Done with elimination
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Stefan Kebekus
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@ -45,6 +45,19 @@ theorem AnalyticOn.stronglyMeromorphicOn
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exact h₁f z hz
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theorem stronglyMeromorphicOn_of_mul_analytic'
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{f g : ℂ → ℂ}
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{U : Set ℂ}
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(h₁g : AnalyticOnNhd ℂ g U)
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(h₂g : ∀ u : U, g u ≠ 0)
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(h₁f : StronglyMeromorphicOn f U) :
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StronglyMeromorphicOn (g * f) U := by
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intro z hz
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rw [mul_comm]
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apply (stronglyMeromorphicAt_of_mul_analytic (h₁g z hz) ?h₂g).mp (h₁f z hz)
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exact h₂g ⟨z, hz⟩
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/- Make strongly MeromorphicOn -/
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noncomputable def MeromorphicOn.makeStronglyMeromorphicOn
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{f : ℂ → ℂ}
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@ -293,62 +293,6 @@ theorem MeromorphicOn.decompose₂
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simpa
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theorem MeromorphicOn.decompose₃
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁U : IsCompact U)
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(h₂U : IsConnected U)
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(h₁f : StronglyMeromorphicOn f U)
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(h₂f : ∃ u : U, f u ≠ 0) :
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∃ g : ℂ → ℂ, (MeromorphicOn g U)
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∧ (AnalyticOn ℂ g U)
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∧ (∀ u : U, g u ≠ 0)
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∧ (f = g * ∏ᶠ u, fun z ↦ (z - u) ^ (h₁f.meromorphicOn.divisor u)) := by
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have h₃f : ∀ u : U, (h₁f u u.2).meromorphicAt.order ≠ ⊤ := MeromorphicOn.order_ne_top h₂U h₁f h₂f
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have h₄f : Set.Finite (Function.support h₁f.meromorphicOn.divisor) := h₁f.meromorphicOn.divisor.finiteSupport h₁U
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let P' : Set U := Subtype.val ⁻¹' Function.support h₁f.meromorphicOn.divisor
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let P := (h₄f.preimage Set.injOn_subtype_val : Set.Finite P').toFinset
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have hP : ∀ p ∈ P, (h₁f p p.2).meromorphicAt.order ≠ ⊤ := by
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intro p hp
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apply h₃f
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obtain ⟨g, h₁g, h₂g, h₃g, h₄g⟩ := MeromorphicOn.decompose₂ h₁f (P := P) hP
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let h := ∏ p ∈ P, fun z => (z - p.1) ^ h₁f.meromorphicOn.divisor p.1
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have h₅g : AnalyticOn ℂ g U := by
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intro z hz
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by_cases h₂z : ⟨z, hz⟩ ∈ P
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· apply AnalyticAt.analyticWithinAt
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exact h₂g ⟨⟨z, hz⟩, h₂z⟩
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· apply AnalyticAt.analyticWithinAt
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rw [analyticAt_of_mul_analytic]
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sorry
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have h₂h : StronglyMeromorphicOn h U := by
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sorry
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have h₁h : MeromorphicOn h U := by
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exact StronglyMeromorphicOn.meromorphicOn h₂h
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have h₃h : h₁h.divisor = h₁f.meromorphicOn.divisor := by
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sorry
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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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· sorry
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· constructor
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· sorry
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· conv =>
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left
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rw [h₄g]
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congr
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simp
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sorry
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theorem MeromorphicOn.decompose₃'
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{f : ℂ → ℂ}
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@ -438,4 +382,111 @@ theorem MeromorphicOn.decompose₃'
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· exact AnalyticOnNhd.analyticOn h₃g
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· constructor
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· exact h₄g
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· sorry
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· have t₀ : StronglyMeromorphicOn (g * ∏ᶠ (u : ℂ), fun z => (z - u) ^ (h₁f.meromorphicOn.divisor u)) U := by
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apply stronglyMeromorphicOn_of_mul_analytic' h₃g h₄g
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apply stronglyMeromorphicOn_ratlPolynomial₃U
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funext z
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by_cases hz : z ∈ U
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· apply Filter.EventuallyEq.eq_of_nhds
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apply StronglyMeromorphicAt.localIdentity (h₁f z hz) (t₀ z hz)
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have h₅g : g =ᶠ[𝓝[≠] z] g' := makeStronglyMeromorphicOn_changeDiscrete h₁g' hz
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have Y' : (g' * ∏ᶠ (u : ℂ), fun z => (z - u) ^ (h₁f.meromorphicOn.divisor u)) =ᶠ[𝓝[≠] z] g * ∏ᶠ (u : ℂ), fun z => (z - u) ^ (h₁f.meromorphicOn.divisor u) := by
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apply Filter.EventuallyEq.symm
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exact Filter.EventuallyEq.mul h₅g (by simp)
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apply Filter.EventuallyEq.trans _ Y'
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unfold g'
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unfold h₁
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let A := h₁f.meromorphicOn.divisor.locallyFiniteInU z hz
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let B := eventually_nhdsWithin_iff.1 A
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 B
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apply eventually_nhdsWithin_iff.2
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rw [eventually_nhds_iff]
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use t
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constructor
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· intro y h₁y h₂y
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let C := h₁t y h₁y h₂y
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rw [mul_assoc]
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simp
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have : (finprod (fun u z => (z - u) ^ d u) y * finprod (fun u z => (z - u) ^ (h₁f.meromorphicOn.divisor u)) y) = 1 := by
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have t₀ : (Function.mulSupport fun u z => (z - u) ^ d u).Finite := by
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rwa [ratlPoly_mulsupport, h₁d]
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rw [finprod_eq_prod _ t₀]
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have t₁ : (Function.mulSupport fun u z => (z - u) ^ h₁f.meromorphicOn.divisor u).Finite := by
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rwa [ratlPoly_mulsupport]
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rw [finprod_eq_prod _ t₁]
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have : (Function.mulSupport fun u z => (z - u) ^ d u) = (Function.mulSupport fun u z => (z - u) ^ h₁f.meromorphicOn.divisor u) := by
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rw [ratlPoly_mulsupport]
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rw [ratlPoly_mulsupport]
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unfold d
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simp
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have : t₀.toFinset = t₁.toFinset := by congr
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rw [this]
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simp
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rw [← Finset.prod_mul_distrib]
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apply Finset.prod_eq_one
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intro x hx
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apply zpow_neg_mul_zpow_self
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have : y ∉ t₁.toFinset := by
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simp
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simp at C
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rw [C]
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simp
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tauto
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by_contra H
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rw [sub_eq_zero] at H
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rw [H] at this
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tauto
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rw [this]
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simp
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· exact ⟨h₂t, h₃t⟩
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· simp
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have : g z = g' z := by
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unfold g
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unfold MeromorphicOn.makeStronglyMeromorphicOn
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simp [hz]
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rw [this]
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unfold g'
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unfold h₁
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simp
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rw [mul_assoc]
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nth_rw 1 [← mul_one (f z)]
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congr
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have t₀ : (Function.mulSupport fun u z => (z - u) ^ d u).Finite := by
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rwa [ratlPoly_mulsupport, h₁d]
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rw [finprod_eq_prod _ t₀]
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have t₁ : (Function.mulSupport fun u z => (z - u) ^ h₁f.meromorphicOn.divisor u).Finite := by
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rwa [ratlPoly_mulsupport]
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rw [finprod_eq_prod _ t₁]
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have : (Function.mulSupport fun u z => (z - u) ^ d u) = (Function.mulSupport fun u z => (z - u) ^ h₁f.meromorphicOn.divisor u) := by
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rw [ratlPoly_mulsupport]
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rw [ratlPoly_mulsupport]
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unfold d
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simp
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have : t₀.toFinset = t₁.toFinset := by congr
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rw [this]
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simp
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rw [← Finset.prod_mul_distrib]
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rw [eq_comm]
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apply Finset.prod_eq_one
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intro x hx
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apply zpow_neg_mul_zpow_self
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have : z ∉ t₁.toFinset := by
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simp
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have : h₁f.meromorphicOn.divisor z = 0 := by
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let A := h₁f.meromorphicOn.divisor.supportInU
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simp at A
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by_contra H
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let B := A z H
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tauto
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rw [this]
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simp
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rfl
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by_contra H
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rw [sub_eq_zero] at H
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rw [H] at this
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tauto
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@ -71,13 +71,9 @@ theorem stronglyMeromorphicOn_ratlPolynomial₃U
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exact stronglyMeromorphicOn_ratlPolynomial₃ d z (trivial)
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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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theorem ratlPoly_mulsupport
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(d : ℂ → ℤ) :
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(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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@ -96,6 +92,13 @@ theorem stronglyMeromorphicOn_divisor_ratlPolynomial₁
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rw [zpow_ne_zero_iff h] at t₁
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tauto
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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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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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@ -114,11 +117,12 @@ theorem stronglyMeromorphicOn_divisor_ratlPolynomial₁
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· apply Filter.Eventually.of_forall
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intro x
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have t₀ : (Function.mulSupport fun u z => (z - u) ^ d u).Finite := by
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rwa [h₂d]
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rwa [ratlPoly_mulsupport d]
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rw [finprod_eq_prod _ t₀]
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have t₁ : h₁d.toFinset = t₀.toFinset := by
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simp
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rwa [eq_comm]
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rw [eq_comm]
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exact ratlPoly_mulsupport d
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rw [t₁]
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simp
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rw [eq_comm]
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