working

Nevanlinna/laplace.lean

@@ -53,35 +53,45 @@ theorem laplace_add {f₁ f₂ : ℂ → F} (h₁ : ContDiff ℝ 2 f₁) (h₂ :
   exact h₂.differentiable one_le_two
 
 
-theorem laplace_add_ContDiffOn {f₁ f₂ : ℂ → F} {s : Set ℂ} (hs : IsOpen s) (h₁ : ContDiffOn ℝ 2 f₁ s) (h₂ : ContDiffOn ℝ 2 f₂ s): ∀ x ∈ s, Complex.laplace (f₁ + f₂) x = (Complex.laplace f₁) x + (Complex.laplace f₂) x := by
+theorem laplace_add_ContDiffOn
+  {f₁ f₂ : ℂ → F}
+  {s : Set ℂ}
+  (hs : IsOpen s)
+  (h₁ : ContDiffOn ℝ 2 f₁ s)
+  (h₂ : ContDiffOn ℝ 2 f₂ s) :
+  ∀ x ∈ s, Complex.laplace (f₁ + f₂) x = (Complex.laplace f₁) x + (Complex.laplace f₂) x := by
 
   unfold Complex.laplace
   simp
   intro x hx
+
   have : partialDeriv ℝ 1 (f₁ + f₂) =ᶠ[nhds x] (partialDeriv ℝ 1 f₁) + (partialDeriv ℝ 1 f₂) := by
     sorry
   rw [partialDeriv_eventuallyEq ℝ this]
+  have t₁ : DifferentiableAt ℝ (partialDeriv ℝ 1 f₁) x := by
+    sorry
+  have t₂ : DifferentiableAt ℝ (partialDeriv ℝ 1 f₂) x := by
+    sorry
+  rw [partialDeriv_add₂_differentiableAt ℝ t₁ t₂]
 
+  have : partialDeriv ℝ Complex.I (f₁ + f₂) =ᶠ[nhds x] (partialDeriv ℝ Complex.I f₁) + (partialDeriv ℝ Complex.I f₂) := by
+    sorry
+  rw [partialDeriv_eventuallyEq ℝ this]
+  have t₃ : DifferentiableAt ℝ (partialDeriv ℝ Complex.I f₁) x := by
+    sorry
+  have t₄ : DifferentiableAt ℝ (partialDeriv ℝ Complex.I f₂) x := by
+    sorry
+  rw [partialDeriv_add₂_differentiableAt ℝ t₃ t₄]
 
-  rw [partialDeriv_add₂]
+  -- I am super confused at this point because the tactic 'ring' does not work.
-
+  -- I do not understand why.
-  rw [partialDeriv_add₂]
+  rw [add_assoc]
-  rw [partialDeriv_add₂]
+  rw [add_assoc]
-  rw [partialDeriv_add₂]
+  rw [add_right_inj (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) x)]
-  exact
+  rw [add_comm]
-    add_add_add_comm (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁))
+  rw [add_assoc]
-      (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂))
+  rw [add_right_inj (partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₁) x)]
-      (partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₁))
+  rw [add_comm]
-      (partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₂))
-
-  exact (partialDeriv_contDiff ℝ h₁ Complex.I).differentiable le_rfl
-  exact (partialDeriv_contDiff ℝ h₂ Complex.I).differentiable le_rfl
-  exact h₁.differentiable one_le_two
-  exact h₂.differentiable one_le_two
-  exact (partialDeriv_contDiff ℝ h₁ 1).differentiable le_rfl
-  exact (partialDeriv_contDiff ℝ h₂ 1).differentiable le_rfl
-  exact h₁.differentiable one_le_two
-  exact h₂.differentiable one_le_two
 
 
 theorem laplace_smul {f : ℂ → F} (h : ContDiff ℝ 2 f) : ∀ v : ℝ, Complex.laplace (v • f) = v • (Complex.laplace f) := by

Nevanlinna/partialDeriv.lean

@@ -54,6 +54,22 @@ theorem partialDeriv_add₂ {f₁ f₂ : E → F} {v : E} (h₁ : Differentiable
     rw [fderiv_add (h₁ w) (h₂ w)]
 
 
+theorem partialDeriv_add₂_differentiableAt
+  {f₁ f₂ : E → F}
+  {v : E}
+  {x : E}
+  (h₁ : DifferentiableAt 𝕜 f₁ x)
+  (h₂ : DifferentiableAt 𝕜 f₂ x)
+  :
+  partialDeriv 𝕜 v (f₁ + f₂) x = (partialDeriv 𝕜 v f₁) x + (partialDeriv 𝕜 v f₂) x := by
+
+  unfold partialDeriv
+  have : f₁ + f₂ = fun y ↦ f₁ y + f₂ y := by rfl
+  rw [this]
+  rw [fderiv_add h₁ h₂]
+  rfl
+
+
 theorem partialDeriv_compContLin {f : E → F} {l : F →L[𝕜] G} {v : E} (h : Differentiable 𝕜 f) : partialDeriv 𝕜 v (l ∘ f) = l ∘ partialDeriv 𝕜 v f := by
   unfold partialDeriv