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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import Mathlib
namespace OAI
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
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
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
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
noncomputable section
open MeasureTheory Set
open scoped Interval
end
end
end
end
end
end OAI
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