| import Mathlib |
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| namespace OAI |
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| /-! Circle Submean Calculus. -/ |
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| section |
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| noncomputable section |
| open Set Filter Topology Metric MeasureTheory |
| namespace PotentialABP |
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| 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 |
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| end PotentialABP |
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| end |
| end |
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| end OAI |
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