apply_bound (A : (E × ℝ) →L[ℝ] F) : ‖leftRestrict A‖ ≤ ‖A‖ := by |
| exact (ContinuousLinearMap.opNorm_comp_le _ _).trans |
| ((mul_le_mul_of_nonneg_left (ContinuousLinearMap.norm_inl_le_one ℝ E ℝ) (norm_nonneg A)).trans_eq (mul_one _)) |
|
|
| theorem leftRestrict_norm : ‖leftRestrict (E := E) (F := F)‖ ≤ 1 := by |
| exact ContinuousLinearMap.opNorm_le_bound _ zero_le_one |
| (fun A => (leftRestrict_apply_bound A).trans_eq (one_mul _).symm) |
|
|
| theorem leftRestrict_hasFDerivAt (f : E × ℝ → F) (hd : Differentiable ℝ f) (x : E) (y : ℝ) : |
| HasFDerivAt (fun z => f (z,y)) (leftRestrict (fderiv ℝ f (x,y))) x := by |
| exact (hd (x,y)).hasFDerivAt.comp x |
| ((hasFDerivAt_id x).prodMk (hasFDerivAt_const y x)) |
|
|
| theorem fderiv_gaussian_integral (f : E × ℝ → F) (hf : ContDiff ℝ 1 f) |
| (hg : HasExpGrowth f) (hg₁ : HasExpGrowth (fderiv ℝ f)) (x : E) : |
| fderiv ℝ (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) x = |
| ∫ y, leftRestrict (fderiv ℝ f (x,y)) ∂gaussianReal 0 1 := by |
| exact (hasFDerivAt_gaussian_integral f _ hf.continuous |
| (leftRestrict.continuous.comp ((hf.fderiv_right (m := 0) (by norm_num)).continuous)) |
| hg (hg₁.clm leftRestrict) |
| (leftRestrict_hasFDerivAt f (hf.differentiable (by norm_num))) x).fderiv |
|
|
| theorem norm_fderiv_gaussian_integral_le (f : E × ℝ → F) (hf : ContDiff ℝ 1 f) |
| (hg : HasExpGrowth f) (hg₁ : HasExpGrowth (fderiv ℝ f)) (x : E) : |
| ‖fderiv ℝ (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) x‖ ≤ |
| ∫ y, ‖fderiv ℝ f (x,y)‖ ∂gaussianReal 0 1 := by |
| rw [fderiv_gaussian_integral f hf hg hg₁ x] |
| apply (norm_integral_le_integral_norm _).trans |
| apply integral_mono |
| · exact ((hg₁.clm leftRestrict).integrable_gaussian_section |
| (leftRestrict.continuous.comp ((hf.fderiv_right (m := 0) (by norm_num)).continuous)) x).norm |
| · exact (hg₁.integrable_gaussian_section ((hf.fderiv_right (m := 0) (by norm_num)).continuous) x).norm |
| · exact fun y => leftRestrict_apply_bound _ |
|
|
| end MarginalDerivatives |
| end SK.Analytic |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
