kebekus
/
nevanlinna
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Update analyticOn_zeroSet.lean

This commit is contained in:
Stefan Kebekus 2024-09-10 11:06:00 +02:00
parent aa79fdb9eb
commit 5dc437751b
1 changed files with 5 additions and 5 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

10
Nevanlinna/analyticOn_zeroSet.lean
View File

@ -213,14 +213,14 @@ theorem discreteZeros
(hU : IsPreconnected U) (hU : IsPreconnected U)
(h₁f : AnalyticOn ℂ f U) (h₁f : AnalyticOn ℂ f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) : (h₂f : ∃ u ∈ U, f u ≠ 0) :
DiscreteTopology ↑(U ∩ f⁻¹' {0}) := by DiscreteTopology ((U.restrict f)⁻¹' {0}) := by
simp_rw [← singletons_open_iff_discrete] simp_rw [← singletons_open_iff_discrete]
simp_rw [Metric.isOpen_singleton_iff] simp_rw [Metric.isOpen_singleton_iff]
intro z intro z
let A := XX hU h₁f h₂f z.2.1 let A := XX hU h₁f h₂f z.1.2
rw [eq_comm] at A rw [eq_comm] at A
rw [AnalyticAt.order_eq_nat_iff] at A rw [AnalyticAt.order_eq_nat_iff] at A
obtain ⟨g, h₁g, h₂g, h₃g⟩ := A obtain ⟨g, h₁g, h₂g, h₃g⟩ := A
@ -265,9 +265,9 @@ theorem discreteZeros
_ < min ε₁ ε₂ := by assumption _ < min ε₁ ε₂ := by assumption
_ ≤ ε₁ := by exact min_le_left ε₁ ε₂ _ ≤ ε₁ := by exact min_le_left ε₁ ε₂
have F := h₂ε₂ y.1 h₂y have F := h₂ε₂ y.1 h₂y
rw [y.2.2] at F have : f y = 0 := by exact y.2
rw [this] at F
simp at F simp at F
have : g y.1 ≠ 0 := by have : g y.1 ≠ 0 := by
@ -285,7 +285,7 @@ theorem finiteZeros
(h₂U : IsCompact U) (h₂U : IsCompact U)
(h₁f : AnalyticOn ℂ f U) (h₁f : AnalyticOn ℂ f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) : (h₂f : ∃ u ∈ U, f u ≠ 0) :
Set.Finite ↑(U ∩ f⁻¹' {0}) := by Set.Finite ((U.restrict f)⁻¹' {0}) := by
have hinter : IsCompact ↑(U ∩ f⁻¹' {0}) := by have hinter : IsCompact ↑(U ∩ f⁻¹' {0}) := by
apply IsCompact.of_isClosed_subset h₂U apply IsCompact.of_isClosed_subset h₂U