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785c727 | 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 | import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
namespace OAI
/-! Passing from derivatives along each initial frame to a derivative in
the space of continuous fields. The fundamental theorem of calculus avoids
any separate uniform remainder assumption. -/
namespace PositiveEinsteinFour.Averages
open MeasureTheory
noncomputable section
variable {Ω E : Type*} [TopologicalSpace Ω] [CompactSpace Ω]
[NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
local instance : NormedAddCommGroup C(Ω, E) := ContinuousMap.instNormedAddCommGroup
local instance : NormedSpace ℝ C(Ω, E) := ContinuousMap.normedSpace
/-- A continuous family of pointwise derivatives is the derivative in the
supremum norm on continuous fields. -/
theorem continuousField_hasDerivAt {F G : ℝ → C(Ω, E)}
(hG : Continuous G)
(hpoint : ∀ t x, HasDerivAt (fun s => F s x) (G t x) t) (t : ℝ) :
HasDerivAt F (G t) t := by
let K : ℝ → C(Ω, E) := fun s => F 0 + ∫ u in (0 : ℝ)..s, G u
have hK (s : ℝ) : HasDerivAt K (G s) s :=
(hG.integral_hasStrictDerivAt 0 s).hasDerivAt.const_add (F 0)
have hFK : F = K := by
funext s
apply ContinuousMap.ext
intro x
have hd (u : ℝ) : HasDerivAt (fun u => F u x - K u x) 0 u := by
let ev : C(Ω, E) →L[ℝ] E := ContinuousMap.evalCLM ℝ x
have he : HasDerivAt (fun u => K u x) (G u x) u :=
ev.hasFDerivAt.comp_hasDerivAt u (hK u)
convert (hpoint u x).sub he using 1
simp
have hc := is_const_of_deriv_eq_zero
(fun u => (hd u).differentiableAt) (fun u => (hd u).deriv) s 0
have hz : F 0 x - K 0 x = 0 := by simp [K]
exact sub_eq_zero.mp (hc.trans hz)
rw [hFK]
exact hK t
/-- Joint continuity in time and the compact parameter gives exactly the
continuity required by `continuousField_hasDerivAt`. -/
theorem continuousField_hasDerivAt_of_joint {F G : ℝ → C(Ω, E)}
(hG : Continuous (fun p : ℝ × Ω => G p.1 p.2))
(hpoint : ∀ t x, HasDerivAt (fun s => F s x) (G t x) t) (t : ℝ) :
HasDerivAt F (G t) t :=
continuousField_hasDerivAt
(ContinuousMap.continuous_of_continuous_uncurry G hG) hpoint t
end
end PositiveEinsteinFour.Averages
end OAI
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