| import Mathlib.Analysis.Calculus.FDeriv.Partial |
| import Mathlib.Analysis.Calculus.ContDiff.Comp |
| import Mathlib.Analysis.Calculus.ContDiff.Operations |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.ODE |
|
|
| def jointDifferential (fx fy : ℝ) : (ℝ × ℝ) →L[ℝ] ℝ := |
| (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight fx + |
| (ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight fy |
|
|
| theorem jointDifferential_eq_coprod (fx fy : ℝ) : |
| jointDifferential fx fy = |
| (ContinuousLinearMap.toSpanSingleton ℝ fx).coprod |
| (ContinuousLinearMap.toSpanSingleton ℝ fy) := by |
| apply ContinuousLinearMap.ext |
| intro v |
| simp [jointDifferential] |
|
|
| theorem hasFDerivAt_of_continuous_partials |
| {s : Set (ℝ × ℝ)} (hs : IsOpen s) {f fx fy : (ℝ × ℝ) → ℝ} |
| (hcx : ContinuousOn fx s) (hcy : ContinuousOn fy s) |
| (hdx : ∀ p ∈ s, HasDerivAt (fun x => f (x, p.2)) (fx p) p.1) |
| (hdy : ∀ p ∈ s, HasDerivAt (fun y => f (p.1, y)) (fy p) p.2) |
| {p : ℝ × ℝ} (hp : p ∈ s) : |
| HasFDerivAt f (jointDifferential (fx p) (fy p)) p := by |
| have hspan : Continuous (ContinuousLinearMap.toSpanSingleton ℝ : |
| ℝ → (ℝ →L[ℝ] ℝ)) := |
| (ContinuousLinearMap.toSpanSingletonCLE (𝕜 := ℝ) (E := ℝ)).continuous |
| have hfirst : ∀ᶠ q in 𝓝 p, |
| HasFDerivAt (fun x => f (x, q.2)) |
| (ContinuousLinearMap.toSpanSingleton ℝ (fx q)) q.1 := by |
| filter_upwards [hs.mem_nhds hp] with q hq |
| exact (hdx q hq).hasFDerivAt |
| have hsecond : ∀ᶠ q in 𝓝 p, |
| HasFDerivAt (fun y => f (q.1, y)) |
| (ContinuousLinearMap.toSpanSingleton ℝ (fy q)) q.2 := by |
| filter_upwards [hs.mem_nhds hp] with q hq |
| exact (hdy q hq).hasFDerivAt |
| have hfull := hasStrictFDerivAt_uncurry_coprod (𝕜 := ℝ) |
| (f := fun x y => f (x, y)) |
| (f₁ := fun x y => ContinuousLinearMap.toSpanSingleton ℝ (fx (x, y))) |
| (f₂ := fun x y => ContinuousLinearMap.toSpanSingleton ℝ (fy (x, y))) |
| hfirst hsecond |
| (hspan.continuousAt.comp (hcx.continuousAt (hs.mem_nhds hp))) |
| (hspan.continuousAt.comp (hcy.continuousAt (hs.mem_nhds hp))) |
| rw [jointDifferential_eq_coprod] |
| exact hfull.hasFDerivAt |
|
|
| theorem joint_smooth_of_partial_derivative_sequence |
| {s : Set (ℝ × ℝ)} (hs : IsOpen s) |
| (H : ℕ → ℕ → (ℝ × ℝ) → ℝ) |
| (hc : ∀ i j, ContinuousOn (H i j) s) |
| (hdx : ∀ i j p, p ∈ s → |
| HasDerivAt (fun x => H i j (x, p.2)) (H (i + 1) j p) p.1) |
| (hdy : ∀ i j p, p ∈ s → |
| HasDerivAt (fun y => H i j (p.1, y)) (H i (j + 1) p) p.2) : |
| ∀ i j, ContDiffOn ℝ ∞ (H i j) s := by |
| have hd (i j : ℕ) (p : ℝ × ℝ) (hp : p ∈ s) : |
| HasFDerivAt (H i j) |
| (jointDifferential (H (i + 1) j p) (H i (j + 1) p)) p := |
| hasFDerivAt_of_continuous_partials hs (hc (i + 1) j) (hc i (j + 1)) |
| (hdx i j) (hdy i j) hp |
| have hnat : ∀ n : ℕ, ∀ i j, ContDiffOn ℝ n (H i j) s := by |
| intro n |
| induction n with |
| | zero => |
| intro i j |
| simpa only [Nat.cast_zero, contDiffOn_zero] using hc i j |
| | succ n hn => |
| intro i j |
| have hnext : ContDiffOn ℝ ((n : ℕ∞ω) + 1) (H i j) s := by |
| rw [contDiffOn_succ_iff_fderiv_of_isOpen hs] |
| refine ⟨fun p hp => (hd i j p hp).differentiableAt.differentiableWithinAt, |
| by simp, ?_⟩ |
| have hfield : ContDiffOn ℝ n |
| (fun p => jointDifferential (H (i + 1) j p) (H i (j + 1) p)) s := |
| (contDiffOn_const.smulRight (hn (i + 1) j)).add |
| (contDiffOn_const.smulRight (hn i (j + 1))) |
| exact hfield.congr (fun p hp => (hd i j p hp).fderiv) |
| simpa only [Nat.cast_add, Nat.cast_one] using hnext |
| intro i j |
| exact contDiffOn_infty.mpr (fun n => hnat n i j) |
|
|
| end SmoothLocal.ODE |
|
|
| end |
|
|
| end OAI |
|
|
