Update meromorphicAt.lean
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Stefan Kebekus
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@ -232,13 +232,14 @@ theorem MeromorphicAt.order_add
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{z₀ : ℂ}
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(hf₁ : MeromorphicAt f₁ z₀)
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(hf₂ : MeromorphicAt f₂ z₀) :
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(hf₁.add hf₂).order ≤ min hf₁.order hf₂.order := by
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min hf₁.order hf₂.order ≤ (hf₁.add hf₂).order := by
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-- Handle the trivial cases where one of the orders equals ⊤
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by_cases h₂f₁: hf₁.order = ⊤
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· rw [h₂f₁]; simp
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rw [hf₁.order_eq_top_iff] at h₂f₁
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have h : f₁ + f₂ =ᶠ[𝓝[≠] z₀] f₂ := by
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-- Optimize this, here an elsewhere
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rw [eventuallyEq_nhdsWithin_iff, eventually_iff_exists_mem]
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rw [eventually_nhdsWithin_iff, eventually_iff_exists_mem] at h₂f₁
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obtain ⟨v, hv⟩ := h₂f₁
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@ -258,7 +259,7 @@ theorem MeromorphicAt.order_add
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obtain ⟨g₂, h₁g₂, h₂g₂, h₃g₂⟩ := hf₂.order_neg_zero_iff.1 h₂f₂
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let n₁ := WithTop.untop' 0 hf₁.order
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let n₂ := WithTop.untop' 0 hf₁.order
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let n₂ := WithTop.untop' 0 hf₂.order
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let n := min n₁ n₂
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have h₁n₁ : 0 ≤ n₁ - n := by
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rw [sub_nonneg]
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@ -277,7 +278,7 @@ theorem MeromorphicAt.order_add
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apply analyticAt_const
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exact h₁g₁
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apply AnalyticAt.mul
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apply AnalyticAt.zpow_nonneg _ h₁n₁
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apply AnalyticAt.zpow_nonneg _ h₁n₂
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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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@ -285,18 +286,37 @@ theorem MeromorphicAt.order_add
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have h₂g : 0 ≤ h₁g.meromorphicAt.order := h₁g.meromorphicAt_order_nonneg
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have : f₁ + f₂ =ᶠ[𝓝[≠] z₀] (fun z ↦ (z - z₀) ^ n) * g := by
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sorry
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rw [eventuallyEq_nhdsWithin_iff, eventually_nhds_iff]
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obtain ⟨t, ht⟩ := eventually_nhds_iff.1 (eventually_nhdsWithin_iff.1 (h₃g₁.and h₃g₂))
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use t
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simp [ht]
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intro y h₁y h₂y
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rw [(ht.1 y h₁y h₂y).1, (ht.1 y h₁y h₂y).2]
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unfold g; simp; rw [mul_add]
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repeat rw [←mul_assoc, ← zpow_add' (by left; exact (sub_ne_zero_of_ne h₂y))]
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simp
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rw [(hf₁.add hf₂).order_congr this]
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have t₀ : MeromorphicAt (fun z ↦ (z - z₀) ^ n) z₀ := by
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sorry
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apply MeromorphicAt.zpow
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apply MeromorphicAt.sub
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apply MeromorphicAt.id
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apply MeromorphicAt.const
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rw [t₀.order_mul h₁g.meromorphicAt]
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have t₁ : t₀.order = n := by
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sorry
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rw [t₀.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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· simp
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rw [t₁]
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-- Exercise in WithTop
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sorry
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unfold n n₁ n₂
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have : hf₁.order ⊓ hf₂.order = (WithTop.untop' 0 hf₁.order ⊓ WithTop.untop' 0 hf₂.order) := by
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rw [←untop'_of_ne_top (d := 0) h₂f₁, ←untop'_of_ne_top (d := 0) h₂f₂]
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simp
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rw [this]
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exact le_add_of_nonneg_right h₂g
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theorem MeromorphicAt.order_add_of_ne_orders
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@ -305,7 +325,134 @@ theorem MeromorphicAt.order_add_of_ne_orders
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(hf₁ : MeromorphicAt f₁ z₀)
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(hf₂ : MeromorphicAt f₂ z₀)
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(hf₁₂ : hf₁.order ≠ hf₂.order) :
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(hf₁.add hf₂).order ≤ hf₁.order + hf₂.order := by
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sorry
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min hf₁.order hf₂.order = (hf₁.add hf₂).order := by
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-- Handle the trivial cases where one of the orders equals ⊤
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by_cases h₂f₁: hf₁.order = ⊤
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· rw [h₂f₁]; simp
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rw [hf₁.order_eq_top_iff] at h₂f₁
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have h : f₁ + f₂ =ᶠ[𝓝[≠] z₀] f₂ := by
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-- Optimize this, here an elsewhere
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rw [eventuallyEq_nhdsWithin_iff, eventually_iff_exists_mem]
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rw [eventually_nhdsWithin_iff, eventually_iff_exists_mem] at h₂f₁
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obtain ⟨v, hv⟩ := h₂f₁
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use v; simp; trivial
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rw [(hf₁.add hf₂).order_congr h]
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by_cases h₂f₂: hf₂.order = ⊤
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· rw [h₂f₂]; simp
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rw [hf₂.order_eq_top_iff] at h₂f₂
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have h : f₁ + f₂ =ᶠ[𝓝[≠] z₀] f₁ := by
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rw [eventuallyEq_nhdsWithin_iff, eventually_iff_exists_mem]
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rw [eventually_nhdsWithin_iff, eventually_iff_exists_mem] at h₂f₂
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obtain ⟨v, hv⟩ := h₂f₂
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use v; simp; trivial
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rw [(hf₁.add hf₂).order_congr h]
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obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := hf₁.order_neg_zero_iff.1 h₂f₁
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obtain ⟨g₂, h₁g₂, h₂g₂, h₃g₂⟩ := hf₂.order_neg_zero_iff.1 h₂f₂
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let n₁ := WithTop.untop' 0 hf₁.order
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let n₂ := WithTop.untop' 0 hf₂.order
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have hn₁₂ : n₁ ≠ n₂ := by
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unfold n₁ n₂
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let A := WithTop.untop'_eq_untop'_iff (d := 0) (x := hf₁.order) (y := hf₂.order)
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let B := A.not
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simp
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rw [B]
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push_neg
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constructor
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· assumption
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· tauto
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let n := min n₁ n₂
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have h₁n₁ : 0 ≤ n₁ - n := by
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rw [sub_nonneg]
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exact Int.min_le_left n₁ n₂
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have h₁n₂ : 0 ≤ n₂ - n := by
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rw [sub_nonneg]
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exact Int.min_le_right n₁ n₂
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let g := (fun z ↦ (z - z₀) ^ (n₁ - n)) * g₁ + (fun z ↦ (z - z₀) ^ (n₂ - n)) * g₂
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have h₁g : AnalyticAt ℂ g z₀ := by
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apply AnalyticAt.add
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apply AnalyticAt.mul
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apply AnalyticAt.zpow_nonneg _ h₁n₁
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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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exact h₁g₁
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apply AnalyticAt.mul
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apply AnalyticAt.zpow_nonneg _ h₁n₂
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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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exact h₁g₂
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have h₂g : 0 ≤ h₁g.meromorphicAt.order := h₁g.meromorphicAt_order_nonneg
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have h₂'g : g z₀ ≠ 0 := by
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unfold g
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simp
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have : n = n₁ ∨ n = n₂ := by
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unfold n
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simp
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by_cases h : n₁ ≤ n₂
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· left; assumption
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· right
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simp at h
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exact le_of_lt h
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rcases this with h|h
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· rw [h]
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have : n₂ - n₁ ≠ 0 := by
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rw [sub_ne_zero, ne_comm]
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apply hn₁₂
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have : (0 : ℂ) ^ (n₂ - n₁) = 0 := by
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rwa [zpow_eq_zero_iff]
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simp [this]
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exact h₂g₁
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· rw [h]
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have : n₁ - n₂ ≠ 0 := by
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rw [sub_ne_zero]
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apply hn₁₂
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have : (0 : ℂ) ^ (n₁ - n₂) = 0 := by
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rwa [zpow_eq_zero_iff]
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simp [this]
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exact h₂g₂
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have h₃g : h₁g.meromorphicAt.order = 0 := by
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let A := h₁g.meromorphicAt_order
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let B := h₁g.order_eq_zero_iff.2 h₂'g
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rw [B] at A
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simpa
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have : f₁ + f₂ =ᶠ[𝓝[≠] z₀] (fun z ↦ (z - z₀) ^ n) * g := by
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rw [eventuallyEq_nhdsWithin_iff, eventually_nhds_iff]
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obtain ⟨t, ht⟩ := eventually_nhds_iff.1 (eventually_nhdsWithin_iff.1 (h₃g₁.and h₃g₂))
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use t
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simp [ht]
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intro y h₁y h₂y
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rw [(ht.1 y h₁y h₂y).1, (ht.1 y h₁y h₂y).2]
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unfold g; simp; rw [mul_add]
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repeat rw [←mul_assoc, ← zpow_add' (by left; exact (sub_ne_zero_of_ne h₂y))]
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simp
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rw [(hf₁.add hf₂).order_congr this]
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have t₀ : MeromorphicAt (fun z ↦ (z - z₀) ^ n) z₀ := by
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apply MeromorphicAt.zpow
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apply MeromorphicAt.sub
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apply MeromorphicAt.id
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apply MeromorphicAt.const
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rw [t₀.order_mul h₁g.meromorphicAt]
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have t₁ : t₀.order = n := by
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rw [t₀.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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· simp
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rw [t₁]
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unfold n n₁ n₂
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have : hf₁.order ⊓ hf₂.order = (WithTop.untop' 0 hf₁.order ⊓ WithTop.untop' 0 hf₂.order) := by
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rw [←untop'_of_ne_top (d := 0) h₂f₁, ←untop'_of_ne_top (d := 0) h₂f₂]
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simp
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rw [this, h₃g]
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simp
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-- might want theorem MeromorphicAt.order_zpow
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