Update complexHarmonic.lean

This commit is contained in:
Stefan Kebekus 2024-05-08 08:27:12 +02:00
parent 631b1bad70
commit 5ce2b83c20
1 changed files with 33 additions and 10 deletions

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@ -5,6 +5,12 @@ 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.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
import Nevanlinna.cauchyRiemann
import Nevanlinna.partialDeriv
@ -40,18 +46,35 @@ theorem holomorphic_is_harmonic {f : } (h : Differentiable f) :
unfold Complex.laplace
rw [CauchyRiemann₄ h]
let l : →L[] := by
--
sorry --(fun x ↦ Complex.I • x)
have : (Complex.I • Real.partialDeriv 1 f) = (l ∘ (Real.partialDeriv 1 f)) := by
sorry
have : ∀ v, ∀ s : , ∀ g : , Differentiable g → Real.partialDeriv v (s • g) = s • (Real.partialDeriv v g) := by
intro v s g hg
let sMuls : →L[] :=
{
toFun := fun x ↦ s * x
map_add' := by
intro x y
ring
map_smul' := by
intro m x
simp
ring
}
have : s • g = sMuls ∘ g := by rfl
rw [this]
rw [partialDeriv_compContLin hg]
rfl
rw [this]
rw [partialDeriv_compContLin]
--rw [partialDeriv_smul₂ fI_is_real_differentiable]
rw [partialDeriv_comm f_is_real_C2 Complex.I 1]
rw [CauchyRiemann₄ h]
rw [partialDeriv_smul₂ fI_is_real_differentiable]
rw [this]
rw [← smul_assoc]
simp
-- Subgoals coming from the application of 'this'
-- Differentiable (Real.partialDeriv 1 f)
exact fI_is_real_differentiable
-- Differentiable (Real.partialDeriv 1 f)
exact fI_is_real_differentiable