Make it work!

Nevanlinna/complexHarmonic.lean

@@ -21,46 +21,37 @@ def Harmonic (f : ℂ → ℂ) : Prop :=
   (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)
 
 
-lemma zwoDiff (f : ℝ × ℝ → ℝ) (h : ContDiff ℝ 2 f) : ∀ z a b : ℝ × ℝ, 0 = 1 := by
-  intro z a b
-
-  let fx  := fun w ↦ (fderiv ℝ f w) a
-  let fxx := fun w ↦ (fderiv ℝ fx w) a
-  let f2  := (fderiv ℝ (fun w => fderiv ℝ f w) z) a a
-
-  have :  iteratedFDeriv ℝ 1 f z ![a] = 0 := by
-    rw [iteratedFDeriv_succ_apply_left]
-    simp
-    let g := iteratedFDeriv ℝ 0 f
-    simp at g
-    sorry
-
-  have : f2 = fxx z := by
-    dsimp [f2, fxx, fx]
-
-    sorry
-
-  sorry
-
-
-lemma derivSymm (f : ℂ → ℂ) (h : Differentiable ℝ f) :
+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)
   have h₀ : ∀ y, HasFDerivAt f (f' y) y := by
+    have h : Differentiable ℝ f := by
+      exact (contDiff_succ_iff_fderiv.1 hf).left
     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 := (contDiff_succ_iff_fderiv.1 hf).right
+    let B := (contDiff_succ_iff_fderiv.1 A).left
+    simp at B
+    exact B z
 
   let A := second_derivative_symmetric h₀ h₁ a b
   dsimp [f'', f'] at A
   apply A
 
 
+lemma l₂ {f : ℂ → ℂ} (hf : ContDiff ℝ 2 f) (z a b : ℂ) :
+    fderiv ℝ (fderiv ℝ f) z b a = fderiv ℝ (fun w ↦ fderiv ℝ f w a) z b := by
+  rw [fderiv_clm_apply]
+  · simp
+  · exact (contDiff_succ_iff_fderiv.1 hf).2.differentiable le_rfl z
+  · simp
+
+
 theorem holomorphic_is_harmonic (f : ℂ → ℂ) :
   Differentiable ℂ f → Harmonic f := by
 
@@ -84,10 +75,17 @@ theorem holomorphic_is_harmonic (f : ℂ → ℂ) :
       intro z
       rw [CauchyRiemann₁ (h z)]
 
+    have t₂₀ : ContDiff ℝ 2 f := by exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
+
+    have t₀₀ : Differentiable ℝ (fun w => (fderiv ℝ f w)) := by
+      let A := (contDiff_succ_iff_fderiv.1 t₂₀).right
+      let B := (contDiff_succ_iff_fderiv.1 A).left
+      exact B
+
     have t₀ : ∀ z, DifferentiableAt ℝ (fun w => (fderiv ℝ f w) 1) z := by
       intro z
-
-      sorry
+      let A := t₀₀
+      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
@@ -97,12 +95,14 @@ theorem holomorphic_is_harmonic (f : ℂ → ℂ) :
       exact t₀ z
     rw [t₁]
 
-    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
+      let B := l₂ t₂₀ z Complex.I  1
+      rw [← B]
+      rw [A]
+      let C := l₂ t₂₀ z 1 Complex.I
+      rw [C]
     rw [t₂]
 
     conv =>