This commit is contained in:
Stefan Kebekus 2024-08-02 11:22:41 +02:00
parent 2e5d008857
commit 03548bc6da
1 changed files with 56 additions and 5 deletions

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@ -337,9 +337,59 @@ theorem primitive_additivity
(R : ) (R : )
(hf : DifferentiableOn f (Metric.ball z₀ R)) (hf : DifferentiableOn f (Metric.ball z₀ R))
(z₁ : ) (z₁ : )
(hz₁ : z₁ ∈ Metric.ball z₀ R) : (hz₁ : z₁ ∈ Metric.ball z₀ R)
∀ z ∈ Metric.ball z₁ (R - ‖z₁‖), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by :
intro z _ ∀ 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
have H : (Set.uIcc z₁.re z.re × Set.uIcc z₀.im z₁.im) ⊆ Metric.ball z₀ R := by
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
apply Real.dist_right_le_of_mem_uIcc
rwa [Set.uIcc_comm]
have A₂ : dist x.re z₁.re ≤ dist z.re z₁.re := by
apply Real.dist_right_le_of_mem_uIcc
rwa [Set.uIcc_comm]
have A₃ : dist z.re z₁.re < R - dist z₁ z₀ := by
have : ∀ x₀ x₁ : , dist x₀.re x₁.re ≤ dist x₀ x₁ := by
intro x₀ x₁
rw [Complex.dist_eq_re_im, Real.dist_eq]
apply Real.le_sqrt_of_sq_le
simp
exact sq_nonneg (x₀.im - x₁.im)
calc dist z.re z₁.re
_ ≤ dist z z₁ := by apply this z z₁
_ < R - dist z₁ z₀ := by exact hz
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
rw [Complex.dist_eq_re_im]
rw [Complex.dist_eq_re_im]
simp
apply Real.sqrt_le_sqrt
simp
exact sq_le_sq.mpr A₁
calc dist x z₀
_ ≤ dist x ⟨z₁.re, x.im⟩ + dist ⟨z₁.re, x.im⟩ z₀ := by apply dist_triangle
_ = dist x.re z₁.re + dist ⟨z₁.re, x.im⟩ z₀ := by rw [Complex.dist_of_im_eq]; rfl
_ ≤ dist z.re z₁.re + dist ⟨z₁.re, x.im⟩ z₀ := by apply add_le_add_right A₂
_ < (R - dist z₁ z₀) + dist ⟨z₁.re, x.im⟩ z₀ := by apply add_lt_add_right A₃
_ ≤ (R - dist z₁ z₀) + dist z₁ z₀ := by exact (add_le_add_iff_left (R - dist z₁ z₀)).mpr B₁
_ = R := by exact sub_add_cancel R (dist z₁ z₀)
unfold primitive unfold primitive
have : (∫ (x : ) in z₀.re..z.re, f { re := x, im := z₀.im }) = (∫ (x : ) in z₀.re..z₁.re, f { re := x, im := z₀.im }) + (∫ (x : ) in z₁.re..z.re, f { re := x, im := z₀.im }) := by have : (∫ (x : ) in z₀.re..z.re, f { re := x, im := z₀.im }) = (∫ (x : ) in z₀.re..z₁.re, f { re := x, im := z₀.im }) + (∫ (x : ) in z₁.re..z.re, f { re := x, im := z₀.im }) := by
@ -363,9 +413,10 @@ theorem primitive_additivity
apply DifferentiableOn.mono hf apply DifferentiableOn.mono hf
intro x hx intro x hx
simp simp
let A₀ : dist x.re z₀.re ≤ dist x.re z₁.re + dist z₁.re z₀.re := by apply dist_triangle let A₀ : dist x.re z₀.re ≤ dist x.re z₁.re + dist z₁.re z₀.re := by apply dist_triangle
let A₁ : dist x.im z₀.im ≤ dist z₁.im z₀.im := by sorry