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.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Basic import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Symmetric 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.cauchyRiemann
import Nevanlinna.partialDeriv import Nevanlinna.partialDeriv
@ -40,18 +46,35 @@ theorem holomorphic_is_harmonic {f : } (h : Differentiable f) :
unfold Complex.laplace unfold Complex.laplace
rw [CauchyRiemann₄ h] rw [CauchyRiemann₄ h]
let l : →L[] := by have : ∀ v, ∀ s : , ∀ g : , Differentiable g → Real.partialDeriv v (s • g) = s • (Real.partialDeriv v g) := by
-- intro v s g hg
sorry --(fun x ↦ Complex.I • x)
have : (Complex.I • Real.partialDeriv 1 f) = (l ∘ (Real.partialDeriv 1 f)) := by let sMuls : →L[] :=
sorry {
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 [this]
rw [partialDeriv_compContLin]
--rw [partialDeriv_smul₂ fI_is_real_differentiable]
rw [partialDeriv_comm f_is_real_C2 Complex.I 1] rw [partialDeriv_comm f_is_real_C2 Complex.I 1]
rw [CauchyRiemann₄ h] rw [CauchyRiemann₄ h]
rw [partialDeriv_smul₂ fI_is_real_differentiable] rw [this]
rw [← smul_assoc] rw [← smul_assoc]
simp 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