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Nevanlinna/complexHarmonic.examples.lean

@@ -138,10 +138,27 @@ theorem log_normSq_of_holomorphicAt_is_harmonicAt
         apply hx.2
 
     rw [HarmonicAt_eventuallyEq this]
+    apply harmonicAt_add_harmonicAt_is_harmonicAt
+    ·
+      sorry
+    · apply holomorphicAt_is_harmonicAt
+      apply HolomorphicAt_comp
+      use Complex.slitPlane
+      constructor
+      · apply IsOpen.mem_nhds
+        exact Complex.isOpen_slitPlane
+        assumption
+      · exact fun z a => Complex.differentiableAt_log a
 
 
 
 
+
+      sorry
+      assumption
+
+
+
     sorry
 
 

Nevanlinna/complexHarmonic.lean

@@ -1,20 +1,4 @@
-import Mathlib.Analysis.Complex.Basic
-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.Instances.RealVectorSpace
-import Nevanlinna.cauchyRiemann
 import Nevanlinna.laplace
-import Nevanlinna.partialDeriv
 
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
 variable {F₁ : Type*} [NormedAddCommGroup F₁] [NormedSpace ℂ F₁] [CompleteSpace F₁]
@@ -162,11 +146,10 @@ theorem harmonicAt_add_harmonicAt_is_harmonicAt
   HarmonicAt (f₁ + f₂) x := by
   constructor
   · exact ContDiffAt.add h₁.1 h₂.1
-  · rw [laplace_add_ContDiffAt]
+  · apply Filter.EventuallyEq.trans (laplace_add_ContDiffAt' h₁.1 h₂.1)
-    intro z hz
+    apply Filter.EventuallyEq.trans (Filter.EventuallyEq.add h₁.2 h₂.2)
-    rw [laplace_add_ContDiffOn hs h₁.1 h₂.1 z hz]
-    rw [h₁.2 z hz, h₂.2 z hz]
     simp
+    rfl
 
 
 theorem harmonic_smul_const_is_harmonic {f : ℂ → F} {c : ℝ} (h : Harmonic f) :

Nevanlinna/holomorphic.lean

@@ -19,6 +19,7 @@ import Nevanlinna.complexHarmonic
 
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F]
+variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℂ G] [CompleteSpace G]
 
 def HolomorphicAt (f : E → F) (x : E) : Prop :=
   ∃ s ∈ nhds x, ∀ z ∈ s, DifferentiableAt ℂ f z
@@ -78,6 +79,31 @@ theorem HolomorphicAt_isOpen
   · exact h₂s
 
 
+theorem HolomorphicAt_comp
+  {g : E → F}
+  {f : F → G}
+  {z : E}
+  (hf : HolomorphicAt f (g z))
+  (hg : HolomorphicAt g z) :
+  HolomorphicAt (f ∘ g) z := by
+  obtain ⟨UE, h₁UE, h₂UE⟩ := hg
+  obtain ⟨UF, h₁UF, h₂UF⟩ := hf
+  use UE ∩ g⁻¹' UF
+  constructor
+  · simp
+    constructor
+    · assumption
+    · apply ContinuousAt.preimage_mem_nhds
+      apply (h₂UE z (mem_of_mem_nhds h₁UE)).continuousAt
+      assumption
+  · intro x hx
+    apply DifferentiableAt.comp
+    apply h₂UF
+    exact hx.2
+    apply h₂UE
+    exact hx.1
+
+
 theorem HolomorphicAt_contDiffAt
   {f : ℂ → F}
   {z : ℂ}

Nevanlinna/laplace.lean

@@ -1,19 +1,4 @@
-import Mathlib.Data.Fin.Tuple.Basic
 import Mathlib.Analysis.Complex.Basic
-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.Data.Complex.Module
-import Mathlib.Data.Complex.Order
-import Mathlib.Data.Complex.Exponential
-import Mathlib.Analysis.RCLike.Basic
-import Mathlib.Order.Filter.Basic
-import Mathlib.Topology.Algebra.InfiniteSum.Module
-import Mathlib.Topology.Basic
-import Mathlib.Topology.Instances.RealVectorSpace
-import Nevanlinna.cauchyRiemann
 import Nevanlinna.partialDeriv
 
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
@@ -169,6 +154,7 @@ theorem laplace_add_ContDiffAt
     apply fun v ↦ (partialDeriv_contDiffAt ℝ h₁ v).differentiableAt le_rfl
     apply fun v ↦ (partialDeriv_contDiffAt ℝ h₂ v).differentiableAt le_rfl
 
+
 theorem laplace_add_ContDiffAt'
   {f₁ f₂ : ℂ → F}
   {x : ℂ}
@@ -177,49 +163,43 @@ theorem laplace_add_ContDiffAt'
   Δ (f₁ + f₂) =ᶠ[nhds x] (Δ f₁) + (Δ f₂):= by
 
   unfold Complex.laplace
-  have : partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) + partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₁) +
-    (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂) + partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₂)) =
-    (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) + (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂)) +
-    (partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₁) + partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₂))) := by
-    group
-  rw [this]
-  apply Filter.EventuallyEq.add
 
-  suffices hyp : partialDeriv ℝ 1 (partialDeriv ℝ 1 (f₁ + f₂)) =ᶠ[nhds x] partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) + partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂) from by
+  rw [add_assoc]
-    have : (fun x => partialDeriv ℝ 1 (partialDeriv ℝ 1 (f₁ + f₂)) x) = partialDeriv ℝ 1 (partialDeriv ℝ 1 (f₁ + f₂)) := by
+  nth_rw  5 [add_comm]
-      rfl
+  rw [add_assoc]
-    rw [this]
+  rw [← add_assoc]
-    have : (fun x => (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) + partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂)) x) = (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁)) + (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂)) := by
-      rfl
-    rwa [this]
 
-  simp
+  have dualPDeriv : ∀ v : ℂ, partialDeriv ℝ v (partialDeriv ℝ v (f₁ + f₂)) =ᶠ[nhds x]
+    partialDeriv ℝ v (partialDeriv ℝ v f₁) + partialDeriv ℝ v (partialDeriv ℝ v f₂) := by
 
+    intro v
     have h₁₁ : ContDiffAt ℝ 1 f₁ x := h₁.of_le one_le_two
     have h₂₁ : ContDiffAt ℝ 1 f₂ x := h₂.of_le one_le_two
-  let A : partialDeriv ℝ 1 (f₁ + f₂)  =ᶠ[nhds x] (partialDeriv ℝ 1 f₁) + (partialDeriv ℝ 1 f₂) := by
-    exact partialDeriv_add₂_contDiffAt ℝ h₁₁ h₂₁
-  let B : partialDeriv ℝ 1 (partialDeriv ℝ 1 (f₁ + f₂)) =ᶠ[nhds x] (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁)) + (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₂)) := by
-    sorry
 
-  sorry
+    have t₁ : partialDeriv ℝ v (partialDeriv ℝ v (f₁ + f₂)) =ᶠ[nhds x] partialDeriv ℝ v (partialDeriv ℝ v f₁ + partialDeriv ℝ v f₂) := by
-  repeat
+      exact partialDeriv_eventuallyEq' ℝ (partialDeriv_add₂_contDiffAt ℝ h₁₁ h₂₁) v
-    rw [partialDeriv_eventuallyEq ℝ (partialDeriv_add₂_contDiffAt ℝ h₁₁ h₂₁)]
-    rw [partialDeriv_add₂_differentiableAt]
 
-  -- I am super confused at this point because the tactic 'ring' does not work.
+    have t₂ : partialDeriv ℝ v (partialDeriv ℝ v f₁ + partialDeriv ℝ v f₂) =ᶠ[nhds x] (partialDeriv ℝ v (partialDeriv ℝ v f₁)) + (partialDeriv ℝ v (partialDeriv ℝ v f₂)) := by
-  -- I do not understand why. So, I need to do things by hand.
+      apply partialDeriv_add₂_contDiffAt ℝ
-  rw [add_assoc]
+      apply partialDeriv_contDiffAt
-  rw [add_assoc]
+      exact h₁
-  rw [add_right_inj (partialDeriv ℝ 1 (partialDeriv ℝ 1 f₁) x)]
+      apply partialDeriv_contDiffAt
-  rw [add_comm]
+      exact h₂
-  rw [add_assoc]
+
-  rw [add_right_inj (partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₁) x)]
+    apply Filter.EventuallyEq.trans t₁ t₂
-  rw [add_comm]
+
+  have eventuallyEq_add
+    {a₁ a₂ b₁ b₂ : ℂ → F}
+    (h : a₁ =ᶠ[nhds x] b₁)
+    (h' : a₂ =ᶠ[nhds x] b₂) : (a₁ + a₂) =ᶠ[nhds x] (b₁ + b₂) := by
+    have {c₁ c₂ : ℂ → F} : (fun x => c₁ x + c₂ x) = c₁ + c₂ := by rfl
+    rw [← this, ← this]
+    exact Filter.EventuallyEq.add h h'
+  apply eventuallyEq_add
+  · exact dualPDeriv 1
+  · nth_rw 2 [add_comm]
+    exact dualPDeriv Complex.I
 
-  repeat
-    apply fun v ↦ (partialDeriv_contDiffAt ℝ h₁ v).differentiableAt le_rfl
-    apply fun v ↦ (partialDeriv_contDiffAt ℝ h₂ v).differentiableAt le_rfl
 
 theorem laplace_smul {f : ℂ → F} : ∀ v : ℝ, Δ (v • f) = v • (Δ f) := by
   intro v

Nevanlinna/partialDeriv.lean

@@ -28,6 +28,16 @@ theorem partialDeriv_eventuallyEq'
   simp
 
 
+theorem partialDeriv_eventuallyEq
+  {f₁ f₂ : E → F}
+  {x : E}
+  (h : f₁ =ᶠ[nhds x] f₂) :
+  ∀ v : E, partialDeriv 𝕜 v f₁ x = partialDeriv 𝕜 v f₂ x := by
+  unfold partialDeriv
+  rw [Filter.EventuallyEq.fderiv_eq h]
+  exact fun v => rfl
+
+
 theorem partialDeriv_smul₁ {f : E → F} {a : 𝕜} {v : E} : partialDeriv 𝕜 (a • v) f = a • partialDeriv 𝕜 v f := by
   unfold partialDeriv
   conv =>
@@ -177,7 +187,12 @@ theorem partialDeriv_contDiff {n : ℕ} {f : E → F} (h : ContDiff 𝕜 (n + 1)
   exact contDiff_const
 
 
-theorem partialDeriv_contDiffAt {n : ℕ} {f : E → F} {x : E} (h : ContDiffAt 𝕜 (n + 1) f x) : ∀ v : E, ContDiffAt 𝕜 n (partialDeriv 𝕜 v f) x := by
+theorem partialDeriv_contDiffAt
+  {n : ℕ}
+  {f : E → F}
+  {x : E}
+  (h : ContDiffAt 𝕜 (n + 1) f x) :
+  ∀ v : E, ContDiffAt 𝕜 n (partialDeriv 𝕜 v f) x := by
 
   unfold partialDeriv
   intro v
@@ -253,12 +268,6 @@ lemma partialDeriv_fderivAt
   exact differentiableAt_const a
 
 
-theorem partialDeriv_eventuallyEq {f₁ f₂ : E → F} {x : E} (h : f₁ =ᶠ[nhds x] f₂) : ∀ v : E, partialDeriv 𝕜 v f₁ x = partialDeriv 𝕜 v f₂ x := by
-  unfold partialDeriv
-  rw [Filter.EventuallyEq.fderiv_eq h]
-  exact fun v => rfl
-
-
 section restrictScalars
 
 theorem partialDeriv_smul'₂