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Nevanlinna/complexHarmonic.lean

@@ -37,7 +37,7 @@ theorem HarmonicOn_of_locally_HarmonicOn {f : ℂ → F} {s : Set ℂ} (h : ∀
     intro x xHyp
     obtain ⟨u, uHyp⟩  := h x xHyp
     use u
-    exact ⟨ uHyp.1, ⟨uHyp.2.1, uHyp.2.2.1⟩⟩ 
+    exact ⟨ uHyp.1, ⟨uHyp.2.1, uHyp.2.2.1⟩⟩
   · intro x xHyp
     obtain ⟨u, uHyp⟩  := h x xHyp
     exact (uHyp.2.2.2) x ⟨xHyp, uHyp.2.1⟩
@@ -96,7 +96,9 @@ theorem harmonicOn_comp_CLM_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} {l :
   · -- Continuous differentiability
     apply ContDiffOn.continuousLinearMap_comp
     exact h.1
-  · rw [laplace_compContLin]
+  · -- Vanishing of Laplace
+
+    rw [laplace_compContLin]
     simp
     intro z zHyp
     rw [h.2 z]
@@ -104,6 +106,8 @@ theorem harmonicOn_comp_CLM_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} {l :
     assumption
 
 
+
+
 theorem harmonic_iff_comp_CLE_is_harmonic {f : ℂ → F₁} {l : F₁ ≃L[ℝ] G₁} :
   Harmonic f ↔ Harmonic (l ∘ f) := by
 

Nevanlinna/laplace.lean

@@ -9,6 +9,7 @@ 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.Instances.RealVectorSpace
 import Nevanlinna.cauchyRiemann
@@ -74,6 +75,46 @@ theorem laplace_compContLin {f : ℂ → F} {l : F →L[ℝ] G} (h : ContDiff
   exact h.differentiable one_le_two
 
 
+theorem laplace_compContLinAt {f : ℂ → F} {l : F →L[ℝ] G} {x : ℂ} (h : ContDiffAt ℝ 2 f x) :
+  Complex.laplace (l ∘ f) x = (l ∘ (Complex.laplace f)) x := by
+
+  have A₂ : ∃ v ∈ nhds x, (IsOpen v) ∧ (x ∈ v) ∧ (ContDiffOn ℝ 2 f v) := by
+    sorry
+  obtain ⟨v, hv₁, hv₂, hv₃, hv₄⟩ := A₂
+
+  have D : ∀ w : ℂ, partialDeriv ℝ w (l ∘ f) =ᶠ[nhds x] l ∘ partialDeriv ℝ w (f) := by
+    intro w
+    apply Filter.eventuallyEq_iff_exists_mem.2
+    use v
+    constructor
+    · exact IsOpen.mem_nhds hv₂ hv₃
+    · intro y hy
+      apply partialDeriv_compContLinAt
+      let V := ContDiffOn.differentiableOn hv₄ one_le_two
+      apply DifferentiableOn.differentiableAt V
+      apply IsOpen.mem_nhds
+      assumption
+      assumption
+
+  unfold Complex.laplace
+  simp
+
+  rw [partialDeriv_eventuallyEq ℝ (D 1) 1]
+  rw [partialDeriv_compContLinAt]
+  rw [partialDeriv_eventuallyEq ℝ (D Complex.I) Complex.I]
+  rw [partialDeriv_compContLinAt]
+  simp
+
+  -- DifferentiableAt ℝ (partialDeriv ℝ Complex.I f) x
+  unfold partialDeriv
+
+  sorry
+
+  -- DifferentiableAt ℝ (partialDeriv ℝ 1 f) x
+
+  sorry
+
+
 theorem laplace_compCLE {f : ℂ → F} {l : F ≃L[ℝ] G} (h : ContDiff ℝ 2 f) :
   Complex.laplace (l ∘ f) = l ∘ (Complex.laplace f) := by
   let l' := (l : F →L[ℝ] G)

Nevanlinna/partialDeriv.lean

@@ -66,6 +66,12 @@ theorem partialDeriv_compContLin {f : E → F} {l : F →L[𝕜] G} {v : E} (h :
   rfl
 
 
+theorem partialDeriv_compContLinAt {f : E → F} {l : F →L[𝕜] G} {v : E} {x : E} (h : DifferentiableAt 𝕜 f x) : (partialDeriv 𝕜 v (l ∘ f)) x = (l ∘ partialDeriv 𝕜 v f) x:= by
+  unfold partialDeriv
+  rw [fderiv.comp x (ContinuousLinearMap.differentiableAt l) h]
+  simp
+
+
 theorem partialDeriv_contDiff {n : ℕ} {f : E → F} (h : ContDiff 𝕜 (n + 1) f) : ∀ v : E, ContDiff 𝕜 n (partialDeriv 𝕜 v f) := by
   unfold partialDeriv
   intro v
@@ -84,6 +90,33 @@ 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
+  unfold partialDeriv
+  intro v
+
+  let A' := (contDiffAt_succ_iff_hasFDerivAt.1 h)
+  obtain ⟨f', ⟨u, hu₁, hu₂⟩ , hf'⟩ := A'
+
+  have : (fun w => (fderiv 𝕜 f w) v) = (fun f => f v) ∘ (fun w => (fderiv 𝕜 f w)) := by
+    rfl
+  rw [this]
+  apply ContDiffAt.comp
+  apply fderiv_clm_apply
+
+  let A := (contDiffAt_succ_iff_fderiv.1 h).right
+  simp at A
+
+  have : (fun w => (fderiv 𝕜 f w) v) = (fun f => f v) ∘ (fun w => (fderiv 𝕜 f w)) := by
+    rfl
+
+  rw [this]
+  refine ContDiff.comp ?hg A
+  refine ContDiff.of_succ ?hg.h
+  refine ContDiff.clm_apply ?hg.h.hf ?hg.h.hg
+  exact contDiff_id
+  exact contDiff_const
+
+
 lemma partialDeriv_fderiv {f : E → F} (hf : ContDiff 𝕜 2 f) (z a b : E) :
     fderiv 𝕜 (fderiv 𝕜 f) z b a = partialDeriv 𝕜 b (partialDeriv 𝕜 a f) z := by
 
@@ -94,6 +127,12 @@ lemma partialDeriv_fderiv {f : E → F} (hf : ContDiff 𝕜 2 f) (z a b : E) :
   · 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
+
+
 section restrictScalars
 
 variable (𝕜 : Type*) [NontriviallyNormedField 𝕜]