Update complexHarmonic.lean

Nevanlinna/complexHarmonic.lean

@@ -3,38 +3,74 @@ import Mathlib.Analysis.Complex.TaylorSeries
 import Mathlib.Analysis.Calculus.LineDeriv.Basic
 import Mathlib.Analysis.Calculus.ContDiff.Defs
 import Mathlib.Analysis.Calculus.FDeriv.Basic
+import Mathlib.Analysis.Calculus.FDeriv.Symmetric
 import Nevanlinna.cauchyRiemann
 
-noncomputable def Complex.laplace : (ℂ → ℝ) → (ℂ → ℝ) := by
+noncomputable def Complex.laplace : (ℂ → ℂ) → (ℂ → ℂ) := by
   intro f
-  let f₁ := fun x ↦ lineDeriv ℝ f x 1
-  let f₁₁ := fun x ↦ lineDeriv ℝ f₁ x 1
-  let f₂ := fun x ↦ lineDeriv ℝ f x Complex.I
-  let f₂₂ := fun x ↦ lineDeriv ℝ f₂ x Complex.I
-  exact f₁₁ + f₂₂
+
+  let fx  := fun w ↦ fderiv ℝ f w 1
+  let fxx := fun z ↦ fderiv ℝ fx z 1
+  let fy  := fun w ↦ fderiv ℝ f w Complex.I
+  let fyy := fun z ↦ fderiv ℝ fy z Complex.I
+  exact fun z ↦ (fxx z) + (fyy z)
 
 
-def Harmonic (f : ℂ → ℝ) : Prop :=
+def Harmonic (f : ℂ → ℂ) : Prop :=
   (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)
 
 
 theorem re_comp_holomorphic_is_harmonic (f : ℂ → ℂ) :
-  Differentiable ℂ f → Harmonic (Complex.reCLM ∘ f) := by
+  Differentiable ℂ f → Harmonic f := by
 
   intro h
 
   constructor
   · -- Complex.reCLM ∘ f is two times real continuously differentiable
-    apply ContDiff.comp
-    · -- Complex.reCLM is two times real continuously differentiable
-      exact ContinuousLinearMap.contDiff Complex.reCLM
-    · -- f is two times real continuously differentiable
-      exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
+    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
 
   · -- Laplace of f is zero
     intro z
     unfold Complex.laplace
     simp
-    let ZZ := (CauchyRiemann₃ (h z)).left
 
-    sorry
+    conv =>
+      left
+      right
+      arg 1
+      arg 2
+      intro z
+      rw [CauchyRiemann₁ (h z)]
+    
+    have t₀ : ∀ z, DifferentiableAt ℝ (fun w => (fderiv ℝ f w) 1) z := by
+      intro z
+      
+      sorry
+
+    have t₁ : ∀ x, (fderiv ℝ (fun w => Complex.I * (fderiv ℝ f w) 1) z) x
+      = Complex.I * ((fderiv ℝ (fun w => (fderiv ℝ f w) 1) z) x) := by
+      intro x
+      rw [fderiv_const_mul]
+      simp
+      exact t₀ z
+    rw [t₁]
+    
+    have t₂ : (fderiv ℝ (fun w => (fderiv ℝ f w) 1) z) Complex.I 
+      = (fderiv ℝ (fun w => (fderiv ℝ f w) Complex.I) z) 1 := by
+      sorry
+    rw [t₂]
+
+    conv =>
+      left
+      right
+      arg 2
+      arg 1
+      arg 2
+      intro z
+      rw [CauchyRiemann₁ (h z)]
+    
+    rw [t₁]
+    
+    rw [← mul_assoc]
+    simp
+