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2fca478 | 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 | import Mathlib
namespace OAI
/-! Circle Submean Calculus. -/
section
noncomputable section
open Set Filter Topology Metric MeasureTheory
namespace PotentialABP
lemma hasDerivAt_intervalIntegral_continuous
{F F' : ℝ → ℝ → ℝ}
(hF : Continuous (Function.uncurry F))
(hF' : Continuous (Function.uncurry F'))
(hderiv : ∀ r θ, HasDerivAt (fun s => F s θ) (F' r θ) r)
(a b r : ℝ) :
HasDerivAt (fun s => ∫ θ in a..b, F s θ) (∫ θ in a..b, F' r θ) r := by
obtain ⟨C, hC⟩ := ((isCompact_closedBall r 1).prod isCompact_uIcc).bddAbove_image
hF'.norm.continuousOn
apply (intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
(bound := fun _ => C) (s := ball r 1) (ball_mem_nhds r (by norm_num)) ?_ ?_ ?_ ?_ ?_ ?_).2
· filter_upwards [] with s
exact (hF.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable
· exact (hF.comp (continuous_const.prodMk continuous_id)).intervalIntegrable a b
· exact (hF'.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable
· filter_upwards [] with θ hθ s hs
exact hC ⟨(s,θ), ⟨ball_subset_closedBall hs, uIoc_subset_uIcc hθ⟩, rfl⟩
· exact intervalIntegrable_const
· filter_upwards [] with θ _ s _
exact hderiv s θ
end PotentialABP
end
end
end OAI
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