| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling.CompactIntegral |
| open Set Function Filter MeasureTheory Metric |
| open scoped Topology |
|
|
| variable {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] |
|
|
| omit [CompleteSpace F] in |
| lemma parametric_hasFDerivAt {f : E × ℝ → F} (hf : ContDiff ℝ 1 f) |
| {s : Set ℝ} (hs : IsCompact s) (x : E) : |
| HasFDerivAt (fun x => ∫t in s,f (x,t)) |
| (∫t in s,(fderiv ℝ f (x,t)).comp (ContinuousLinearMap.inl ℝ E ℝ)) x := by |
| let D := fun p : E × ℝ => (fderiv ℝ f p).comp (ContinuousLinearMap.inl ℝ E ℝ) |
| have hD : Continuous D := (hf.continuous_fderiv (by norm_num)).clm_comp continuous_const |
| obtain ⟨C,hC⟩ := ((isCompact_closedBall x 1).prod hs).exists_bound_of_continuousOn hD.continuousOn |
| apply hasFDerivAt_integral_of_dominated_of_fderiv_le |
| (F':=fun y t => D (y,t)) (bound:=fun _ => C) (ball_mem_nhds x zero_lt_one) |
| · exact Eventually.of_forall fun y => (hf.continuous.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| · exact (hf.continuous.comp (continuous_const.prodMk continuous_id)).continuousOn.integrableOn_compact hs |
| · exact (hD.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| · filter_upwards [ae_restrict_mem hs.measurableSet] with t ht y hy |
| exact hC (y,t) ⟨ball_subset_closedBall hy,ht⟩ |
| · exact integrableOn_const hs.measure_ne_top |
| · filter_upwards [] with t y _hy |
| exact ((hf.differentiable (by norm_num) (y,t)).hasFDerivAt).comp y |
| ((hasFDerivAt_id (𝕜:=ℝ) y).prodMk (hasFDerivAt_const (𝕜:=ℝ) t y)) |
|
|
| theorem contDiff_parametric {f : E × ℝ → F} (n : ℕ) |
| (hf : ContDiff ℝ (n:WithTop ℕ∞) f) {s : Set ℝ} (hs : IsCompact s) : |
| ContDiff ℝ (n:WithTop ℕ∞) (fun x => ∫t in s,f (x,t)) := by |
| induction n generalizing F with |
| | zero => exact contDiff_zero.mpr (continuous_parametric_integral_of_continuous (f:=fun x t => f (x,t)) hf.continuous hs) |
| | succ n ih => |
| let D := fun p : E × ℝ => (fderiv ℝ f p).comp (ContinuousLinearMap.inl ℝ E ℝ) |
| have hD : ContDiff ℝ (n:WithTop ℕ∞) D := (hf.fderiv_right (by simp)).clm_comp contDiff_const |
| have hd (x : E) := parametric_hasFDerivAt (hf.of_le (by exact_mod_cast Nat.succ_pos n)) hs x |
| rw [show ((n+1:ℕ):WithTop ℕ∞)=(n:WithTop ℕ∞)+1 by simp,contDiff_succ_iff_fderiv] |
| refine ⟨fun x => (hd x).differentiableAt,?_⟩ |
| refine ⟨by simp,?_⟩ |
| have he : fderiv ℝ (fun x => ∫t in s,f (x,t))=fun x => ∫t in s,D (x,t) := funext fun x => (hd x).fderiv |
| rw [he] |
| exact ih hD |
|
|
| theorem smooth_parametric {f : E × ℝ → F} (hf : ContDiff ℝ (⊤:ℕ∞) f) |
| {s : Set ℝ} (hs : IsCompact s) : |
| ContDiff ℝ (⊤:ℕ∞) (fun x => ∫t in s,f (x,t)) := by |
| apply contDiff_infty.mpr |
| intro n |
| exact contDiff_parametric n (hf.of_le (WithTop.coe_le_coe.mpr le_top)) hs |
| end LogConcaveSampling.CompactIntegral |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
