Update complexHarmonic.lean

Nevanlinna/complexHarmonic.lean

@@ -46,9 +46,14 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
     unfold Complex.laplace
     rw [CauchyRiemann₄ h]
 
+    -- This lemma says that partial derivatives commute with complex scalar
+    -- multiplication. This is a consequence of partialDeriv_compContLin once we
+    -- note that complex scalar multiplication is continuous ℝ-linear.
     have : ∀ v, ∀ s : ℂ, ∀ g : ℂ → ℂ, Differentiable ℝ g →  Real.partialDeriv v (s • g) = s • (Real.partialDeriv v g) := by
       intro v s g hg
 
+      -- Present scalar multiplication as a continuous ℝ-linear map. This is
+      -- horrible, there must be better ways to do that.
       let sMuls : ℂ →L[ℝ] ℂ :=
       {
         toFun := fun x ↦ s * x
@@ -61,8 +66,11 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
           ring
       }
 
+      -- Bring the goal into a form that is recognized by
+      -- partialDeriv_compContLin.
       have : s • g = sMuls ∘ g := by rfl
       rw [this]
+
       rw [partialDeriv_compContLin hg]
       rfl