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This commit is contained in:
Stefan Kebekus 2024-08-06 11:13:21 +02:00
parent c799170843
commit 6bdb910b7c
1 changed files with 16 additions and 7 deletions
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23
Nevanlinna/holomorphic_primitive2.lean
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@ -515,9 +515,9 @@ theorem primitive_additivity'
: :
primitive z₀ f =ᶠ[nhds z₁] fun z ↦ (primitive z₁ f z) + (primitive z₀ f z₁) := by primitive z₀ f =ᶠ[nhds z₁] fun z ↦ (primitive z₁ f z) + (primitive z₀ f z₁) := by
let d := fun ε ↦ √((z₁.re - z₀.re + ε) ^ 2 + (z₁.im - z₀.im + ε) ^ 2) let d := fun ε ↦ √((dist z₁.re z₀.re + ε) ^ 2 + (dist z₁.im z₀.im + ε) ^ 2)
have h₀d : Continuous d := by continuity 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 h₁d : ∀ ε, 0 ≤ d ε := fun ε ↦ Real.sqrt_nonneg ((dist z₁.re z₀.re + ε) ^ 2 + (dist z₁.im z₀.im + ε) ^ 2)
obtain ⟨ε, h₀ε, h₁ε⟩ : ∃ ε > 0, d ε < R := by obtain ⟨ε, h₀ε, h₁ε⟩ : ∃ ε > 0, d ε < R := by
let Omega := d⁻¹' Metric.ball 0 R let Omega := d⁻¹' Metric.ball 0 R
@ -525,6 +525,11 @@ theorem primitive_additivity'
have lem₀Ω : IsOpen Omega := IsOpen.preimage h₀d Metric.isOpen_ball have lem₀Ω : IsOpen Omega := IsOpen.preimage h₀d Metric.isOpen_ball
have lem₁Ω : 0 ∈ Omega := by have lem₁Ω : 0 ∈ Omega := by
dsimp [Omega, d]; simp dsimp [Omega, d]; simp
have : dist z₁.re z₀.re = |z₁.re - z₀.re| := by exact rfl
rw [this]
have : dist z₁.im z₀.im = |z₁.im - z₀.im| := by exact rfl
rw [this]
simp
rw [← Complex.dist_eq_re_im]; simp rw [← Complex.dist_eq_re_im]; simp
exact hz₁ exact hz₁
obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 lem₀Ω 0 lem₁Ω obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 lem₀Ω 0 lem₁Ω
@ -570,13 +575,17 @@ theorem primitive_additivity'
have : dist x.im z₀.im < ry := Metric.mem_ball.mp hx.2 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 have t₀ : √((x.re - z₀.re) ^ 2 + (x.im - z₀.im) ^ 2) < √( rx ^ 2 + ry ^ 2) := by
sorry rw [Real.sqrt_lt_sqrt_iff]
have t₁ : √( rx ^ 2 + ry ^ 2) = d ε := by apply add_lt_add
sorry · dsimp [rx]
sorry
· sorry
calc √((x.re - z₀.re) ^ 2 + (x.im - z₀.im) ^ 2) calc √((x.re - z₀.re) ^ 2 + (x.im - z₀.im) ^ 2)
_ < √( rx ^ 2 + ry ^ 2) := by exact t₀ _ < √( rx ^ 2 + ry ^ 2) := by
_ = d ε := by exact t₁ exact t₀
_ = d ε := by dsimp [d, rx, ry]
_ < R := by exact h₁ε _ < R := by exact h₁ε
have h'z₁ : z₁ ∈ (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry) := by have h'z₁ : z₁ ∈ (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry) := by