390 lines
10 KiB
Plaintext
390 lines
10 KiB
Plaintext
import Init.Classical
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import Mathlib.Analysis.Analytic.Meromorphic
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import Mathlib.Topology.ContinuousOn
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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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by_cases hf : AnalyticAt ℂ f z
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· exact hf.order.toNat
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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_localDescription
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(h : z₀ ∈ Function.support (zeroDivisor f)) :
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∃ (g : ℂ → ℂ), AnalyticAt ℂ g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhds z₀, f z = (z - z₀) ^ (zeroDivisor f z₀) • g z := by
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have A : zeroDivisor f ↑z₀ ≠ 0 := by exact h
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let B := zeroDivisor_eq_ord_AtZeroDivisorSupport h
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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 h) m
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let F := hm
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rw [E] at F
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have : m = zeroDivisor f z₀ := by
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rw [B, hm]
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simp
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rwa [this] at F
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theorem zeroDivisor_zeroSet
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(h : z₀ ∈ Function.support (zeroDivisor f)) :
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f z₀ = 0 := by
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obtain ⟨g, _, _, h₃⟩ := zeroDivisor_localDescription h
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rw [Filter.Eventually.self_of_nhds h₃]
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simp
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left
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exact h
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theorem zeroDivisor_support_iff
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{f : ℂ → ℂ}
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{z₀ : ℂ} :
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z₀ ∈ Function.support (zeroDivisor f) ↔
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f z₀ = 0 ∧
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AnalyticAt ℂ f z₀ ∧
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∃ (g : ℂ → ℂ), AnalyticAt ℂ g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhds z₀, f z = (z - z₀) ^ (zeroDivisor f z₀) • g z := by
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constructor
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· intro hz
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constructor
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· exact zeroDivisor_zeroSet hz
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· constructor
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· exact analyticAtZeroDivisorSupport hz
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· exact zeroDivisor_localDescription hz
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· intro ⟨h₁, h₂, h₃⟩
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have : zeroDivisor f z₀ = (h₂.order).toNat := by
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unfold zeroDivisor
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simp [h₂]
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simp [this]
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simp [(h₂.order_eq_nat_iff (zeroDivisor f z₀)).2 h₃]
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := h₃
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rw [Filter.Eventually.self_of_nhds h₃g] at h₁
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simp [h₂g] at h₁
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assumption
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theorem topOnPreconnected
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ z ∈ U, f z ≠ 0) :
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∀ (hz : z ∈ U), (h₁f z hz).order ≠ ⊤ := by
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by_contra H
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push_neg at H
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obtain ⟨z', hz'⟩ := H
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rw [AnalyticAt.order_eq_top_iff] at hz'
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rw [← AnalyticAt.frequently_zero_iff_eventually_zero (h₁f z z')] at hz'
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have A := AnalyticOn.eqOn_zero_of_preconnected_of_frequently_eq_zero h₁f hU z' hz'
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tauto
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theorem supportZeroSet
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ z ∈ U, f z ≠ 0) :
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U ∩ Function.support (zeroDivisor f) = U ∩ f⁻¹' {0} := by
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ext x
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constructor
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· intro hx
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constructor
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· exact hx.1
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exact zeroDivisor_zeroSet hx.2
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· simp
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intro h₁x h₂x
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constructor
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· exact h₁x
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· let A := zeroDivisor_support_iff (f := f) (z₀ := x)
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simp at A
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rw [A]
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constructor
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· exact h₂x
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· constructor
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· exact h₁f x h₁x
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· have B := AnalyticAt.order_eq_nat_iff (h₁f x h₁x) (zeroDivisor f x)
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simp at B
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rw [← B]
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dsimp [zeroDivisor]
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simp [h₁f x h₁x]
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refine Eq.symm (ENat.coe_toNat ?h.mpr.right.right.right.a)
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exact topOnPreconnected hU h₁f h₂f h₁x
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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_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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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ z ∈ U, f z ≠ 0) -- not needed!
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(h₂U : IsCompact U) :
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Set.Finite (U ∩ Function.support (zeroDivisor f)) := by
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have hinter : IsCompact (U ∩ Function.support (zeroDivisor f)) := by
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apply IsCompact.of_isClosed_subset h₂U
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rw [supportZeroSet]
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apply h₁f.continuousOn.preimage_isClosed_of_isClosed
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exact IsCompact.isClosed h₂U
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exact isClosed_singleton
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assumption
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assumption
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assumption
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exact Set.inter_subset_left
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apply hinter.finite
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apply DiscreteTopology.of_subset (s := Function.support (zeroDivisor f))
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exact discreteZeros (f := f)
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exact Set.inter_subset_right
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theorem AnalyticOn.order_eq_nat_iff
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{f : ℂ → ℂ}
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{U : Set ℂ}
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{z₀ : ℂ}
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(hf : AnalyticOn ℂ f U)
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(hz₀ : z₀ ∈ U)
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(n : ℕ) :
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(hf z₀ hz₀).order = ↑n → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z ∈ U, f z = (z - z₀) ^ n • g z := by
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intro hn
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obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
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let g : ℂ → ℂ := fun z ↦ if hz : z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
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have t₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by
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rw [eventually_nhds_iff]
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rw [eventually_nhds_iff] at h₃gloc
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₃gloc
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use t
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constructor
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· intro y h₁y
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by_cases h₂y : y = z₀
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· dsimp [g]; simp [h₂y]
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· dsimp [g]; simp [h₂y]
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rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
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exact h₁t y h₁y
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norm_num
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rw [sub_eq_zero]
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tauto
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· constructor
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· assumption
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· assumption
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have t₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
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intro hz₁
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rw [eventually_nhds_iff]
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use {z₀}ᶜ
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constructor
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· intro y hy
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simp at hy
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dsimp [g]
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simp [hy]
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· constructor
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· exact isOpen_compl_singleton
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· tauto
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use g
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constructor
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· -- AnalyticOn ℂ g U
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intro z h₁z
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by_cases h₂z : z = z₀
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· rw [h₂z]
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apply AnalyticAt.congr h₁gloc
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exact Filter.EventuallyEq.symm t₀
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· simp_rw [eq_comm] at t₁
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apply AnalyticAt.congr _ (t₁ h₂z)
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apply AnalyticAt.div
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exact hf z h₁z
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apply AnalyticAt.pow
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apply AnalyticAt.sub
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apply analyticAt_id
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exact analyticAt_const
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simp
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rw [sub_eq_zero]
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tauto
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theorem eliminatingZeros
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁U : IsPreconnected U)
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(h₂U : IsCompact U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ z ∈ U, f z ≠ 0) :
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∃ F : ℂ → ℂ, (AnalyticOn ℂ F U) ∧ (f = F * ∏ᶠ a ∈ (U ∩ (zeroDivisor f).support), fun z ↦ (z - a) ^ (zeroDivisor f a)) := by
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have hs := zeroDivisor_finiteOnCompact h₁U h₁f h₂f h₂U
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rw [finprod_mem_eq_finite_toFinset_prod _ hs]
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let A := hs.card_eq
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let F : ℂ → ℂ := by
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intro z
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if hz : z ∈ U ∩ (zeroDivisor f).support then
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exact (Classical.choose (zeroDivisor_support_iff.1 hz.2).2.2) z
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else
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exact (f z) / ∏ i ∈ hs.toFinset, (z - i) ^ zeroDivisor f i
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use F
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constructor
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· intro z h₁z
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by_cases h₂z : z ∈ (zeroDivisor f).support
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· -- case: z ∈ Function.support (zeroDivisor f)
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have : z ∈ U ∩ (zeroDivisor f).support := by exact Set.mem_inter h₁z h₂z
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sorry
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· sorry
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have : MeromorphicOn F U := by
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apply MeromorphicOn.div
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exact AnalyticOn.meromorphicOn h₁f
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apply AnalyticOn.meromorphicOn
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apply Finset.analyticOn_prod
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intro n hn
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apply AnalyticOn.pow
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apply AnalyticOn.sub
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exact analyticOn_id ℂ
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exact analyticOn_const
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--use F
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sorry
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