Update complexHarmonic.lean

Nevanlinna/complexHarmonic.lean

@@ -126,6 +126,22 @@ theorem logabs_of_holomorphic_is_harmonic
   have f_is_real_C2 : ContDiff ℝ 2 f :=
     ContDiff.restrict_scalars ℝ (Differentiable.contDiff h₁)
 
+  -- Complex.log ∘ f is real C²
+  have t₀ : Differentiable ℂ (Complex.log ∘ f) := by
+    intro z
+    apply DifferentiableAt.comp
+    exact Complex.differentiableAt_log (h₃ z)
+    exact h₁ z
+
+  have t₂ : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = ⇑(starRingEnd ℂ) ∘ Complex.log ∘ f := by
+    funext z
+    unfold Function.comp
+    rw [Complex.log_conj]
+    exact Complex.slitPlane_arg_ne_pi (h₃ z)
+
+  have t₃ : ⇑(starRingEnd ℂ) ∘ Complex.log ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by
+    rfl
+
   -- The norm square is z * z.conj
   have normSq_conj : ∀ (z : ℂ), (starRingEnd ℂ) z * z = ↑‖z‖ ^ 2 := Complex.conj_mul'
 
@@ -145,6 +161,16 @@ theorem logabs_of_holomorphic_is_harmonic
     apply ContinuousLinearMap.contDiff Complex.imCLM
     apply ContinuousLinearMap.contDiff Complex.imCLM
 
+  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
+
   constructor
   · -- logabs f is real C²
     have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by
@@ -191,7 +217,8 @@ theorem logabs_of_holomorphic_is_harmonic
       intro z
       rw [laplace_compContLin]
       simp
-      sorry
+      -- ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f)
+      exact t₄
     conv =>
       intro z
       rw [this z]
@@ -231,32 +258,31 @@ theorem logabs_of_holomorphic_is_harmonic
     rw [this]
     rw [laplace_add]
 
-    have : Differentiable ℂ (Complex.log ∘ f) := by
-      intro z
-      apply DifferentiableAt.comp
-      exact Complex.differentiableAt_log (h₃ z)
-      exact h₁ z
-
     have t₁: Complex.laplace (Complex.log ∘ f) = 0 := by
-      let A := holomorphic_is_harmonic this
+      let A := holomorphic_is_harmonic t₀
       funext z
       exact A.2 z
     rw [t₁]
     simp
 
-    have : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = ⇑(starRingEnd ℂ) ∘ Complex.log ∘ f := by
-      funext z
-      unfold Function.comp
-      rw [Complex.log_conj]
-      exact Complex.slitPlane_arg_ne_pi (h₃ z)
-    rw [this]
+    rw [t₂]
 
-    have : ⇑(starRingEnd ℂ) ∘ Complex.log ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by
-      rfl
-    rw [this]
+    rw [t₃]
     rw [laplace_compCLE]
     rw [t₁]
     simp
-    sorry
-    sorry
-    sorry
+
+    -- ContDiff ℝ 2 (Complex.log ∘ f)
+    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff t₀)
+
+    -- ContDiff ℝ 2 (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f)
+    rw [t₂, t₃]
+    apply ContDiff.comp
+    exact ContinuousLinearEquiv.contDiff Complex.conjCLE
+    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff t₀)
+
+    -- ContDiff ℝ 2 (Complex.log ∘ f)
+    exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff t₀)
+
+    -- ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f)
+    exact t₄