| import Mathlib.Topology.ContinuousMap.Compact |
| import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus |
| import Mathlib.Analysis.Calculus.ContDiff.Deriv |
| import Mathlib.Tactic.Abel |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped Topology Interval ContDiff |
|
|
| namespace SmoothLocal.ODE |
|
|
| variable {X E : Type*} [TopologicalSpace X] [CompactSpace X] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
|
|
| theorem hasDerivAt_continuousMap_of_eval |
| {a b : ℝ} {F P : ℝ → C(X, E)} |
| (hP : ContinuousOn P (Ioo a b)) |
| (hpoint : ∀ t ∈ Ioo a b, ∀ x : X, |
| HasDerivAt (fun s => F s x) (P t x) t) |
| {t : ℝ} (ht : t ∈ Ioo a b) : HasDerivAt F (P t) t := by |
| have hmeas : StronglyMeasurableAtFilter P (𝓝 t) volume := |
| ContinuousOn.stronglyMeasurableAtFilter isOpen_Ioo hP t ht |
| have hprim : HasDerivAt (fun y => F t + ∫ s in t..y, P s) (P t) t := |
| (intervalIntegral.integral_hasDerivAt_right |
| (IntervalIntegrable.refl : IntervalIntegrable P volume t t) |
| hmeas (hP.continuousAt (Ioo_mem_nhds ht.1 ht.2))).const_add (F t) |
| have heq : F =ᶠ[𝓝 t] (fun y => F t + ∫ s in t..y, P s) := by |
| apply Filter.mem_of_superset (Ioo_mem_nhds ht.1 ht.2) |
| intro y hy |
| have hseg : uIcc t y ⊆ Ioo a b := by |
| intro s hs |
| exact ⟨lt_of_lt_of_le (lt_min ht.1 hy.1) hs.1, |
| lt_of_le_of_lt hs.2 (max_lt ht.2 hy.2)⟩ |
| have hPI : IntervalIntegrable P volume t y := (hP.mono hseg).intervalIntegrable |
| apply ContinuousMap.ext |
| intro x |
| have hPIx : IntervalIntegrable (fun s => P s x) volume t y := |
| ((ContinuousMap.evalCLM ℝ x).continuous.comp_continuousOn |
| (hP.mono hseg)).intervalIntegrable |
| have hscalar : (∫ s in t..y, P s x) = F y x - F t x := |
| intervalIntegral.integral_eq_sub_of_hasDerivAt |
| (fun s hs => hpoint s (hseg hs) x) hPIx |
| change F y x = F t x + (ContinuousMap.evalCLM ℝ x) (∫ s in t..y, P s) |
| rw [← (ContinuousMap.evalCLM ℝ x).intervalIntegral_comp_comm hPI] |
| change F y x = F t x + ∫ s in t..y, P s x |
| rw [hscalar] |
| abel |
| exact hprim.congr_of_eventuallyEq heq |
|
|
| theorem continuousMap_smooth_of_eval_derivative_sequence |
| {a b : ℝ} (F : ℕ → ℝ → C(X, E)) |
| (hcont : ∀ n, ContinuousOn (F n) (Ioo a b)) |
| (hpoint : ∀ n t, t ∈ Ioo a b → ∀ x : X, |
| HasDerivAt (fun s => F n s x) (F (n + 1) t x) t) : |
| ∀ n, ContDiffOn ℝ ∞ (F n) (Ioo a b) := by |
| have hd (n : ℕ) (t : ℝ) (ht : t ∈ Ioo a b) : |
| HasDerivAt (F n) (F (n + 1) t) t := |
| hasDerivAt_continuousMap_of_eval (hcont (n + 1)) (hpoint n) ht |
| have hnat : ∀ m : ℕ, ∀ n : ℕ, ContDiffOn ℝ m (F n) (Ioo a b) := by |
| intro m |
| induction m with |
| | zero => |
| intro n |
| simpa only [Nat.cast_zero, contDiffOn_zero] using hcont n |
| | succ m hm => |
| intro n |
| have hnext : ContDiffOn ℝ ((m : ℕ∞ω) + 1) (F n) (Ioo a b) := by |
| rw [contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo] |
| refine ⟨fun t ht => (hd n t ht).differentiableAt.differentiableWithinAt, |
| by simp, ?_⟩ |
| exact (hm (n + 1)).congr (fun t ht => (hd n t ht).deriv) |
| simpa only [Nat.cast_add, Nat.cast_one] using hnext |
| intro n |
| exact contDiffOn_infty.mpr (fun m => hnat m n) |
|
|
| theorem smooth_curry_of_joint_derivative_sequence |
| {a b : ℝ} (H : ℕ → ℝ → X → E) |
| (hjoint : ∀ n, ContinuousOn (Function.uncurry (H n)) |
| ((Ioo a b) ×ˢ (Set.univ : Set X))) |
| (hpoint : ∀ n t, t ∈ Ioo a b → ∀ x : X, |
| HasDerivAt (fun s => H n s x) (H (n + 1) t x) t) : |
| ∀ n, ContDiffOn ℝ ∞ (fun t => ContinuousMap.mkD (H n t) 0) (Ioo a b) := by |
| let F : ℕ → ℝ → C(X, E) := fun n t => ContinuousMap.mkD (H n t) 0 |
| have hslice (n : ℕ) (t : ℝ) (ht : t ∈ Ioo a b) : Continuous (H n t) := |
| (hjoint n).comp_continuous (continuous_const.prodMk continuous_id) |
| (fun _ => ⟨ht, Set.mem_univ _⟩) |
| have heval (n : ℕ) (t : ℝ) (ht : t ∈ Ioo a b) (x : X) : F n t x = H n t x := by |
| dsimp only [F] |
| rw [ContinuousMap.mkD_of_continuous (hslice n t ht)] |
| rfl |
| have hcont (n : ℕ) : ContinuousOn (F n) (Ioo a b) := |
| ContinuousMap.continuousOn_mkD_of_uncurry (H n) 0 (hjoint n) |
| have hpointF (n : ℕ) (t : ℝ) (ht : t ∈ Ioo a b) (x : X) : |
| HasDerivAt (fun s => F n s x) (F (n + 1) t x) t := by |
| have hevent : (fun s => F n s x) =ᶠ[𝓝 t] (fun s => H n s x) := |
| Filter.mem_of_superset (Ioo_mem_nhds ht.1 ht.2) |
| (fun s hs => heval n s hs x) |
| rw [heval (n + 1) t ht x] |
| exact (hpoint n t ht x).congr_of_eventuallyEq hevent |
| exact continuousMap_smooth_of_eval_derivative_sequence F hcont hpointF |
|
|
| end SmoothLocal.ODE |
|
|
| end |
|
|
| end OAI |
|
|
