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Nevanlinna/complexHarmonic.lean

@@ -4,12 +4,12 @@ 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.Analysis.RCLike.Basic
 import Mathlib.Topology.Algebra.InfiniteSum.Module
 import Mathlib.Topology.Instances.RealVectorSpace
 import Nevanlinna.cauchyRiemann
@@ -254,13 +254,6 @@ theorem holomorphicOn_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} (hs : IsOpe
   have f_is_real_C2 : ContDiffOn ℝ 2 f s :=
     ContDiffOn.restrict_scalars ℝ (DifferentiableOn.contDiffOn h hs)
 
-  have fI_is_real_differentiable : DifferentiableOn ℝ (partialDeriv ℝ 1 f) s := by
-    intro z hz
-    apply DifferentiableAt.differentiableWithinAt
-    let ZZ := (f_is_real_C2 z hz).contDiffAt (IsOpen.mem_nhds hs hz)
-    let AA := partialDeriv_contDiffAt ℝ ZZ 1
-    exact AA.differentiableAt (by rfl)
-
   constructor
   · -- f is two times real continuously differentiable
     exact f_is_real_C2
@@ -269,8 +262,6 @@ theorem holomorphicOn_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} (hs : IsOpe
     unfold Complex.laplace
     intro z hz
     simp
-    have : ∀ z ∈ s, partialDeriv ℝ Complex.I f z = Complex.I • partialDeriv ℝ 1 f z := by
-      sorry
     have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
       unfold Filter.EventuallyEq
       unfold Filter.Eventually
@@ -289,17 +280,33 @@ theorem holomorphicOn_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} (hs : IsOpe
     rw [partialDeriv_eventuallyEq ℝ this Complex.I]
     rw [partialDeriv_smul'₂]
 
-    rw [partialDeriv_comm f_is_real_C2 Complex.I 1]
+    simp
-    rw [CauchyRiemann₄ h]
+    rw [partialDeriv_commOn hs f_is_real_C2 Complex.I 1 z hz]
+
+    have : Complex.I • partialDeriv ℝ 1 (partialDeriv ℝ Complex.I f) z = Complex.I • (partialDeriv ℝ 1 (partialDeriv ℝ Complex.I f) z) := by
+      rfl
     rw [this]
+    have : partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by
+      unfold Filter.EventuallyEq
+      unfold Filter.Eventually
+      simp
+      refine mem_nhds_iff.mpr ?_
+      use s
+      constructor
+      · intro x hx
+        simp
+        apply CauchyRiemann₅
+        apply DifferentiableOn.differentiableAt h
+        exact IsOpen.mem_nhds hs hx
+      · constructor
+        · exact hs
+        · exact hz
+    rw [partialDeriv_eventuallyEq ℝ this 1]
+    rw [partialDeriv_smul'₂]
+    simp
     rw [← smul_assoc]
     simp
 
-    -- Subgoals coming from the application of 'this'
-    -- Differentiable ℝ (Real.partialDeriv 1 f)
-    exact fI_is_real_differentiable
-    -- Differentiable ℝ (Real.partialDeriv 1 f)
-    exact fI_is_real_differentiable
 
 
 theorem re_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
@@ -367,11 +374,13 @@ theorem log_normSq_of_holomorphicOn_is_harmonicOn
   rw [HarmonicOn_congr hs this]
   simp
 
-  apply harmonicOn_add_harmonicOn_is_harmonicOn
+  apply harmonicOn_add_harmonicOn_is_harmonicOn hs
-  exact hs
+
   have : (fun x => Complex.log ((starRingEnd ℂ) (f x))) = (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) := by
     rfl
   rw [this]
+
+  -- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) s
   have : ∀ z ∈ s, (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) z = (Complex.conjCLE ∘ Complex.log ∘ f) z := by
     intro z hz
     unfold Function.comp
@@ -383,9 +392,24 @@ theorem log_normSq_of_holomorphicOn_is_harmonicOn
   rw [← harmonicOn_iff_comp_CLE_is_harmonicOn]
 
   apply holomorphicOn_is_harmonicOn
-  intro z
+  exact hs
+
+  intro z hz
+  apply DifferentiableAt.differentiableWithinAt
   apply DifferentiableAt.comp
-  exact Complex.differentiableAt_log (h₃ z)
+
+
+
+  exact Complex.differentiableAt_log (h₃ z hz)
+  apply DifferentiableOn.differentiableAt h₁ -- (h₁ z hz)
+  exact IsOpen.mem_nhds hs hz
+  exact hs
+
+  -- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) s
+  apply holomorphicOn_is_harmonicOn hs
+  apply?
+
+
   exact h₁ z
 
 

Nevanlinna/partialDeriv.lean

@@ -1,6 +1,8 @@
 import Mathlib.Analysis.Calculus.FDeriv.Basic
 import Mathlib.Analysis.Calculus.FDeriv.Symmetric
 import Mathlib.Analysis.Calculus.ContDiff.Basic
+import Mathlib.Topology.Basic
+import Mathlib.Topology.Defs.Filter
 
 
 variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
@@ -168,6 +170,23 @@ lemma partialDeriv_fderiv {f : E → F} (hf : ContDiff 𝕜 2 f) (z a b : E) :
   · simp
 
 
+lemma partialDeriv_fderivOn
+  {s : Set E}
+  {f : E → F}
+  (hs : IsOpen s)
+  (hf : ContDiffOn 𝕜 2 f s) (a b : E) :
+  ∀ z ∈ s, fderiv 𝕜 (fderiv 𝕜 f) z b a = partialDeriv 𝕜 b (partialDeriv 𝕜 a f) z := by
+
+  intro z hz
+  unfold partialDeriv
+  rw [fderiv_clm_apply]
+  · simp
+  · convert DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
+    apply ContDiffOn.differentiableOn _ (Submonoid.oneLE.proof_2 ℕ∞)
+    exact ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 hf).2
+  · 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]
@@ -268,3 +287,42 @@ theorem partialDeriv_comm
   rw [← partialDeriv_fderiv ℝ h z v₂ v₁]
   rw [derivSymm]
   rw [partialDeriv_fderiv ℝ h z v₁ v₂]
+
+
+theorem partialDeriv_commOn
+  {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+  {s : Set E}
+  {f : E → F}
+  (hs : IsOpen s)
+  (h : ContDiffOn ℝ 2 f s) :
+  ∀ v₁ v₂ : E, ∀ z ∈ s, partialDeriv ℝ v₁ (partialDeriv ℝ v₂ f) z = partialDeriv ℝ v₂ (partialDeriv ℝ v₁ f) z := by
+
+  intro v₁ v₂ z hz
+
+  have derivSymm :
+    (fderiv ℝ (fun w => fderiv ℝ f w) z) v₁ v₂ = (fderiv ℝ (fun w => fderiv ℝ f w) z) v₂ v₁ := by
+
+    let f' := fderiv ℝ f
+    have h₀ : ∀ y ∈ s, HasFDerivAt f (f' y) y := by
+      intro y hy
+      apply DifferentiableAt.hasFDerivAt
+      apply DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hy)
+      apply h.differentiableOn one_le_two
+
+    let f'' := (fderiv ℝ f' z)
+    have h₁ : HasFDerivAt f' f'' z := by
+      apply DifferentiableAt.hasFDerivAt
+      apply DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
+      apply ContDiffOn.differentiableOn _ (Submonoid.oneLE.proof_2 ℕ∞)
+      exact ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 h).2
+
+    have h₀' : ∀ᶠ (y : E) in nhds z, HasFDerivAt f (f' y) y := by
+      apply eventually_nhds_iff.mpr
+      use s
+
+    exact second_derivative_symmetric_of_eventually h₀' h₁ v₁ v₂
+
+  rw [← partialDeriv_fderivOn ℝ hs h v₂ v₁ z hz]
+  rw [derivSymm]
+  rw [← partialDeriv_fderivOn ℝ hs h v₁ v₂ z hz]