66 lines
2.7 KiB
Lean4
66 lines
2.7 KiB
Lean4
import Mathlib.Analysis.Meromorphic.Basic
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import Mathlib.Analysis.Calculus.Deriv.Mul
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import Mathlib.Analysis.Calculus.Deriv.ZPow
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import Mathlib.Analysis.Calculus.Deriv.Shift
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open MeromorphicOn Real Set Classical Topology
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set_option checkBinderAnnotations false
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theorem eventually_eventually_nhdsWithin₁
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{α : Type*} [TopologicalSpace α] [T1Space α] {a : α} {p : α → Prop} :
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(∀ᶠ (y : α) in 𝓝[≠] a, ∀ᶠ x in 𝓝 y, p x) ↔ ∀ᶠ x in 𝓝[≠] a, p x := by
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nth_rw 2 [← eventually_eventually_nhdsWithin]
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constructor
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· intro h
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filter_upwards [h] with y hy
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exact eventually_nhdsWithin_of_eventually_nhds hy
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· intro h
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filter_upwards [h, eventually_nhdsWithin_of_forall fun _ a ↦ a] with y h₁y h₂y
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simp_all [IsOpen.nhdsWithin_eq]
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theorem Filter.EventuallyEq.nhdsNE_deriv
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{𝕜 : Type u} [NontriviallyNormedField 𝕜]
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{F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {x : 𝕜}
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(h : f₁ =ᶠ[𝓝[≠] x] f) :
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deriv f₁ =ᶠ[𝓝[≠] x] deriv f := by
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rw [eventuallyEq_nhdsWithin_iff, eventually_nhds_iff] at *
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := h
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use t
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simp_all only [mem_compl_iff, mem_singleton_iff, and_self, and_true]
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intro y h₁y h₂y
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rw [Filter.EventuallyEq.deriv_eq]
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apply eventually_nhds_iff.2
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use t \ {x}
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simp_all [IsOpen.sdiff h₂t isClosed_singleton]
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variable
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{𝕜 : Type*} [NontriviallyNormedField 𝕜]
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{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E]
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{U : Set 𝕜} {f g : 𝕜 → E} {a : WithTop E} {a₀ : E}
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/-- Derivatives of meromorphic functions are meromorphic. -/
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@[fun_prop]
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theorem meromorphicAt_deriv {f : 𝕜 → E} {x : 𝕜} (h : MeromorphicAt f x) :
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MeromorphicAt (deriv f) x := by
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rw [MeromorphicAt.iff_eventuallyEq_zpow_smul_analyticAt] at h
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obtain ⟨n, g, h₁g, h₂g⟩ := h
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have : deriv (fun z ↦ (z - x) ^ n • g z)
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=ᶠ[nhdsWithin x {x}ᶜ] fun z₀ ↦ (n * (z₀ - x) ^ (n - 1)) • g z₀ + (z₀ - x) ^ n • deriv g z₀ := by
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have τ₀ : ∀ᶠ (y : 𝕜) in nhdsWithin x {x}ᶜ, AnalyticAt 𝕜 g y := by
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exact eventually_nhdsWithin_of_eventually_nhds h₁g.eventually_analyticAt
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filter_upwards [eventually_nhdsWithin_of_eventually_nhds h₁g.eventually_analyticAt,
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eventually_nhdsWithin_of_forall fun _ a ↦ a] with z₀ h₁ h₂
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rw [deriv_smul (DifferentiableAt.zpow (by fun_prop) (by simp_all [sub_ne_zero_of_ne h₂])) (by fun_prop),
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add_comm, deriv_comp_sub_const (f := fun z ↦ z ^ n)]
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aesop
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have : deriv f =ᶠ[nhdsWithin x {x}ᶜ] deriv (fun z ↦ (z - x) ^ n • g z) := by
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sorry
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apply MeromorphicAt.congr _ this.symm
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fun_prop
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