Update complexHarmonic.lean

Nevanlinna/complexHarmonic.lean

@@ -26,6 +26,17 @@ def Harmonic (f : ℂ → F) : Prop :=
   (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)
 
 
+theorem harmonic_add_harmonic_is_harmonic {f₁ f₂ : ℂ → F₁} (h₁ : Harmonic f₁) (h₂ : Harmonic f₂) :
+  Harmonic (f₁ + f₂) := by
+  constructor
+  · exact ContDiff.add h₁.1 h₂.1
+  · rw [laplace_add h₁.1 h₂.1]
+    simp
+    intro z
+    rw [h₁.2 z, h₂.2 z]
+    simp
+
+
 theorem harmonic_comp_CLM_is_harmonic {f : ℂ → F₁} {l : F₁ →L[ℝ] G} (h : Harmonic f) :
   Harmonic (l ∘ f) := by
 
@@ -138,96 +149,18 @@ theorem log_normSq_of_holomorphic_is_harmonic
   (h₃ : ∀ z, f z ∈ Complex.slitPlane) :
   Harmonic (Real.log ∘ Complex.normSq ∘ f) := by
 
+  -- Suffices to show Harmonic (⇑Complex.ofRealCLM ∘ Real.log ∘ ⇑Complex.normSq ∘ f)
+  let F := Real.log ∘ Complex.normSq ∘ f
+  have : Harmonic (Complex.ofRealCLM ∘ F) → Harmonic F := by
+    intro hyp
+    have t₁ : Harmonic (Complex.reCLM ∘ Complex.ofRealCLM ∘ F) := harmonic_comp_CLM_is_harmonic hyp
+    have t₂ : Complex.reCLM ∘ Complex.ofRealCLM ∘ F = F := rfl
+    rw [t₂] at t₁
+    exact t₁
+  apply this
+  dsimp [F]
 
-
+  -- Suffices to show Harmonic (Complex.log ∘ Complex.ofRealCLM ∘ Complex.normSq ∘ f)
-  /- We start with a number of lemmas on regularity of all the functions involved -/
-
-  -- The norm square is real C²
-  have normSq_is_real_C2 : ContDiff ℝ 2 Complex.normSq := by
-    unfold Complex.normSq
-    simp
-    conv =>
-      arg 3
-      intro x
-      rw [← Complex.reCLM_apply, ← Complex.imCLM_apply]
-    apply ContDiff.add
-    apply ContDiff.mul
-    apply ContinuousLinearMap.contDiff Complex.reCLM
-    apply ContinuousLinearMap.contDiff Complex.reCLM
-    apply ContDiff.mul
-    apply ContinuousLinearMap.contDiff Complex.imCLM
-    apply ContinuousLinearMap.contDiff Complex.imCLM
-
-  -- f is real C²
-  have f_is_real_C2 : ContDiff ℝ 2 f :=
-    ContDiff.restrict_scalars ℝ (Differentiable.contDiff h₁)
-
-  -- Complex.log ∘ f is real C²
-  have log_f_is_holomorphic : Differentiable ℂ (Complex.log ∘ f) := by
-    intro z
-    apply DifferentiableAt.comp
-    exact Complex.differentiableAt_log (h₃ z)
-    exact h₁ z
-
-  -- Real.log |f|² is real C²
-  have t₄ : ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f) := by
-    rw [contDiff_iff_contDiffAt]
-    intro z
-    apply ContDiffAt.comp
-    apply Real.contDiffAt_log.mpr
-    simp
-    exact h₂ z
-    apply ContDiff.comp_contDiffAt z normSq_is_real_C2
-    exact ContDiff.contDiffAt f_is_real_C2
-
-  have t₂ : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by
-    funext z
-    unfold Function.comp
-    rw [Complex.log_conj]
-    rfl
-    exact Complex.slitPlane_arg_ne_pi (h₃ z)
-
-  constructor
-  · -- logabs f is real C²
-    have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by
-      funext z
-      simp
-      unfold Complex.abs
-      simp
-      rw [Real.log_sqrt]
-      rw [div_eq_inv_mul (Real.log (Complex.normSq (f z))) 2]
-      exact Complex.normSq_nonneg (f z)
-    rw [this]
-
-    have : (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) = (fun z ↦ (2 : ℝ)⁻¹ • ((Real.log ∘ ⇑Complex.normSq ∘ f) z)) := by
-      exact rfl
-    rw [this]
-    apply ContDiff.const_smul
-    exact t₄
-
-  · -- Laplace vanishes
-    have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by
-      funext z
-      simp
-      unfold Complex.abs
-      simp
-      rw [Real.log_sqrt]
-      rw [div_eq_inv_mul (Real.log (Complex.normSq (f z))) 2]
-      exact Complex.normSq_nonneg (f z)
-    rw [this]
-    rw [laplace_smul]
-    simp
-
-    have : ∀ (z : ℂ), Complex.laplace (Real.log ∘ ⇑Complex.normSq ∘ f) z = 0 ↔ Complex.laplace (Complex.ofRealCLM ∘ Real.log ∘ ⇑Complex.normSq ∘ f) z = 0 := by
-      intro z
-      rw [laplace_compContLin]
-      simp
-      -- ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f)
-      exact t₄
-    conv =>
-      intro z
-      rw [this z]
-
   have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ Complex.ofRealCLM ∘ Complex.normSq ∘ f := by
     unfold Function.comp
     funext z
@@ -235,12 +168,16 @@ theorem log_normSq_of_holomorphic_is_harmonic
     exact Complex.normSq_nonneg (f z)
   rw [this]
 
+  -- Suffices to show Harmonic (Complex.log ∘ (⇑(starRingEnd ℂ) ∘ f * f))
   have : Complex.ofRealCLM ∘ ⇑Complex.normSq ∘ f = ((starRingEnd ℂ) ∘ f) * f := by
     funext z
     simp
     exact Complex.normSq_eq_conj_mul_self
   rw [this]
 
+
+  -- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f)
+  -- THIS IS WHERE WE USE h₃
   have : Complex.log ∘ (⇑(starRingEnd ℂ) ∘ f * f) = Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f := by
     unfold Function.comp
     funext z
@@ -261,28 +198,23 @@ theorem log_normSq_of_holomorphic_is_harmonic
     exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z)
     exact h₂ z
   rw [this]
-    rw [laplace_add]
 
-    rw [t₂, laplace_compCLE]
+  apply harmonic_add_harmonic_is_harmonic
+  have : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by
+    funext z
+    unfold Function.comp
+    rw [Complex.log_conj]
+    rfl
+    exact Complex.slitPlane_arg_ne_pi (h₃ z)
+  rw [this]
+  rw [← harmonic_iff_comp_CLE_is_harmonic]
+
+  repeat
+    apply holomorphic_is_harmonic
     intro z
-    simp
+    apply DifferentiableAt.comp
-    rw [(holomorphic_is_harmonic log_f_is_holomorphic).2 z]
+    exact Complex.differentiableAt_log (h₃ z)
-    simp
+    exact h₁ z
-
-    -- ContDiff ℝ 2 (Complex.log ∘ f)
-    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff log_f_is_holomorphic)
-
-    -- ContDiff ℝ 2 (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f)
-    rw [t₂]
-    apply ContDiff.comp
-    exact ContinuousLinearEquiv.contDiff Complex.conjCLE
-    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff log_f_is_holomorphic)
-
-    -- ContDiff ℝ 2 (Complex.log ∘ f)
-    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff log_f_is_holomorphic)
-
-    -- ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f)
-    exact t₄
 
 
 theorem logabs_of_holomorphic_is_harmonic