Update complexHarmonic.lean

Nevanlinna/complexHarmonic.lean

@@ -7,14 +7,49 @@ import Mathlib.Analysis.Calculus.FDeriv.Basic
 import Mathlib.Analysis.Calculus.FDeriv.Symmetric
 import Nevanlinna.cauchyRiemann
 
+noncomputable def Real.partialDeriv : ℂ → (ℂ → ℂ) → (ℂ → ℂ) := by
+  intro v
+  intro f
+  exact fun w ↦ (fderiv ℝ f w) v
+
+theorem CauchyRiemann₄ {f : ℂ → ℂ} : (Differentiable ℂ f)
+  → Real.partialDeriv Complex.I f = Complex.I • Real.partialDeriv 1 f := by
+  intro h
+  unfold Real.partialDeriv
+  
+  conv =>
+    left
+    intro w
+    rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
+    simp
+    rw [← mul_one Complex.I]
+    rw [← smul_eq_mul]
+    rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f w) Complex.I 1]
+  conv =>
+    right
+    right
+    intro w    
+    rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
+
+theorem partialDeriv_smul {f : ℂ → ℂ } {a v : ℂ } : Real.partialDeriv v (a • f) = a • Real.partialDeriv v f := by
+  unfold Real.partialDeriv
+  have : a • f = fun y ↦ a • f y := by rfl
+  conv =>
+    left
+    intro w
+    rw [this]
+    rw [fderiv_const_smul]
+    
+  
+  sorry
+
 noncomputable def Complex.laplace : (ℂ → ℂ) → (ℂ → ℂ) := by
   intro f
-
-  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)
+  let fx  := Real.partialDeriv 1 f
+  let fxx := Real.partialDeriv 1 fx
+  let fy  := Real.partialDeriv Complex.I f 
+  let fyy := Real.partialDeriv Complex.I fy
+  exact fxx + fyy
 
 
 def Harmonic (f : ℂ → ℂ) : Prop :=
@@ -25,7 +60,7 @@ lemma derivSymm (f : ℂ → ℂ) (hf : ContDiff ℝ 2 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)
+  let f' := fderiv ℝ f
   have h₀ : ∀ y, HasFDerivAt f (f' y) y := by
     have h : Differentiable ℝ f := by
       exact (contDiff_succ_iff_fderiv.1 hf).left
@@ -67,6 +102,11 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
   have f'_is_differentiable : Differentiable ℝ (fderiv ℝ f) :=
     (contDiff_succ_iff_fderiv.1 f'_is_real_C1).left
 
+  -- Partial derivative in direction 1
+  let f_1 := fun w ↦ (fderiv ℝ f w) 1
+
+  -- Partial derivative in direction I
+  let f_I := fun w ↦ (fderiv ℝ f w) Complex.I
 
   constructor
   · -- f is two times real continuously differentiable
@@ -76,38 +116,25 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
     intro z
     unfold Complex.laplace
 
-    simp
-
-    conv =>
-      left
-      right
-      arg 1
-      arg 2
-      intro z
-      rw [CauchyRiemann₁ (h z)]
+    rw [CauchyRiemann₄ h]
 
 
-    have t₀ : ∀ z, DifferentiableAt ℝ (fun w ↦ (fderiv ℝ f w) 1) z := by
-      intro z
-      let A := f'_is_differentiable
+
+    have t₁a : (fderiv ℝ (fun w ↦ Complex.I * (fderiv ℝ f w) 1) z) 
+      = Complex.I • (fderiv ℝ f_1 z) := by
+      rw [fderiv_const_mul]
       fun_prop
 
-    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]
+    rw [t₁a]
 
-    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 f_is_real_C2 z 1 Complex.I
-      let B := l₂ f_is_real_C2 z Complex.I  1
+    have t₂ : (fderiv ℝ f_1 z) Complex.I = (fderiv ℝ f_I z) 1 := by
+      let B := l₂ f_is_real_C2 z Complex.I 1
       rw [← B]
+      let A := derivSymm f f_is_real_C2 z 1 Complex.I
       rw [A]
       let C := l₂ f_is_real_C2 z 1 Complex.I
       rw [C]
+    simp
     rw [t₂]
 
     conv =>
@@ -117,9 +144,11 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
       arg 1
       arg 2
       intro z
+      simp [f_I]
       rw [CauchyRiemann₁ (h z)]
+    
+    rw [t₁a]
 
-    rw [t₁]
-
+    simp
     rw [← mul_assoc]
     simp