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This commit is contained in:
Stefan Kebekus 2024-08-03 07:20:01 +02:00
parent c376bd3c46
commit 8a8e0769c2
1 changed files with 8 additions and 13 deletions
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21
Nevanlinna/holomorphic_primitive2.lean
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@ -334,10 +334,10 @@ theorem primitive_additivity
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
(f : ℂ → E) (f : ℂ → E)
(z₀ : ℂ) (z₀ : ℂ)
(R : ℝ) (rx ry : ℝ)
(hf : DifferentiableOn ℂ f (Metric.ball z₀ R)) (hf : DifferentiableOn ℂ f (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry))
(z₁ : ℂ) (z₁ : ℂ)
(hz₁ : z₁ ∈ Metric.ball z₀ R) (hz₁ : z₁ ∈ (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry))
: :
∀ z ∈ Metric.ball z₁ (R - dist z₁ z₀), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by ∀ z ∈ Metric.ball z₁ (R - dist z₁ z₀), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by
intro z hz intro z hz
@ -345,19 +345,15 @@ theorem primitive_additivity
have H : (Set.uIcc z₁.re z.re ×ℂ Set.uIcc z₀.im z₁.im) ⊆ Metric.ball z₀ R := by have H : (Set.uIcc z₁.re z.re ×ℂ Set.uIcc z₀.im z₁.im) ⊆ Metric.ball z₀ R := by
intro x hx intro x hx
have a₀ : x.re ∈ Set.uIcc z₁.re z.re := by exact (Complex.mem_reProdIm.1 hx).1
have a₁ : x.im ∈ Set.uIcc z₀.im z₁.im := by exact (Complex.mem_reProdIm.1 hx).2
have A₀ : dist x.re z₀.re ≤ dist x.re z₁.re + dist z₁.re z₀.re := by apply dist_triangle
have A₁ : dist x.im z₀.im ≤ dist z₁.im z₀.im := by have A₁ : dist x.im z₀.im ≤ dist z₁.im z₀.im := by
apply Real.dist_right_le_of_mem_uIcc apply Real.dist_right_le_of_mem_uIcc
rwa [Set.uIcc_comm] rw [Set.uIcc_comm]
exact (Complex.mem_reProdIm.1 hx).2
have A₂ : dist x.re z₁.re ≤ dist z.re z₁.re := by have A₂ : dist x.re z₁.re ≤ dist z.re z₁.re := by
apply Real.dist_right_le_of_mem_uIcc apply Real.dist_right_le_of_mem_uIcc
rwa [Set.uIcc_comm] rw [Set.uIcc_comm]
exact (Complex.mem_reProdIm.1 hx).1
have A₃ : dist z.re z₁.re < R - dist z₁ z₀ := by have A₃ : dist z.re z₁.re < R - dist z₁ z₀ := by
have : ∀ x₀ x₁ : ℂ, dist x₀.re x₁.re ≤ dist x₀ x₁ := by have : ∀ x₀ x₁ : ℂ, dist x₀.re x₁.re ≤ dist x₀ x₁ := by
@ -372,8 +368,6 @@ theorem primitive_additivity
simp simp
have B₀ : dist x z₀ ≤ dist x ⟨z₁.re, x.im⟩ + dist ⟨z₁.re, x.im⟩ z₀ := by apply dist_triangle
have B₁ : dist ⟨z₁.re, x.im⟩ z₀ ≤ dist z₁ z₀ := by have B₁ : dist ⟨z₁.re, x.im⟩ z₀ ≤ dist z₁ z₀ := by
rw [Complex.dist_eq_re_im] rw [Complex.dist_eq_re_im]
rw [Complex.dist_eq_re_im] rw [Complex.dist_eq_re_im]
@ -474,6 +468,7 @@ theorem primitive_additivity
intro w hw intro w hw
simp simp
sorry --apply integrability₂ f hf sorry --apply integrability₂ f hf
sorry --apply integrability₂ f hf sorry --apply integrability₂ f hf