| 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 |
|
|
