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Update analyticOn_zeroSet.lean

This commit is contained in:
Stefan Kebekus 2024-09-10 11:30:27 +02:00
parent cd58c18a78
commit f732c82f92
1 changed files with 2 additions and 14 deletions
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16
Nevanlinna/analyticOn_zeroSet.lean
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@ -309,15 +309,7 @@ theorem AnalyticOnCompact.eliminateZeros
(h₂f : ∃ u ∈ U, f u ≠ 0) : (h₂f : ∃ u ∈ U, f u ≠ 0) :
∃ (g : ℂ → ℂ) (A : Finset U), AnalyticOn ℂ g U ∧ (∀ z ∈ U, g z ≠ 0) ∧ ∀ z, f z = (∏ a ∈ A, (z - a) ^ (h₁f a a.2).order.toNat) • g z := by ∃ (g : ℂ → ℂ) (A : Finset U), AnalyticOn ℂ 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 → ℂ := Subtype.val let A := (finiteZeros h₁U h₂U h₁f h₂f).toFinset
let A₁ := ι⁻¹' (U ∩ f⁻¹' {0})
have : A₁.Finite := by
apply Set.Finite.preimage
exact Set.injOn_subtype_val
exact finiteZeros h₁U h₂U h₁f h₂f
let A := this.toFinset
let n : ℂ → ℕ := by let n : ℂ → ℕ := by
intro z intro z
@ -354,14 +346,10 @@ theorem AnalyticOnCompact.eliminateZeros
· exact h₂g ⟨z, h₁z⟩ h₂z · exact h₂g ⟨z, h₁z⟩ h₂z
· have : f z ≠ 0 := by · have : f z ≠ 0 := by
by_contra C by_contra C
have : ⟨z, h₁z⟩ ∈ ↑A₁ := by
dsimp [A₁, ι]
simp
exact C
have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by have : ⟨z, h₁z⟩ ∈ ↑A.toSet := by
dsimp [A] dsimp [A]
simp simp
exact this exact C
tauto tauto
rw [inter z] at this rw [inter z] at this
exact right_ne_zero_of_smul this exact right_ne_zero_of_smul this