152 lines
3.7 KiB
Plaintext
152 lines
3.7 KiB
Plaintext
import Mathlib.Analysis.Analytic.IsolatedZeros
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import Nevanlinna.holomorphic
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noncomputable def zeroDivisor
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(f : ℂ → ℂ) :
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ℂ → ℕ := by
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intro z
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if hf : AnalyticAt ℂ f z then
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exact hf.order.toNat
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else
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exact 0
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theorem analyticAtZeroDivisorSupport
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{f : ℂ → ℂ}
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{z : ℂ}
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(h : z ∈ Function.support (zeroDivisor f)) :
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AnalyticAt ℂ f z := by
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by_contra h₁f
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simp at h
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dsimp [zeroDivisor] at h
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simp [h₁f] at h
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theorem zeroDivisor_eq_ord_AtZeroDivisorSupport
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{f : ℂ → ℂ}
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{z : ℂ}
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(h : z ∈ Function.support (zeroDivisor f)) :
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zeroDivisor f z = (analyticAtZeroDivisorSupport h).order.toNat := by
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unfold zeroDivisor
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simp [analyticAtZeroDivisorSupport h]
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lemma toNatEqSelf_iff {n : ℕ∞} : n.toNat = n ↔ ∃ m : ℕ, m = n := by
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constructor
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· intro H₁
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rw [← ENat.some_eq_coe, ← WithTop.ne_top_iff_exists]
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by_contra H₂
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rw [H₂] at H₁
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simp at H₁
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· intro H
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obtain ⟨m, hm⟩ := H
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rw [← hm]
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simp
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lemma natural_if_toNatNeZero {n : ℕ∞} : n.toNat ≠ 0 → ∃ m : ℕ, m = n := by
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rw [← ENat.some_eq_coe, ← WithTop.ne_top_iff_exists]
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contrapose; simp; tauto
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theorem zeroDivisor_in_zeroSet
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{f : ℂ → ℂ}
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{z : ℂ}
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(h : z ∈ Function.support (zeroDivisor f)) :
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f z = 0 := by
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sorry
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theorem discreteZeros
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{f : ℂ → ℂ} :
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DiscreteTopology (Function.support (zeroDivisor f)) := by
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simp_rw [← singletons_open_iff_discrete, Metric.isOpen_singleton_iff]
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intro z
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have A : zeroDivisor f ↑z ≠ 0 := by exact z.2
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let B := zeroDivisor_eq_ord_AtZeroDivisorSupport z.2
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rw [B] at A
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have C := natural_if_toNatNeZero A
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obtain ⟨m, hm⟩ := C
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have h₂m : m ≠ 0 := by
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rw [← hm] at A
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simp at A
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assumption
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rw [eq_comm] at hm
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let E := AnalyticAt.order_eq_nat_iff (analyticAtZeroDivisorSupport z.2) m
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rw [E] at hm
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := hm
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rw [Metric.eventually_nhds_iff_ball] at h₃g
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have : ∃ ε > 0, ∀ y ∈ Metric.ball (↑z) ε, g y ≠ 0 := by
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have h₄g : ContinuousAt g z := AnalyticAt.continuousAt h₁g
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have : {0}ᶜ ∈ nhds (g z) := by
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exact compl_singleton_mem_nhds_iff.mpr h₂g
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let F := h₄g.preimage_mem_nhds this
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rw [Metric.mem_nhds_iff] at F
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obtain ⟨ε, h₁ε, h₂ε⟩ := F
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use ε
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constructor; exact h₁ε
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intro y hy
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let G := h₂ε hy
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simp at G
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exact G
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obtain ⟨ε₁, h₁ε₁⟩ := this
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obtain ⟨ε₂, h₁ε₂, h₂ε₂⟩ := h₃g
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use min ε₁ ε₂
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constructor
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· have : 0 < min ε₁ ε₂ := by
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rw [lt_min_iff]
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exact And.imp_right (fun _ => h₁ε₂) h₁ε₁
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exact this
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intro y
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intro h₁y
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have h₂y : ↑y ∈ Metric.ball (↑z) ε₂ := by
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simp
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calc dist y z
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_ < min ε₁ ε₂ := by assumption
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_ ≤ ε₂ := by exact min_le_right ε₁ ε₂
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have h₃y : ↑y ∈ Metric.ball (↑z) ε₁ := by
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simp
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calc dist y z
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_ < min ε₁ ε₂ := by assumption
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_ ≤ ε₁ := by exact min_le_left ε₁ ε₂
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let F := h₂ε₂ y.1 h₂y
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rw [zeroDivisor_in_zeroSet y.2] at F
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simp at F
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simp [h₂m] at F
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have : g y.1 ≠ 0 := by
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exact h₁ε₁.2 y h₃y
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simp [this] at F
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ext
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rwa [sub_eq_zero] at F
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theorem zeroDivisor_finiteOnCompact
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{f : ℂ → ℂ}
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{s : Set ℂ}
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(hs : IsCompact s) :
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Set.Finite (s ∩ Function.support (zeroDivisor f)) := by
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sorry
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theorem eliminatingZeros
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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{R : ℝ}
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(h₁f : ∀ z ∈ Metric.ball z₀ R, HolomorphicAt f z)
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(h₂f : ∃ z ∈ Metric.ball z₀ R, f z ≠ 0) :
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∃ F : ℂ → ℂ, ∀ z ∈ Metric.ball z₀ R, (HolomorphicAt F z) ∧ (f z = (F z) * ∏ᶠ a ∈ Metric.ball z₀ R, (z - a) ^ (zeroDivisor f a) ) := by
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sorry
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