Make it work!

This commit is contained in:
Stefan Kebekus 2024-05-06 10:09:49 +02:00
parent c44f4fe3b0
commit e5383eff34
1 changed files with 29 additions and 29 deletions

View File

@ -21,46 +21,37 @@ def Harmonic (f : ) : Prop :=
(ContDiff 2 f) ∧ (∀ z, Complex.laplace f z = 0) (ContDiff 2 f) ∧ (∀ z, Complex.laplace f z = 0)
lemma zwoDiff (f : × ) (h : ContDiff 2 f) : ∀ z a b : × , 0 = 1 := by lemma derivSymm (f : ) (hf : ContDiff 2 f) :
intro z a b
let fx := fun w ↦ (fderiv f w) a
let fxx := fun w ↦ (fderiv fx w) a
let f2 := (fderiv (fun w => fderiv f w) z) a a
have : iteratedFDeriv 1 f z ![a] = 0 := by
rw [iteratedFDeriv_succ_apply_left]
simp
let g := iteratedFDeriv 0 f
simp at g
sorry
have : f2 = fxx z := by
dsimp [f2, fxx, fx]
sorry
sorry
lemma derivSymm (f : ) (h : Differentiable f) :
∀ z a b : , (fderiv (fun w => fderiv f w) z) a b = (fderiv (fun w => fderiv f w) z) b a := by ∀ z a b : , (fderiv (fun w => fderiv f w) z) a b = (fderiv (fun w => fderiv f w) z) b a := by
intro z a b intro z a b
let f' := fun w => (fderiv f w) let f' := fun w => (fderiv f w)
have h₀ : ∀ y, HasFDerivAt f (f' y) y := by have h₀ : ∀ y, HasFDerivAt f (f' y) y := by
have h : Differentiable f := by
exact (contDiff_succ_iff_fderiv.1 hf).left
exact fun y => DifferentiableAt.hasFDerivAt (h y) exact fun y => DifferentiableAt.hasFDerivAt (h y)
let f'' := (fderiv f' z) let f'' := (fderiv f' z)
have h₁ : HasFDerivAt f' f'' z := by have h₁ : HasFDerivAt f' f'' z := by
apply DifferentiableAt.hasFDerivAt apply DifferentiableAt.hasFDerivAt
sorry let A := (contDiff_succ_iff_fderiv.1 hf).right
let B := (contDiff_succ_iff_fderiv.1 A).left
simp at B
exact B z
let A := second_derivative_symmetric h₀ h₁ a b let A := second_derivative_symmetric h₀ h₁ a b
dsimp [f'', f'] at A dsimp [f'', f'] at A
apply A apply A
lemma l₂ {f : } (hf : ContDiff 2 f) (z a b : ) :
fderiv (fderiv f) z b a = fderiv (fun w ↦ fderiv f w a) z b := by
rw [fderiv_clm_apply]
· simp
· exact (contDiff_succ_iff_fderiv.1 hf).2.differentiable le_rfl z
· simp
theorem holomorphic_is_harmonic (f : ) : theorem holomorphic_is_harmonic (f : ) :
Differentiable f → Harmonic f := by Differentiable f → Harmonic f := by
@ -84,10 +75,17 @@ theorem holomorphic_is_harmonic (f : ) :
intro z intro z
rw [CauchyRiemann₁ (h z)] rw [CauchyRiemann₁ (h z)]
have t₂₀ : ContDiff 2 f := by exact ContDiff.restrict_scalars (Differentiable.contDiff h)
have t₀₀ : Differentiable (fun w => (fderiv f w)) := by
let A := (contDiff_succ_iff_fderiv.1 t₂₀).right
let B := (contDiff_succ_iff_fderiv.1 A).left
exact B
have t₀ : ∀ z, DifferentiableAt (fun w => (fderiv f w) 1) z := by have t₀ : ∀ z, DifferentiableAt (fun w => (fderiv f w) 1) z := by
intro z intro z
let A := t₀₀
sorry fun_prop
have t₁ : ∀ x, (fderiv (fun w => Complex.I * (fderiv f w) 1) z) x have t₁ : ∀ x, (fderiv (fun w => Complex.I * (fderiv f w) 1) z) x
= Complex.I * ((fderiv (fun w => (fderiv f w) 1) z) x) := by = Complex.I * ((fderiv (fun w => (fderiv f w) 1) z) x) := by
@ -97,12 +95,14 @@ theorem holomorphic_is_harmonic (f : ) :
exact t₀ z exact t₀ z
rw [t₁] rw [t₁]
have t₂₀ : Differentiable f := by sorry
have t₂ : (fderiv (fun w => (fderiv f w) 1) z) Complex.I have t₂ : (fderiv (fun w => (fderiv f w) 1) z) Complex.I
= (fderiv (fun w => (fderiv f w) Complex.I) z) 1 := by = (fderiv (fun w => (fderiv f w) Complex.I) z) 1 := by
let A := derivSymm f t₂₀ z 1 Complex.I let A := derivSymm f t₂₀ z 1 Complex.I
let B := l₂ t₂₀ z Complex.I 1
sorry rw [← B]
rw [A]
let C := l₂ t₂₀ z 1 Complex.I
rw [C]
rw [t₂] rw [t₂]
conv => conv =>