Update analyticOn_zeroSet.lean

This commit is contained in:
Stefan Kebekus 2024-08-20 09:14:54 +02:00
parent 1ca46cf454
commit b0d663530b
1 changed files with 28 additions and 2 deletions

View File

@ -101,9 +101,9 @@ theorem AnalyticOn.order_eq_nat_iff'
{A : Finset U}
(hf : AnalyticOn f U)
(n : ) :
(∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ), AnalyticOn g U ∧ (∀ a : A, g a ≠ 0) ∧ ∀ z, f z = (∏ a : A, (z - a) ^ (n a)) • g z := by
(∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ), AnalyticOn g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by
apply Finset.induction (α := U) (p := fun A ↦ (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ), AnalyticOn g U ∧ (∀ a : A, g a ≠ 0) ∧ ∀ z, f z = (∏ a : A, (z - a) ^ (n a)) • g z)
apply Finset.induction (α := U) (p := fun A ↦ (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ), AnalyticOn g U ∧ (∀ a ∈ A, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z)
-- case empty
simp
@ -115,3 +115,29 @@ theorem AnalyticOn.order_eq_nat_iff'
intro b₀ B hb iHyp
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
obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticOn.order_eq_nat_iff h₁g₀ b₀.2 (n b₀)).1 this
use g₁
constructor
· exact h₁g₁
· constructor
· intro a h₁a
by_cases h₂a : a = b₀
· rwa [h₂a]
· let A' := Finset.mem_of_mem_insert_of_ne h₁a h₂a
let B' := h₃g₁ a
let C' := h₂g₀ a A'
rw [B'] at C'
exact right_ne_zero_of_smul C'
· intro z
let A' := h₃g₀ z
rw [h₃g₁ z] at A'
rw [A']
rw [← smul_assoc]
congr
simp
rw [Finset.prod_insert]
ring
exact hb