This commit is contained in:
Stefan Kebekus 2024-08-06 10:40:54 +02:00
parent 819037fb01
commit c799170843
1 changed files with 52 additions and 10 deletions

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@ -515,15 +515,44 @@ theorem primitive_additivity'
:
primitive z₀ f =ᶠ[nhds z₁] fun z ↦ (primitive z₁ f z) + (primitive z₀ f z₁) := by
let ε := R - dist z₀ z₁
let d := fun ε ↦ √((z₁.re - z₀.re + ε) ^ 2 + (z₁.im - z₀.im + ε) ^ 2)
have h₀d : Continuous d := by continuity
have h₁d : ∀ ε, 0 ≤ d ε := fun ε ↦ Real.sqrt_nonneg ((z₁.re - z₀.re + ε) ^ 2 + (z₁.im - z₀.im + ε) ^ 2)
have hε : 0 < ε := by
dsimp [ε]
simp
exact Metric.mem_ball'.mp hz₁
obtain ⟨ε, h₀ε, h₁ε⟩ : ∃ ε > 0, d ε < R := by
let Omega := d⁻¹' Metric.ball 0 R
let rx := dist z₀.re z₁.re + ε/(2 : )
let ry := dist z₀.im z₁.im + ε/(2 : )
have lem₀Ω : IsOpen Omega := IsOpen.preimage h₀d Metric.isOpen_ball
have lem₁Ω : 0 ∈ Omega := by
dsimp [Omega, d]; simp
rw [← Complex.dist_eq_re_im]; simp
exact hz₁
obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 lem₀Ω 0 lem₁Ω
have h'₁ε : 0 < ε := by exact h₁ε
let ε' := (2 : )⁻¹ * ε
have h₀ε' : ε' ∈ Omega := by
apply h₂ε
dsimp [ε']; simp
have : |ε| = ε := by apply abs_of_pos h₁ε
rw [this]
apply (inv_mul_lt_iff zero_lt_two).mpr
linarith
have h₁ε' : 0 < ε' := by
apply Real.mul_pos _ h₁ε
apply inv_pos.mpr
exact zero_lt_two
use ε'
constructor
· exact h₁ε'
· dsimp [Omega] at h₀ε'; simp at h₀ε'
rwa [abs_of_nonneg (h₁d ε')] at h₀ε'
let rx := dist z₁.re z₀.re + ε
let ry := dist z₁.im z₀.im + ε
have h'ry : 0 < ry := by
dsimp [ry]
@ -535,13 +564,26 @@ theorem primitive_additivity'
apply hf.mono
intro x hx
simp
sorry
rw [Complex.dist_eq_re_im]
have : dist x.re z₀.re < rx := Metric.mem_ball.mp hx.1
have : dist x.im z₀.im < ry := Metric.mem_ball.mp hx.2
have t₀ : √((x.re - z₀.re) ^ 2 + (x.im - z₀.im) ^ 2) < √( rx ^ 2 + ry ^ 2) := by
sorry
have t₁ : √( rx ^ 2 + ry ^ 2) = d ε := by
sorry
calc √((x.re - z₀.re) ^ 2 + (x.im - z₀.im) ^ 2)
_ < √( rx ^ 2 + ry ^ 2) := by exact t₀
_ = d ε := by exact t₁
_ < R := by exact h₁ε
have h'z₁ : z₁ ∈ (Metric.ball z₀.re rx × Metric.ball z₀.im ry) := by
dsimp [rx, ry]
constructor
· rw [dist_comm]; simp; exact hε
· rw [dist_comm]; simp; exact hε
· simp; exact hε
· simp; exact hε
obtain ⟨εx, hεx, εy, hεy, hε⟩ := primitive_additivity h'f h'ry h'z₁