| import Mathlib |
|
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| namespace OAI |
|
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| section |
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology Interval |
| namespace SK.Analytic |
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| theorem hasFDerivAt_intervalIntegral |
| {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] |
| (f : E → ℝ → F) (f₁ : E → ℝ → E →L[ℝ] F) |
| (hf : Continuous (Function.uncurry f)) (hf₁ : Continuous (Function.uncurry f₁)) |
| (hd : ∀ x y, HasFDerivAt (fun z => f z y) (f₁ x y) x) |
| (a b : ℝ) (x : E) : |
| HasFDerivAt (fun z => ∫ y in a..b, f z y) (∫ y in a..b, f₁ x y) x := by |
| obtain ⟨C, hC⟩ := ((isCompact_closedBall x 1).prod (show IsCompact (uIcc a b) from isCompact_uIcc)).exists_bound_of_continuousOn |
| hf₁.continuousOn |
| apply intervalIntegral.hasFDerivAt_integral_of_dominated_of_fderiv_le |
| (F' := f₁) (bound := fun _ => C) (s := Metric.closedBall x 1) |
| (Metric.closedBall_mem_nhds x (by norm_num)) |
| · exact Filter.Eventually.of_forall fun z => (hf.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| · exact (hf.comp (continuous_const.prodMk continuous_id)).intervalIntegrable a b |
| · exact (hf₁.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| · exact Filter.Eventually.of_forall fun y hy z hz => hC (z, y) ⟨hz, uIoc_subset_uIcc hy⟩ |
| · exact intervalIntegrable_const |
| · exact Filter.Eventually.of_forall fun y _ z _ => hd z y |
|
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| theorem contDiff_intervalIntegral |
| {E F : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] |
| (n : ℕ) (f : E × ℝ → F) (hf : ContDiff ℝ n f) (a b : ℝ) : |
| ContDiff ℝ n (fun x => ∫ y in a..b, f (x, y)) := by |
| induction n generalizing F with |
| | zero => |
| apply contDiff_zero.mpr |
| exact intervalIntegral.continuous_parametric_intervalIntegral_of_continuous' |
| (f := fun x y => f (x,y)) (by convert! hf.continuous using 1) a b |
| | succ n ih => |
| let f₁ : E × ℝ → E →L[ℝ] F := fun p => |
| (fderiv ℝ f p).comp (ContinuousLinearMap.inl ℝ E ℝ) |
| have hf₁ : ContDiff ℝ n f₁ := by |
| apply ContDiff.clm_comp _ contDiff_const |
| exact hf.fderiv_right (by simp) |
| have hdiff : ∀ x y, HasFDerivAt (fun z => f (z, y)) (f₁ (x,y)) x := by |
| intro x y |
| have hd := (hf.differentiable (by simp) (x, y)).hasFDerivAt |
| exact hd.comp x ((hasFDerivAt_id x).prodMk (hasFDerivAt_const y x)) |
| have hI : ∀ x, HasFDerivAt (fun z => ∫ y in a..b, f (z,y)) |
| (∫ y in a..b, f₁ (x,y)) x := by |
| intro x |
| exact hasFDerivAt_intervalIntegral (fun x y => f (x,y)) |
| (fun x y => f₁ (x,y)) hf.continuous hf₁.continuous hdiff a b x |
| rw [show (↑(n + 1) : WithTop ℕ∞) = (n : WithTop ℕ∞) + 1 by simp, |
| contDiff_succ_iff_fderiv] |
| refine ⟨fun x => (hI x).differentiableAt, ?_, ?_⟩ |
| · simp |
| · have heq : fderiv ℝ (fun z => ∫ y in a..b, f (z,y)) = |
| (fun x => ∫ y in a..b, f₁ (x,y)) := funext fun x => (hI x).fderiv |
| rw [heq] |
| exact ih f₁ hf₁ |
|
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| theorem hasDerivAt_intervalIntegral |
| {F : Type} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] |
| (f f₁ : ℝ → ℝ → F) |
| (hf : Continuous (Function.uncurry f)) (hf₁ : Continuous (Function.uncurry f₁)) |
| (hd : ∀ x y, HasDerivAt (fun z => f z y) (f₁ x y) x) |
| (a b x : ℝ) : |
| HasDerivAt (fun z => ∫ y in a..b, f z y) (∫ y in a..b, f₁ x y) x := by |
| obtain ⟨C, hC⟩ := ((isCompact_closedBall x 1).prod |
| (show IsCompact (uIcc a b) from isCompact_uIcc)).exists_bound_of_continuousOn hf₁.continuousOn |
| apply (intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le |
| (F' := f₁) (bound := fun _ => C) (s := Metric.closedBall x 1) |
| (Metric.closedBall_mem_nhds x (by norm_num)) ?_ ?_ ?_ ?_ ?_ ?_).2 |
| · exact Filter.Eventually.of_forall fun z => (hf.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| · exact (hf.comp (continuous_const.prodMk continuous_id)).intervalIntegrable a b |
| · exact (hf₁.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| · exact Filter.Eventually.of_forall fun y hy z hz => hC (z, y) ⟨hz, uIoc_subset_uIcc hy⟩ |
| · exact intervalIntegrable_const |
| · exact Filter.Eventually.of_forall fun y _ z _ => hd z y |
|
|
| end SK.Analytic |
|
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| noncomputable section |
| open MeasureTheory Set |
| open scoped Interval |
|
|
| end |
| end |
| end |
| end |
| end |
|
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| end OAI |
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