Working

Nevanlinna/complexHarmonic.examples.lean

@@ -30,19 +30,15 @@ theorem holomorphicAt_is_harmonicAt
   (hf : HolomorphicAt f z) :
   HarmonicAt f z := by
 
-  obtain ⟨s, hs, hz, h'f⟩ := HolomorphicAt_iff.1 hf
+  let t := {x | HolomorphicAt f x}
+  have ht : IsOpen t := HolomorphicAt_isOpen f
+  have hz : z ∈ t := by exact hf
 
   constructor
   · -- ContDiffAt ℝ 2 f z
-    apply ContDiffOn.contDiffAt _ ((IsOpen.mem_nhds_iff hs).2 hz)
+    exact HolomorphicAt_contDiffAt hf
-    suffices h : ContDiffOn ℂ 2 f s from by
+
-      apply ContDiffOn.restrict_scalars ℝ h
-    apply DifferentiableOn.contDiffOn _ hs
-    intro w hw
-    apply DifferentiableAt.differentiableWithinAt
-    exact h'f w hw
   · -- Δ f =ᶠ[nhds z] 0
-    obtain ⟨t, ht, hz, h'f⟩ := HolomorphicAt_isOpen.1 hf
     apply Filter.eventuallyEq_iff_exists_mem.2
     use t
     constructor
@@ -50,15 +46,12 @@ theorem holomorphicAt_is_harmonicAt
     · intro w hw
       unfold Complex.laplace
       simp
-
+      rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ hw) Complex.I]
-      rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ (h'f w hw)) Complex.I]
       rw [partialDeriv_smul'₂]
       simp
 
-      have f_is_real_C2 : ContDiffOn ℝ 2 f t :=
+      rw [partialDeriv_commAt (HolomorphicAt_contDiffAt hw) Complex.I 1]
-        ContDiffOn.restrict_scalars ℝ (DifferentiableOn.contDiffOn h'f ht)
+      rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ hw) 1]
-      rw [partialDeriv_commOn ht f_is_real_C2 Complex.I 1 w hw]
-      rw [partialDeriv_eventuallyEq ℝ (CauchyRiemann'₆ (h'f w hw)) 1]
       rw [partialDeriv_smul'₂]
       simp
 

Nevanlinna/holomorphic.lean

@@ -1,9 +1,24 @@
 import Mathlib.Analysis.Complex.Basic
-import Nevanlinna.partialDeriv
+import Mathlib.Analysis.Complex.TaylorSeries
+import Mathlib.Analysis.Calculus.LineDeriv.Basic
+import Mathlib.Analysis.Calculus.ContDiff.Defs
+import Mathlib.Analysis.Calculus.FDeriv.Basic
+import Mathlib.Analysis.Calculus.FDeriv.Symmetric
+import Mathlib.Analysis.RCLike.Basic
+import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
+import Mathlib.Data.Complex.Module
+import Mathlib.Data.Complex.Order
+import Mathlib.Data.Complex.Exponential
+import Mathlib.Data.Fin.Tuple.Basic
+import Mathlib.Topology.Algebra.InfiniteSum.Module
+import Mathlib.Topology.Defs.Filter
+import Mathlib.Topology.Instances.RealVectorSpace
+import Nevanlinna.cauchyRiemann
+import Nevanlinna.laplace
+import Nevanlinna.complexHarmonic
 
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
-variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
+variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F]
-
 
 def HolomorphicAt (f : E → F) (x : E) : Prop :=
   ∃ s ∈ nhds x, ∀ z ∈ s, DifferentiableAt ℂ f z
@@ -32,8 +47,17 @@ theorem HolomorphicAt_iff
     · assumption
 
 
-theorem HolomorphicAt_isOpen'
+theorem HolomorphicAt_differentiableAt
-  {f : E → F} :
+  {f : E → F}
+  {x : E} :
+  HolomorphicAt f x →  DifferentiableAt ℂ f x := by
+  intro hf
+  obtain ⟨s, _, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hf
+  exact h₃s x h₂s
+
+
+theorem HolomorphicAt_isOpen
+  (f : E → F) :
   IsOpen { x : E | HolomorphicAt f x } := by
   
   rw [← subset_interior_iff_isOpen]
@@ -53,37 +77,26 @@ theorem HolomorphicAt_isOpen'
       · exact h₃s
   · exact h₂s
 
-/-
+
-theorem HolomorphicAt_isOpen
+theorem HolomorphicAt_contDiffAt
-  {f : E → F}
+  {f : ℂ → F}
-  {x : E} :
+  {z : ℂ}
-  HolomorphicAt f x ↔ ∃ s : Set E, IsOpen s ∧ x ∈ s ∧ (∀ z ∈ s, HolomorphicAt f z) := by
+  (hf : HolomorphicAt f z) :
-  constructor
+  ContDiffAt ℝ 2 f z := by
-  · intro hf
+
-    obtain ⟨t, h₁t, h₂t⟩ := hf
+  let t := {x | HolomorphicAt f x}
-    obtain ⟨s, h₁s, h₂s, h₃s⟩ := mem_nhds_iff.1 h₁t
+  have ht : IsOpen t := HolomorphicAt_isOpen f
-    use s
+  have hz : z ∈ t := by exact hf
-    constructor
+
-    · assumption
+  -- ContDiffAt ℝ 2 f z
-    · constructor
+  apply ContDiffOn.contDiffAt _ ((IsOpen.mem_nhds_iff ht).2 hz)
-      · assumption
+  suffices h : ContDiffOn ℂ 2 f t from by
-      · intro z hz
+    apply ContDiffOn.restrict_scalars ℝ h
-        apply HolomorphicAt_iff.2
+  apply DifferentiableOn.contDiffOn _ ht
-        use s
+  intro w hw
-        constructor
+  apply DifferentiableAt.differentiableWithinAt
-        · assumption
+  exact HolomorphicAt_differentiableAt hw
-        · constructor
+
-          · assumption
-          · exact fun w hw ↦ h₂t w (h₁s hw)
-  · intro hyp
-    obtain ⟨s, h₁s, h₂s, hf⟩ := hyp
-    use s
-    constructor
-    · apply (IsOpen.mem_nhds_iff h₁s).2 h₂s
-    · intro z hz
-      obtain ⟨t, ht⟩ := (hf z hz)
-      exact ht.2 z (mem_of_mem_nhds ht.1)
--/
 
 theorem CauchyRiemann'₅
   {f : ℂ → F}