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This commit is contained in:
Stefan Kebekus 2024-08-05 10:56:41 +02:00
parent c44f7e2efd
commit 9294d89ef1
1 changed files with 20 additions and 47 deletions

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@ -335,7 +335,6 @@ theorem primitive_additivity
(f : → E)
(z₀ : )
(rx ry : )
(hrx : 0 < rx)
(hry : 0 < ry)
(hf : DifferentiableOn f (Metric.ball z₀.re rx × Metric.ball z₀.im ry))
(z₁ : )
@ -347,50 +346,6 @@ theorem primitive_additivity
use ry - dist z₀.im z₁.im
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 A₁ : dist x.im z₀.im ≤ dist z₁.im z₀.im := by
apply Real.dist_right_le_of_mem_uIcc
rw [Set.uIcc_comm]
exact (Complex.mem_reProdIm.1 hx).2
have A₂ : dist x.re z₁.re ≤ dist z.re z₁.re := by
apply Real.dist_right_le_of_mem_uIcc
rw [Set.uIcc_comm]
exact (Complex.mem_reProdIm.1 hx).1
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 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
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
@ -545,8 +500,26 @@ theorem primitive_additivity
have H' : DifferentiableOn f (Set.uIcc z₁.re z.re × Set.uIcc z₀.im z₁.im) := by
apply DifferentiableOn.mono hf
intro x hx
exact H hx
constructor
· simp
calc dist x.re z₀.re
_ ≤ dist x.re z₁.re + dist z₁.re z₀.re := by exact dist_triangle x.re z₁.re z₀.re
_ ≤ dist z.re z₁.re + dist z₁.re z₀.re := by
apply (add_le_add_iff_right (dist z₁.re z₀.re)).mpr
rw [Set.uIcc_comm] at hx
apply Real.dist_right_le_of_mem_uIcc (Complex.mem_reProdIm.1 hx).1
_ < (rx - dist z₀.re z₁.re) + dist z₁.re z₀.re := by
apply (add_lt_add_iff_right (dist z₁.re z₀.re)).mpr
apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).1
_ = rx := by
rw [dist_comm]
simp
· simp
calc dist x.im z₀.im
_ ≤ dist z₀.im z₁.im := by rw [dist_comm]; exact Real.dist_left_le_of_mem_uIcc (Complex.mem_reProdIm.1 hx).2
_ < ry := by
rw [dist_comm]
exact Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz₁).2
let A := Complex.integral_boundary_rect_eq_zero_of_differentiableOn f ⟨z₁.re, z₀.im⟩ ⟨z.re, z₁.im⟩ H'
have {x : } {w : } : ↑x + w.im * Complex.I = { re := x, im := w.im } := by