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Nevanlinna.lean

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-def hello := "world"
+import Nevanlinna.cauchyRiemann

Nevanlinna/complexHarmonic.lean

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+import Mathlib.Analysis.Complex.Basic
+import Mathlib.Analysis.Calculus.LineDeriv.Basic
+import Mathlib.Analysis.Calculus.ContDiff.Defs
+import Mathlib.Analysis.Calculus.FDeriv.Basic
+import Nevanlinna.cauchyRiemann
+
+noncomputable def Complex.laplace : (ℂ → ℝ) → (ℂ → ℝ) := 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
+
+
+def Harmonic (f : ℂ → ℝ) : 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
+
+  intro h
+
+  constructor
+  · -- f is two times real continuously differentiable
+    sorry
+
+  · -- Laplace of f is zero
+    intro z
+    unfold Complex.laplace
+    simp
+    let ZZ := (CauchyRiemann₃ (h z)).left
+    rw [ZZ]
+    
+    sorry
+

nevanlinna/complexHarmonic.lean

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-import Mathlib.Analysis.Complex.Basic
-import Mathlib.Analysis.Calculus.LineDeriv.Basic
-import Mathlib.Analysis.Calculus.ContDiff.Defs
-import Mathlib.Analysis.Calculus.FDeriv.Basic
-
-noncomputable def Real.laplace : (ℝ × ℝ → ℝ) → (ℝ × ℝ → ℝ) := by
-  intro f
-  let f₁ := fun x ↦ lineDeriv ℝ f x ⟨1,0⟩
-  let f₁₁ := fun x ↦ lineDeriv ℝ f₁ x ⟨1,0⟩
-  let f₂ := fun x ↦ lineDeriv ℝ f x ⟨0,1⟩
-  let f₂₂ := fun x ↦ lineDeriv ℝ f₂ x ⟨0,1⟩
-  exact f₁₁ + f₂₂
-
-noncomputable def Complex.laplace : (ℂ → ℝ) → (ℂ → ℝ) := 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
-
-
-theorem xx : ∀ f : ℂ → , f = 0  := by
-  intro f
-
-  let f₁ := fun x ↦ lineDeriv ℝ f x 1
-  let f₂ := fun x ↦ lineDeriv ℝ f x Complex.I
-
-  have : ∀ z, fderiv ℂ f z = 0 := by
-    sorry
-
-
-  have : (fun x ↦ lineDeriv ℝ f x 1) = (fun x ↦ lineDeriv ℝ f x Complex.I) := by
-
-    unfold lineDeriv
-
-    sorry
-
-  sorry