Update analyticOn_zeroSet.lean

This commit is contained in:
Stefan Kebekus 2024-08-20 11:06:24 +02:00
parent 8bc46c2874
commit 688347a837
1 changed files with 41 additions and 4 deletions

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@ -1,3 +1,4 @@
import Mathlib.Analysis.Analytic.Linear
import Init.Classical import Init.Classical
import Mathlib.Analysis.Analytic.Meromorphic import Mathlib.Analysis.Analytic.Meromorphic
import Mathlib.Topology.ContinuousOn import Mathlib.Topology.ContinuousOn
@ -146,7 +147,7 @@ theorem AnalyticOn.order_of_mul
· exact Set.mem_inter h₃t₁ h₃t₂ · exact Set.mem_inter h₃t₁ h₃t₂
theorem AnalyticOn.order_eq_nat_iff' theorem AnalyticOn.eliminateZeros
{f : } {f : }
{U : Set } {U : Set }
{A : Finset U} {A : Finset U}
@ -169,9 +170,45 @@ theorem AnalyticOn.order_eq_nat_iff'
have : (h₁g₀ b₀ b₀.2).order = n b₀ := by have : (h₁g₀ b₀ b₀.2).order = n b₀ := by
let A := hBinsert b₀ (Finset.mem_insert_self b₀ B) rw [← hBinsert b₀ (Finset.mem_insert_self b₀ B)]
exact A
let φ := fun z ↦ (∏ a ∈ B, (z - a.1) ^ n a.1)
have : f = fun z ↦ φ z * g₀ z := by
funext z
rw [h₃g₀ z]
rfl
simp_rw [this]
have h₁φ : AnalyticAt φ b₀ := by
dsimp [φ]
apply Finset.analyticAt_prod
intro b hb
apply AnalyticAt.pow
apply AnalyticAt.sub
apply analyticAt_id
have h₂φ : h₁φ.order = (0 : ) := by
rw [AnalyticAt.order_eq_nat_iff h₁φ 0]
use φ
constructor
· assumption
· constructor
· dsimp [φ]
push_neg
rw [Finset.prod_ne_zero_iff]
intro a ha
have AA : b₀.1 - a ≠ 0 := by
sorry sorry
simp [AA]
· simp
rw [AnalyticOn.order_of_mul h₁φ (h₁g₀ b₀ b₀.2)]
rw [h₂φ]
simp
obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ b₀.2 (n b₀)).1 this obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ b₀.2 (n b₀)).1 this