working

Nevanlinna/complexHarmonic.lean

@@ -35,14 +35,43 @@ theorem HarmonicOn_of_locally_HarmonicOn {f : ℂ → F} {s : Set ℂ} (h : ∀
   constructor
   · apply contDiffOn_of_locally_contDiffOn
     intro x xHyp
-    obtain ⟨u, uHyp⟩  := h x xHyp
+    obtain ⟨u, uHyp⟩ := h x xHyp
     use u
     exact ⟨ uHyp.1, ⟨uHyp.2.1, uHyp.2.2.1⟩⟩
   · intro x xHyp
-    obtain ⟨u, uHyp⟩  := h x xHyp
+    obtain ⟨u, uHyp⟩ := h x xHyp
     exact (uHyp.2.2.2) x ⟨xHyp, uHyp.2.1⟩
 
 
+theorem HarmonicOn_congr {f₁ f₂ : ℂ → F} {s : Set ℂ} (hs : IsOpen s) (hf₁₂ : ∀ x ∈ s, f₁ x = f₂ x) :
+  HarmonicOn f₁ s ↔ HarmonicOn f₂ s := by
+  constructor
+  · intro h₁
+    constructor
+    · apply ContDiffOn.congr h₁.1
+      intro x hx
+      rw [eq_comm]
+      exact hf₁₂ x hx
+    · intro z hz
+      have : f₁ =ᶠ[nhds z] f₂ := by 
+        unfold Filter.EventuallyEq
+        unfold Filter.Eventually
+        simp
+        apply?
+        sorry
+      rw [← laplace_eventuallyEq this]
+      exact h₁.2 z hz
+  · intro h₁
+    constructor
+    · apply ContDiffOn.congr h₁.1
+      intro x hx
+      exact hf₁₂ x hx
+    · intro z hz
+      have : f₁ =ᶠ[nhds z] f₂ := by sorry
+      rw [laplace_eventuallyEq this]
+      exact h₁.2 z hz
+
+
 theorem harmonic_add_harmonic_is_harmonic {f₁ f₂ : ℂ → F} (h₁ : Harmonic f₁) (h₂ : Harmonic f₂) :
   Harmonic (f₁ + f₂) := by
   constructor
@@ -239,8 +268,8 @@ theorem log_normSq_of_holomorphicOn_is_harmonicOn
     · exact Real.pi_nonneg
     exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z hz)
     exact h₂ z hz
-  
+  rw [HarmonicOn_ext this]
-  rw [this]
+  simp
 
   apply harmonic_add_harmonic_is_harmonic
   have : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by

Nevanlinna/laplace.lean

@@ -24,6 +24,13 @@ noncomputable def Complex.laplace : (ℂ → F) → (ℂ → F) :=
   fun f ↦ partialDeriv ℝ 1 (partialDeriv ℝ 1 f) + partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f)
 
 
+theorem laplace_eventuallyEq {f₁ f₂ : ℂ → F} {x : ℂ} (h : f₁ =ᶠ[nhds x] f₂) : Complex.laplace f₁ x = Complex.laplace f₂ x := by
+  unfold Complex.laplace
+  simp
+  rw [partialDeriv_eventuallyEq ℝ (partialDeriv_eventuallyEq' ℝ h 1) 1]
+  rw [partialDeriv_eventuallyEq ℝ (partialDeriv_eventuallyEq' ℝ h Complex.I) Complex.I]
+
+
 theorem laplace_add {f₁ f₂ : ℂ → F} (h₁ : ContDiff ℝ 2 f₁)  (h₂ : ContDiff ℝ 2 f₂): Complex.laplace (f₁ + f₂) = (Complex.laplace f₁) + (Complex.laplace f₂) := by
   unfold Complex.laplace
   rw [partialDeriv_add₂]

Nevanlinna/partialDeriv.lean

@@ -128,6 +128,15 @@ theorem partialDeriv_eventuallyEq {f₁ f₂ : E → F} {x : E} (h : f₁ =ᶠ[n
   exact fun v => rfl
 
 
+theorem partialDeriv_eventuallyEq' {f₁ f₂ : E → F} {x : E} (h : f₁ =ᶠ[nhds x] f₂) : ∀ v : E, partialDeriv 𝕜 v f₁ =ᶠ[nhds x] partialDeriv 𝕜 v f₂ := by
+  unfold partialDeriv
+  intro v
+  let A : fderiv 𝕜 f₁ =ᶠ[nhds x] fderiv 𝕜 f₂ := Filter.EventuallyEq.fderiv h
+  apply Filter.EventuallyEq.comp₂ 
+  exact A
+  simp
+
+
 section restrictScalars
 
 variable (𝕜 : Type*) [NontriviallyNormedField 𝕜]