From bfdc7f6d2ac0799eeba0280d42793646082b4777 Mon Sep 17 00:00:00 2001
From: Stefan Kebekus <kebekus@users.noreply.github.com>
Date: Fri, 3 May 2024 12:23:09 +0200
Subject: [PATCH] Update complexHarmonic.lean

---
 Nevanlinna/complexHarmonic.lean | 64 +++++++++++++++++++++++++++------
 1 file changed, 53 insertions(+), 11 deletions(-)

diff --git a/Nevanlinna/complexHarmonic.lean b/Nevanlinna/complexHarmonic.lean
index a991fff..176a71f 100644
--- a/Nevanlinna/complexHarmonic.lean
+++ b/Nevanlinna/complexHarmonic.lean
@@ -9,16 +9,55 @@ import Nevanlinna.cauchyRiemann
 noncomputable def Complex.laplace : (ℂ → ℂ) → (ℂ → ℂ) := by
   intro f
 
-  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
+  let fx  := fun w ↦ (fderiv ℝ f w) 1
+  let fxx := fun w ↦ (fderiv ℝ fx w) 1
+  let fy  := fun w ↦ (fderiv ℝ f w) Complex.I
+  let fyy := fun w ↦ (fderiv ℝ fy w) Complex.I
   exact fun z ↦ (fxx z) + (fyy z)
 
 
 def Harmonic (f : ℂ → ℂ) : Prop :=
   (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)
 
+#check second_derivative_symmetric
+
+lemma zwoDiff (f : ℝ × ℝ → ℝ) (h : ContDiff ℝ 2 f) : ∀ z a b : ℂ, 0 = 1 := by
+  intro z a b
+
+  let fx  := fun w ↦ (fderiv ℝ f w) 1
+  let fxx := fun w ↦ (fderiv ℝ fx w) 1
+  let f2  := (fderiv ℝ (fun w => fderiv ℝ f w) z) 1 1
+
+  have : iteratedFDeriv ℝ (1 + 1) f = 0 := by
+    rw [iteratedFDeriv_succ_eq_comp_left]
+
+    simp
+    sorry
+
+  have : f2 = fxx z := by
+    dsimp [f2, fxx, fx]
+
+    sorry
+
+  sorry
+
+
+lemma derivSymm (f : ℂ → ℂ) (h : Differentiable ℝ f) :
+  ∀ z a b : ℂ, (fderiv ℝ (fun w => fderiv ℝ f w) z) a b = (fderiv ℝ (fun w => fderiv ℝ f w) z) b a := by
+  intro z a b
+
+  let f' := fun w => (fderiv ℝ f w)
+  have h₀ : ∀ y, HasFDerivAt f (f' y) y := by
+    exact fun y => DifferentiableAt.hasFDerivAt (h y)
+
+  let f'' := (fderiv ℝ f' z)
+  have h₁ : HasFDerivAt f' f'' z := by
+    apply DifferentiableAt.hasFDerivAt
+    sorry
+
+  let A := second_derivative_symmetric h₀ h₁ a b
+  dsimp [f'', f'] at A
+  apply A
 
 theorem re_comp_holomorphic_is_harmonic (f : ℂ → ℂ) :
   Differentiable ℂ f → Harmonic f := by
@@ -32,6 +71,7 @@ theorem re_comp_holomorphic_is_harmonic (f : ℂ → ℂ) :
   · -- Laplace of f is zero
     intro z
     unfold Complex.laplace
+
     simp
 
     conv =>
@@ -41,10 +81,10 @@ theorem re_comp_holomorphic_is_harmonic (f : ℂ → ℂ) :
       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
@@ -54,9 +94,12 @@ theorem re_comp_holomorphic_is_harmonic (f : ℂ → ℂ) :
       simp
       exact t₀ z
     rw [t₁]
-    
-    have t₂ : (fderiv ℝ (fun w => (fderiv ℝ f w) 1) z) Complex.I 
+
+    have t₂₀ : Differentiable ℝ f := by sorry
+    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
+      let A := derivSymm f t₂₀ z 1 Complex.I
+
       sorry
     rw [t₂]
 
@@ -68,9 +111,8 @@ theorem re_comp_holomorphic_is_harmonic (f : ℂ → ℂ) :
       arg 2
       intro z
       rw [CauchyRiemann₁ (h z)]
-    
+
     rw [t₁]
-    
+
     rw [← mul_assoc]
     simp
-    
\ No newline at end of file