Update analyticOn_zeroSet.lean

This commit is contained in:
Stefan Kebekus 2024-08-20 10:16:37 +02:00
parent b0d663530b
commit 8bc46c2874
1 changed files with 57 additions and 1 deletions

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@ -2,6 +2,7 @@ import Init.Classical
import Mathlib.Analysis.Analytic.Meromorphic
import Mathlib.Topology.ContinuousOn
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Analytic.Constructions
import Nevanlinna.holomorphic
@ -95,6 +96,56 @@ theorem AnalyticOn.order_eq_nat_iff
exact ⟨h₁g z₀ hz₀, ⟨h₂g, Filter.eventually_of_forall h₃g⟩⟩
theorem AnalyticOn.order_of_mul
{f₁ f₂ : }
{z₀ : }
(hf₁ : AnalyticAt f₁ z₀)
(hf₂ : AnalyticAt f₂ z₀) :
(AnalyticAt.mul hf₁ hf₂).order = hf₁.order + hf₂.order := by
by_cases h₂f₁ : hf₁.order =
· simp [h₂f₁]
rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₁
obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₁
use t
constructor
· intro y hy
rw [h₁t y hy]
ring
· exact ⟨h₂t, h₃t⟩
· by_cases h₂f₂ : hf₂.order =
· simp [h₂f₂]
rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₂
obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₂
use t
constructor
· intro y hy
rw [h₁t y hy]
ring
· exact ⟨h₂t, h₃t⟩
· obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticAt.order_eq_nat_iff hf₁ ↑hf₁.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₁))
obtain ⟨g₂, h₁g₂, h₂g₂, h₃g₂⟩ := (AnalyticAt.order_eq_nat_iff hf₂ ↑hf₂.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₂))
rw [← ENat.coe_toNat h₂f₁, ← ENat.coe_toNat h₂f₂, ← ENat.coe_add]
rw [AnalyticAt.order_eq_nat_iff (AnalyticAt.mul hf₁ hf₂) ↑(hf₁.order.toNat + hf₂.order.toNat)]
use g₁ * g₂
constructor
· exact AnalyticAt.mul h₁g₁ h₁g₂
· constructor
· simp; tauto
· obtain ⟨t₁, h₁t₁, h₂t₁, h₃t₁⟩ := eventually_nhds_iff.1 h₃g₁
obtain ⟨t₂, h₁t₂, h₂t₂, h₃t₂⟩ := eventually_nhds_iff.1 h₃g₂
rw [eventually_nhds_iff]
use t₁ ∩ t₂
constructor
· intro y hy
rw [h₁t₁ y hy.1, h₁t₂ y hy.2]
simp; ring
· constructor
· exact IsOpen.inter h₂t₁ h₂t₂
· exact Set.mem_inter h₃t₁ h₃t₂
theorem AnalyticOn.order_eq_nat_iff'
{f : }
{U : Set }
@ -116,7 +167,12 @@ theorem AnalyticOn.order_eq_nat_iff'
intro hBinsert
obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀⟩ := iHyp (fun a ha ↦ hBinsert a (Finset.mem_insert_of_mem ha))
have : (h₁g₀ b₀ b₀.2).order = n b₀ := by sorry
have : (h₁g₀ b₀ b₀.2).order = n b₀ := by
let A := hBinsert b₀ (Finset.mem_insert_self b₀ B)
exact A
sorry
obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ b₀.2 (n b₀)).1 this
use g₁