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Update holomorphic_zero.lean

This commit is contained in:
Stefan Kebekus 2024-08-19 08:30:02 +02:00
parent 7b1c861a92
commit 97293e3a60
1 changed files with 23 additions and 4 deletions
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Nevanlinna/holomorphic_zero.lean
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@ -1,3 +1,4 @@
import Mathlib.Topology.ContinuousOn
import Mathlib.Analysis.Analytic.IsolatedZeros import Mathlib.Analysis.Analytic.IsolatedZeros
import Nevanlinna.holomorphic import Nevanlinna.holomorphic
@ -243,10 +244,28 @@ theorem discreteZeros
theorem zeroDivisor_finiteOnCompact theorem zeroDivisor_finiteOnCompact
{f : ℂ → ℂ} {f : ℂ → ℂ}
{s : Set ℂ} {U : Set ℂ}
(hs : IsCompact s) : (hU : IsPreconnected U)
Set.Finite (s ∩ Function.support (zeroDivisor f)) := by (h₁f : AnalyticOn ℂ f U)
sorry (h₂f : ∃ z ∈ U, f z ≠ 0)
(h₂U : IsCompact U) :
Set.Finite (U ∩ Function.support (zeroDivisor f)) := by
have hinter : IsCompact (U ∩ Function.support (zeroDivisor f)) := by
apply IsCompact.of_isClosed_subset h₂U
rw [supportZeroSet]
apply h₁f.continuousOn.preimage_isClosed_of_isClosed
exact IsCompact.isClosed h₂U
exact isClosed_singleton
assumption
assumption
assumption
exact Set.inter_subset_left
apply hinter.finite
apply DiscreteTopology.of_subset (s := Function.support (zeroDivisor f))
exact discreteZeros (f := f)
exact Set.inter_subset_right
theorem eliminatingZeros theorem eliminatingZeros