Update complexHarmonic.lean

Nevanlinna/complexHarmonic.lean

@@ -128,13 +128,20 @@ theorem logabs_of_holomorphic_is_harmonic
   have normSq_is_real_C2 : ContDiff ℝ 2 Complex.normSq := by
     unfold Complex.normSq
     simp
+    conv =>
+      arg 3
+      intro x
+      rw [← Complex.reCLM_apply, ← Complex.imCLM_apply]
     apply ContDiff.add
     apply ContDiff.mul
-    sorry
+    apply ContinuousLinearMap.contDiff Complex.reCLM
+    apply ContinuousLinearMap.contDiff Complex.reCLM
+    apply ContDiff.mul
+    apply ContinuousLinearMap.contDiff Complex.imCLM
+    apply ContinuousLinearMap.contDiff Complex.imCLM
 
   constructor
   · -- logabs f is real C²
-
     have : (fun z ↦ Real.log ‖f z‖) = (Real.log ∘ Complex.normSq ∘ f) / 2 := by
       funext z
       simp
@@ -143,8 +150,10 @@ theorem logabs_of_holomorphic_is_harmonic
       rw [Real.log_sqrt]
       exact Complex.normSq_nonneg (f z)
     rw [this]
-    have : Real.log ∘ ⇑Complex.normSq ∘ f / 2 = (fun z ↦ (1 / 2) • ((Real.log ∘ ⇑Complex.normSq ∘ f) z)) := by
-      sorry
+    have : Real.log ∘ ⇑Complex.normSq ∘ f / 2 = (fun z ↦ (1 / (2 : ℝ)) • ((Real.log ∘ ⇑Complex.normSq ∘ f) z)) := by
+      funext z
+      simp
+      exact div_eq_inv_mul (Real.log (Complex.normSq (f z))) 2
     rw [this]
 
     apply contDiff_iff_contDiffAt.2
@@ -160,4 +169,5 @@ theorem logabs_of_holomorphic_is_harmonic
     exact ContDiff.contDiffAt f_is_real_C2
 
   · -- Laplace vanishes
+  
     sorry