Update stronglyMeromorphicOn_ratlPolynomial.lean
This commit is contained in:
Stefan Kebekus
parent
5f85934bbe
commit
0e88e97295
@ -1,344 +1,167 @@
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import Mathlib.Analysis.Analytic.Meromorphic
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import Nevanlinna.analyticAt
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import Nevanlinna.divisor
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import Nevanlinna.meromorphicAt
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import Nevanlinna.meromorphicOn
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import Nevanlinna.meromorphicOn_divisor
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import Nevanlinna.stronglyMeromorphicOn
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import Nevanlinna.mathlibAddOn
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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theorem MeromorphicOn.decompose₁
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theorem analyticAt_ratlPolynomial₁
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{z : ℂ}
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(d : ℂ → ℤ)
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(P : Finset ℂ) :
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z ∉ P → AnalyticAt ℂ (∏ u ∈ P, fun z ↦ (z - u) ^ d u) z := by
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intro hz
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rw [Finset.prod_fn]
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apply Finset.analyticAt_prod
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intro u hu
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apply AnalyticAt.zpow
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apply AnalyticAt.sub
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apply analyticAt_id
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apply analyticAt_const
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rw [sub_ne_zero, ne_comm]
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exact ne_of_mem_of_not_mem hu hz
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theorem stronglyMeromorphicOn_ratlPolynomial₂
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(d : ℂ → ℤ)
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(P : Finset ℂ) :
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StronglyMeromorphicOn (∏ u ∈ P, fun z ↦ (z - u) ^ d u) ⊤ := by
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intro z hz
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by_cases h₂z : z ∈ P
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· rw [← Finset.mul_prod_erase P _ h₂z]
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right
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use d z
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use ∏ x ∈ P.erase z, fun z => (z - x) ^ d x
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constructor
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· have : z ∉ P.erase z := Finset.not_mem_erase z P
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apply analyticAt_ratlPolynomial₁ d (P.erase z) this
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· constructor
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· simp only [Finset.prod_apply]
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rw [Finset.prod_ne_zero_iff]
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intro u hu
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apply zpow_ne_zero
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rw [sub_ne_zero]
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by_contra hCon
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rw [hCon] at hu
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let A := Finset.not_mem_erase u P
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tauto
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· exact Filter.Eventually.of_forall (congrFun rfl)
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· apply AnalyticAt.stronglyMeromorphicAt
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exact analyticAt_ratlPolynomial₁ d P (z := z) h₂z
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theorem stronglyMeromorphicOn_ratlPolynomial₃
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(d : ℂ → ℤ) :
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StronglyMeromorphicOn (∏ᶠ u, fun z ↦ (z - u) ^ d u) ⊤ := by
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by_cases hd : (Function.mulSupport fun u z => (z - u) ^ d u).Finite
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· rw [finprod_eq_prod _ hd]
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apply stronglyMeromorphicOn_ratlPolynomial₂ d hd.toFinset
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· rw [finprod_of_infinite_mulSupport hd]
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apply AnalyticOn.stronglyMeromorphicOn
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apply analyticOnNhd_const
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theorem stronglyMeromorphicOn_divisor_ratlPolynomial₁
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{z : ℂ}
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(d : ℂ → ℤ)
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(h₁d : Set.Finite d.support) :
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(((stronglyMeromorphicOn_ratlPolynomial₃ d).meromorphicOn) z trivial).order = d z := by
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have h₂d : (Function.mulSupport fun u z ↦ (z - u) ^ d u) = d.support := by
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ext u
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constructor
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· intro h
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simp at h
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simp
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by_contra hCon
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rw [hCon] at h
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simp at h
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tauto
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· intro h
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simp
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by_contra hCon
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let A := congrFun hCon u
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simp at A
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have t₁ : (0 : ℂ) ^ d u ≠ 0 := by
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exact ne_zero_of_eq_one A
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rw [zpow_ne_zero_iff h] at t₁
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tauto
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rw [MeromorphicAt.order_eq_int_iff]
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use ∏ x ∈ h₁d.toFinset.erase z, fun z => (z - x) ^ d x
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constructor
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· have : z ∉ h₁d.toFinset.erase z := Finset.not_mem_erase z h₁d.toFinset
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apply analyticAt_ratlPolynomial₁ d (h₁d.toFinset.erase z) this
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· constructor
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· simp only [Finset.prod_apply]
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rw [Finset.prod_ne_zero_iff]
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intro u hu
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apply zpow_ne_zero
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rw [sub_ne_zero]
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by_contra hCon
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rw [hCon] at hu
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let A := Finset.not_mem_erase u h₁d.toFinset
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tauto
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· apply Filter.Eventually.of_forall
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intro x
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simp
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rw [← Finset.mul_prod_erase]
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sorry
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by_cases hz : z ∈ d.support
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· sorry
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· use (∏ᶠ u, fun z ↦ (z - u) ^ d u)
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constructor
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·
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apply?
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simp [hz]
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sorry
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· rw [Function.nmem_support.mp fun a => hz (h₂d a)]
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rw [MeromorphicOn.divisor]
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simp [hz]
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theorem stronglyMeromorphicOn_divisor_ratlPolynomial
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{U : Set ℂ}
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(d : ℂ → ℤ)
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(h₁d : Set.Finite d.support)
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(h₂d : d.support ⊆ U) :
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(stronglyMeromorphicOn_ratlPolynomial₃ d (U := U)).meromorphicOn.divisor = d := by
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funext z
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by_cases hz : z ∈ U
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· rw [MeromorphicOn.divisor]
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simp [hz]
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sorry
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· rw [Function.nmem_support.mp fun a => hz (h₂d a)]
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rw [MeromorphicOn.divisor]
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simp [hz]
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theorem makeStronglyMeromorphicOn_changeDiscrete'
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{f : ℂ → ℂ}
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{U : Set ℂ}
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{z₀ : ℂ}
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(h₁f : MeromorphicOn f U)
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(h₂f : StronglyMeromorphicAt f z₀)
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(h₃f : h₂f.meromorphicAt.order ≠ ⊤)
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(hf : MeromorphicOn f U)
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(hz₀ : z₀ ∈ U) :
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∃ g : ℂ → ℂ, (MeromorphicOn g U)
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∧ (AnalyticAt ℂ g z₀)
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∧ (g z₀ ≠ 0)
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∧ (f = g * fun z ↦ (z - z₀) ^ (h₁f.divisor z₀)) := by
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let h₁ := fun z ↦ (z - z₀) ^ (-h₁f.divisor z₀)
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have h₁h₁ : MeromorphicOn h₁ U := by
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apply MeromorphicOn.zpow
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apply AnalyticOnNhd.meromorphicOn
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apply AnalyticOnNhd.sub
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exact analyticOnNhd_id
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exact analyticOnNhd_const
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let n : ℤ := (-h₁f.divisor z₀)
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have h₂h₁ : (h₁h₁ z₀ hz₀).order = n := by
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simp_rw [(h₁h₁ z₀ hz₀).order_eq_int_iff]
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use 1
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constructor
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· apply analyticAt_const
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· constructor
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· simp
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· apply eventually_nhdsWithin_of_forall
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intro z hz
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simp
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let g₁ := f * h₁
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have h₁g₁ : MeromorphicOn g₁ U := by
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apply h₁f.mul h₁h₁
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have h₂g₁ : (h₁g₁ z₀ hz₀).order = 0 := by
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rw [(h₁f z₀ hz₀).order_mul (h₁h₁ z₀ hz₀)]
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rw [h₂h₁]
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unfold n
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rw [MeromorphicOn.divisor_def₂ h₁f hz₀ h₃f]
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conv =>
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left
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left
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rw [Eq.symm (WithTop.coe_untop (h₁f z₀ hz₀).order h₃f)]
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have
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(a b c : ℤ)
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(h : a + b = c) :
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(a : WithTop ℤ) + (b : WithTop ℤ) = (c : WithTop ℤ) := by
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rw [← h]
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simp
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rw [this ((h₁f z₀ hz₀).order.untop h₃f) (-(h₁f z₀ hz₀).order.untop h₃f) 0]
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simp
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ring
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let g := (h₁g₁ z₀ hz₀).makeStronglyMeromorphicAt
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have h₂g : StronglyMeromorphicAt g z₀ := by
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exact StronglyMeromorphicAt_of_makeStronglyMeromorphic (h₁g₁ z₀ hz₀)
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have h₁g : MeromorphicOn g U := by
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intro z hz
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by_cases h₁z : z = z₀
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· rw [h₁z]
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apply h₂g.meromorphicAt
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· apply (h₁g₁ z hz).congr
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rw [eventuallyEq_nhdsWithin_iff]
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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 h₁y h₂y
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let A := m₁ (h₁g₁ z₀ hz₀) y h₁y
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unfold g
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rw [← A]
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· constructor
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· exact isOpen_compl_singleton
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· exact h₁z
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have h₃g : (h₁g z₀ hz₀).order = 0 := by
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unfold g
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let B := m₂ (h₁g₁ z₀ hz₀)
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let A := (h₁g₁ z₀ hz₀).order_congr B
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rw [← A]
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rw [h₂g₁]
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have h₄g : AnalyticAt ℂ g z₀ := by
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apply h₂g.analytic
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rw [h₃g]
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use g
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constructor
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· exact h₁g
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· constructor
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· exact h₄g
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· constructor
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· exact (h₂g.order_eq_zero_iff).mp h₃g
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· funext z
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by_cases hz : z = z₀
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· rw [hz]
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simp
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by_cases h : h₁f.divisor z₀ = 0
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· simp [h]
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have h₂h₁ : h₁ = 1 := by
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funext w
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unfold h₁
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simp [h]
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have h₃g₁ : g₁ = f := by
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unfold g₁
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rw [h₂h₁]
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simp
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have h₄g₁ : StronglyMeromorphicAt g₁ z₀ := by
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rwa [h₃g₁]
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let A := h₄g₁.makeStronglyMeromorphic_id
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unfold g
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rw [← A, h₃g₁]
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· have : (0 : ℂ) ^ h₁f.divisor z₀ = (0 : ℂ) := by
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exact zero_zpow (h₁f.divisor z₀) h
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rw [this]
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simp
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let A := h₂f.order_eq_zero_iff.not
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simp at A
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rw [← A]
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unfold MeromorphicOn.divisor at h
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simp [hz₀] at h
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exact h.1
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· simp
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let B := m₁ (h₁g₁ z₀ hz₀) z hz
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unfold g
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rw [← B]
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unfold g₁ h₁
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simp [hz]
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rw [mul_assoc]
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rw [inv_mul_cancel₀]
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simp
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apply zpow_ne_zero
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rwa [sub_ne_zero]
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hf.makeStronglyMeromorphicOn =ᶠ[𝓝 z₀] (hf z₀ hz₀).makeStronglyMeromorphicAt := by
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apply Mnhds
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let A := makeStronglyMeromorphicOn_changeDiscrete hf hz₀
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apply Filter.EventuallyEq.trans A
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exact m₂ (hf z₀ hz₀)
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unfold MeromorphicOn.makeStronglyMeromorphicOn
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simp [hz₀]
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theorem MeromorphicOn.decompose₂
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theorem StronglyMeromorphicOn_of_makeStronglyMeromorphicOn
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{f : ℂ → ℂ}
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{U : Set ℂ}
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{P : Finset U}
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(hf : StronglyMeromorphicOn f U) :
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(∀ p ∈ P, (hf p p.2).meromorphicAt.order ≠ ⊤) →
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∃ g : ℂ → ℂ, (MeromorphicOn g U)
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∧ (∀ p : P, AnalyticAt ℂ g p)
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∧ (∀ p : P, g p ≠ 0)
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∧ (f = g * ∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) := by
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apply Finset.induction (p := fun (P : Finset U) ↦
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(∀ p ∈ P, (hf p p.2).meromorphicAt.order ≠ ⊤) →
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∃ g : ℂ → ℂ, (MeromorphicOn g U)
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∧ (∀ p : P, AnalyticAt ℂ g p)
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∧ (∀ p : P, g p ≠ 0)
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∧ (f = g * ∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)))
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-- case empty
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simp
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exact hf.meromorphicOn
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-- case insert
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intro u P hu iHyp
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intro hOrder
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obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀, h₄g₀⟩ := iHyp (fun p hp ↦ hOrder p (Finset.mem_insert_of_mem hp))
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have h₀ : AnalyticAt ℂ (∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) u := by
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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
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funext w
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simp
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rw [this]
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apply Finset.analyticAt_prod
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intro p hp
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apply AnalyticAt.zpow
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apply AnalyticAt.sub
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apply analyticAt_id
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apply analyticAt_const
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by_contra hCon
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rw [sub_eq_zero] at hCon
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have : p.1 = u := by
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exact SetCoe.ext (_root_.id (Eq.symm hCon))
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rw [← this] at hu
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simp [hp] at hu
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have h₁ : (∏ p : P, fun z ↦ (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) u ≠ 0 := by
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simp only [Finset.prod_apply]
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rw [Finset.prod_ne_zero_iff]
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intro p hp
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apply zpow_ne_zero
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by_contra hCon
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rw [sub_eq_zero] at hCon
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have : p.1 = u := by
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exact SetCoe.ext (_root_.id (Eq.symm hCon))
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rw [← this] at hu
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simp [hp] at hu
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have h₅g₀ : StronglyMeromorphicAt g₀ u := by
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rw [stronglyMeromorphicAt_of_mul_analytic h₀ h₁]
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rw [← h₄g₀]
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exact hf u u.2
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have h₆g₀ : (h₁g₀ u u.2).order ≠ ⊤ := by
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by_contra hCon
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let A := (h₁g₀ u u.2).order_mul h₀.meromorphicAt
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simp_rw [← h₄g₀, hCon] at A
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simp at A
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let B := hOrder u (Finset.mem_insert_self u P)
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tauto
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obtain ⟨g, h₁g, h₂g, h₃g, h₄g⟩ := h₁g₀.decompose₁ h₅g₀ h₆g₀ u.2
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use g
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· constructor
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· exact h₁g
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· constructor
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· intro ⟨v₁, v₂⟩
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simp
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simp at v₂
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rcases v₂ with hv|hv
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· rwa [hv]
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· let A := h₂g₀ ⟨v₁, hv⟩
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rw [h₄g] at A
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rw [← analyticAt_of_mul_analytic] at A
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simp at A
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exact A
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--
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simp
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apply AnalyticAt.zpow
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apply AnalyticAt.sub
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apply analyticAt_id
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apply analyticAt_const
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by_contra hCon
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rw [sub_eq_zero] at hCon
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have : v₁ = u := by
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exact SetCoe.ext hCon
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rw [this] at hv
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tauto
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--
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apply zpow_ne_zero
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simp
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by_contra hCon
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rw [sub_eq_zero] at hCon
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have : v₁ = u := by
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exact SetCoe.ext hCon
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rw [this] at hv
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tauto
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· constructor
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· intro ⟨v₁, v₂⟩
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simp
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simp at v₂
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rcases v₂ with hv|hv
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· rwa [hv]
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· by_contra hCon
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let A := h₃g₀ ⟨v₁,hv⟩
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rw [h₄g] at A
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simp at A
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tauto
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· conv =>
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left
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rw [h₄g₀, h₄g]
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simp
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rw [mul_assoc]
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congr
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rw [Finset.prod_insert]
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simp
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congr
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have : (hf u u.2).meromorphicAt.order = (h₁g₀ u u.2).order := by
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have h₅g₀ : f =ᶠ[𝓝 u.1] (g₀ * ∏ p : P, fun z => (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) := by
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exact Eq.eventuallyEq h₄g₀
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have h₆g₀ : f =ᶠ[𝓝[≠] u.1] (g₀ * ∏ p : P, fun z => (z - p.1.1) ^ (hf.meromorphicOn.divisor p.1.1)) := by
|
||||
exact eventuallyEq_nhdsWithin_of_eqOn fun ⦃x⦄ a => congrFun h₄g₀ x
|
||||
rw [(hf u u.2).meromorphicAt.order_congr h₆g₀]
|
||||
let C := (h₁g₀ u u.2).order_mul h₀.meromorphicAt
|
||||
rw [C]
|
||||
let D := h₀.order_eq_zero_iff.2 h₁
|
||||
let E := h₀.meromorphicAt_order
|
||||
rw [E, D]
|
||||
simp
|
||||
have : hf.meromorphicOn.divisor u = h₁g₀.divisor u := by
|
||||
unfold MeromorphicOn.divisor
|
||||
simp
|
||||
rw [this]
|
||||
rw [this]
|
||||
--
|
||||
simpa
|
||||
|
||||
|
||||
theorem MeromorphicOn.decompose₃
|
||||
{f : ℂ → ℂ}
|
||||
{U : Set ℂ}
|
||||
(h₁U : IsCompact U)
|
||||
(h₂U : IsConnected U)
|
||||
(h₁f : StronglyMeromorphicOn f U)
|
||||
(h₂f : ∃ u : U, f u ≠ 0) :
|
||||
∃ g : ℂ → ℂ, (MeromorphicOn g U)
|
||||
∧ (AnalyticOn ℂ g U)
|
||||
∧ (∀ u : U, g u ≠ 0)
|
||||
∧ (f = g * ∏ᶠ u, fun z ↦ (z - u) ^ (h₁f.meromorphicOn.divisor u)) := by
|
||||
|
||||
have h₃f : ∀ u : U, (h₁f u u.2).meromorphicAt.order ≠ ⊤ := by
|
||||
apply MeromorphicOn.order_ne_top h₂U h₁f h₂f
|
||||
|
||||
have h₄f : Set.Finite (Function.support h₁f.meromorphicOn.divisor) := by
|
||||
exact h₁f.meromorphicOn.divisor.finiteSupport h₁U
|
||||
|
||||
let P' : Set U := Subtype.val ⁻¹' Function.support h₁f.meromorphicOn.divisor
|
||||
let P := (h₄f.preimage Set.injOn_subtype_val : Set.Finite P').toFinset
|
||||
have hP : ∀ p ∈ P, (h₁f p p.2).meromorphicAt.order ≠ ⊤ := by
|
||||
intro p hp
|
||||
apply h₃f
|
||||
|
||||
obtain ⟨g, h₁g, h₂g, h₃g, h₄g⟩ := MeromorphicOn.decompose₂ h₁f (P := P) hP
|
||||
|
||||
let h := ∏ p ∈ P, fun z => (z - p.1) ^ h₁f.meromorphicOn.divisor p.1
|
||||
|
||||
have h₂h : StronglyMeromorphicOn h U := by
|
||||
unfold h
|
||||
apply stronglyMeromorphicOn_ratlPolynomial₂
|
||||
have h₁h : MeromorphicOn h U := by
|
||||
exact StronglyMeromorphicOn.meromorphicOn h₂h
|
||||
have h₃h : h₁h.divisor = h₁f.meromorphicOn.divisor := by
|
||||
sorry
|
||||
|
||||
|
||||
use g
|
||||
constructor
|
||||
· exact h₁g
|
||||
· constructor
|
||||
· sorry
|
||||
· constructor
|
||||
· sorry
|
||||
· conv =>
|
||||
left
|
||||
rw [h₄g]
|
||||
congr
|
||||
simp
|
||||
sorry
|
||||
(hf : MeromorphicOn f U) :
|
||||
StronglyMeromorphicOn hf.makeStronglyMeromorphicOn U := by
|
||||
intro z₀ hz₀
|
||||
rw [stronglyMeromorphicAt_congr (makeStronglyMeromorphicOn_changeDiscrete' hf hz₀)]
|
||||
exact StronglyMeromorphicAt_of_makeStronglyMeromorphic (hf z₀ hz₀)
|
||||
|
||||
Loading…
Reference in New Issue
Block a user