Update analyticOn_zeroSet.lean

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Stefan Kebekus 2024-08-20 16:56:25 +02:00
parent efe088f9b5
commit 867b88bf5a
1 changed files with 27 additions and 82 deletions

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@ -346,7 +346,6 @@ theorem finiteZeros
rfl
theorem AnalyticOnCompact.eliminateZeros
{f : }
{U : Set }
@ -354,95 +353,41 @@ theorem AnalyticOnCompact.eliminateZeros
(h₂U : IsCompact U)
(h₁f : AnalyticOn f U)
(h₂f : ∃ u ∈ U, f u ≠ 0) :
∃ (g : ), AnalyticOn g U ∧ (∀ a ∈ U, g a ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (n a)) • g z := by
∃ (g : ) (A : Finset U), AnalyticOn g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (h₁f a a.2).order.toNat) • g z := by
let ι : U → := Subtype.val
let A := U ∩ f ⁻¹' {0}
let A := ι⁻¹' (U ∩ f⁻¹' {0})
by sorry -- (finiteZeros h₁U h₂U h₁f h₂f).toFinset
let B := AnalyticOn.eliminateZeros h₁f
have t₁ : (U ∩ f⁻¹' {0}).Finite := by
sorry
have : A₁.Finite := by
apply Set.Finite.preimage
exact Set.injOn_subtype_val
exact t₁
let A := this.toFinset
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)
let n : := by
intro z
by_cases hz : z ∈ U
· exact (h₁f z hz).order.toNat
· exact 0
-- case empty
simp
use f
simp
exact hf
have hn : ∀ a ∈ A, (h₁f a a.2).order = n a := by
sorry
-- case insert
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
rw [← hBinsert b₀ (Finset.mem_insert_self b₀ B)]
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 _
apply AnalyticAt.pow
apply AnalyticAt.sub
apply analyticAt_id
exact analyticAt_const
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
simp
have : ¬ (b₀.1 - a.1 = 0) := by
by_contra C
rw [sub_eq_zero] at C
rw [SetCoe.ext C] at hb
tauto
tauto
· simp
rw [AnalyticAt.order_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
use g₁
obtain ⟨g, h₁g, h₂g, h₃g⟩ := AnalyticOn.eliminateZeros (A := A) h₁f n hn
use g
use A
constructor
· exact h₁g
· exact h₁g
· 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'
· sorry
· intro z
let A' := h₃g₀ z
rw [h₃g₁ z] at A'
rw [A']
rw [← smul_assoc]
rw [h₃g z]
congr
simp
rw [Finset.prod_insert]
ring
exact hb
funext a
congr
dsimp [n]
simp [a.2]