nevanlinna/Nevanlinna/stronglyMeromorphicOn_elimi...

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import Mathlib.Analysis.Analytic.Meromorphic
import Nevanlinna.analyticAt
import Nevanlinna.divisor
import Nevanlinna.meromorphicAt
import Nevanlinna.meromorphicOn_divisor
import Nevanlinna.stronglyMeromorphicOn
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import Nevanlinna.mathlibAddOn
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open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral
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theorem MeromorphicOn.decompose₁
{f : }
{U : Set }
{z₀ : }
(h₁f : MeromorphicOn f U)
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(h₂f : StronglyMeromorphicAt f z₀)
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(h₃f : h₂f.meromorphicAt.order ≠ )
(hz₀ : z₀ ∈ U) :
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∃ g : , (MeromorphicOn g U)
∧ (AnalyticAt g z₀)
∧ (g z₀ ≠ 0)
∧ (f = g * fun z ↦ (z - z₀) ^ (h₁f.divisor z₀)) := by
let h₁ := fun z ↦ (z - z₀) ^ (-h₁f.divisor z₀)
have h₁h₁ : MeromorphicOn h₁ U := by
apply MeromorphicOn.zpow
apply AnalyticOnNhd.meromorphicOn
apply AnalyticOnNhd.sub
exact analyticOnNhd_id
exact analyticOnNhd_const
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let n : := (-h₁f.divisor z₀)
have h₂h₁ : (h₁h₁ z₀ hz₀).order = n := by
simp_rw [(h₁h₁ z₀ hz₀).order_eq_int_iff]
use 1
constructor
· apply analyticAt_const
· constructor
· simp
· apply eventually_nhdsWithin_of_forall
intro z hz
simp
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let g₁ := f * h₁
have h₁g₁ : MeromorphicOn g₁ U := by
apply h₁f.mul h₁h₁
have h₂g₁ : (h₁g₁ z₀ hz₀).order = 0 := by
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rw [(h₁f z₀ hz₀).order_mul (h₁h₁ z₀ hz₀)]
rw [h₂h₁]
unfold n
rw [MeromorphicOn.divisor_def₂ h₁f hz₀ h₃f]
conv =>
left
left
rw [Eq.symm (WithTop.coe_untop (h₁f z₀ hz₀).order h₃f)]
have
(a b c : )
(h : a + b = c) :
(a : WithTop ) + (b : WithTop ) = (c : WithTop ) := by
rw [← h]
simp
rw [this ((h₁f z₀ hz₀).order.untop h₃f) (-(h₁f z₀ hz₀).order.untop h₃f) 0]
simp
ring
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let g := (h₁g₁ z₀ hz₀).makeStronglyMeromorphicAt
have h₂g : StronglyMeromorphicAt g z₀ := by
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exact StronglyMeromorphicAt_of_makeStronglyMeromorphic (h₁g₁ z₀ hz₀)
have h₁g : MeromorphicOn g U := by
intro z hz
by_cases h₁z : z = z₀
· rw [h₁z]
apply h₂g.meromorphicAt
· apply (h₁g₁ z hz).congr
rw [eventuallyEq_nhdsWithin_iff]
rw [eventually_nhds_iff]
use {z₀}ᶜ
constructor
· intro y h₁y h₂y
let A := m₁ (h₁g₁ z₀ hz₀) y h₁y
unfold g
rw [← A]
· constructor
· exact isOpen_compl_singleton
· exact h₁z
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have h₃g : (h₁g z₀ hz₀).order = 0 := by
unfold g
let B := m₂ (h₁g₁ z₀ hz₀)
let A := (h₁g₁ z₀ hz₀).order_congr B
rw [← A]
rw [h₂g₁]
have h₄g : AnalyticAt g z₀ := by
apply h₂g.analytic
rw [h₃g]
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use g
constructor
· exact h₁g
· constructor
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· exact h₄g
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· constructor
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· exact (h₂g.order_eq_zero_iff).mp h₃g
· funext z
by_cases hz : z = z₀
· rw [hz]
simp
by_cases h : h₁f.divisor z₀ = 0
· simp [h]
have h₂h₁ : h₁ = 1 := by
funext w
unfold h₁
simp [h]
have h₃g₁ : g₁ = f := by
unfold g₁
rw [h₂h₁]
simp
have h₄g₁ : StronglyMeromorphicAt g₁ z₀ := by
rwa [h₃g₁]
let A := h₄g₁.makeStronglyMeromorphic_id
unfold g
rw [← A, h₃g₁]
· have : (0 : ) ^ h₁f.divisor z₀ = (0 : ) := by
exact zero_zpow (h₁f.divisor z₀) h
rw [this]
simp
let A := h₂f.order_eq_zero_iff.not
simp at A
rw [← A]
unfold MeromorphicOn.divisor at h
simp [hz₀] at h
exact h.1
· simp
let B := m₁ (h₁g₁ z₀ hz₀) z hz
unfold g
rw [← B]
unfold g₁ h₁
simp [hz]
rw [mul_assoc]
rw [inv_mul_cancel₀]
simp
apply zpow_ne_zero
rwa [sub_ne_zero]
theorem MeromorphicOn.decompose₂
{f : }
{U : Set }
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{P : Finset U}
(hf : StronglyMeromorphicOn f U) :
(∀ p ∈ P, (hf p p.2).meromorphicAt.order ≠ ) →
∃ g : , (MeromorphicOn g U)
∧ (∀ p : P, AnalyticAt g p)
∧ (∀ p : P, g p ≠ 0)
∧ (f = g * ∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) := by
apply Finset.induction (p := fun (P : Finset U) ↦
(∀ p ∈ P, (hf p p.2).meromorphicAt.order ≠ ) →
∃ g : , (MeromorphicOn g U)
∧ (∀ p : P, AnalyticAt g p)
∧ (∀ p : P, g p ≠ 0)
∧ (f = g * ∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)))
-- case empty
simp
exact hf.meromorphicOn
-- case insert
intro u P hu iHyp
intro hOrder
obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀, h₄g₀⟩ := iHyp (fun p hp ↦ hOrder p (Finset.mem_insert_of_mem hp))
have h₅g₀ : StronglyMeromorphicAt g₀ u := by
rw [stronglyMeromorphicAt_of_mul_analytic (g := ∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) (z₀ := u) (f := g₀)]
rw [← h₄g₀]
exact hf u u.2
--
have : (∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) = (fun z => ∏ p : P, (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) := by
funext w
simp
rw [this]
apply Finset.analyticAt_prod
intro p hp
apply AnalyticAt.zpow
apply AnalyticAt.sub
apply analyticAt_id
apply analyticAt_const
--
by_contra hCon
rw [sub_eq_zero] at hCon
have : p.1 = u := by
exact SetCoe.ext (_root_.id (Eq.symm hCon))
rw [← this] at hu
simp [hp] at hu
--
simp only [Finset.prod_apply]
rw [Finset.prod_ne_zero_iff]
intro p hp
apply zpow_ne_zero
by_contra hCon
rw [sub_eq_zero] at hCon
have : p.1 = u := by
exact SetCoe.ext (_root_.id (Eq.symm hCon))
rw [← this] at hu
simp [hp] at hu
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have h₆g₀ : (h₁g₀ u u.2).order ≠ := by
sorry
obtain ⟨g, h₁g, h₂g, h₃g, h₄g⟩ := h₁g₀.decompose₁ h₅g₀ h₆g₀ u.2
use g
· constructor
· exact h₁g
· constructor
· sorry
· constructor
· sorry
· sorry