114 lines
2.9 KiB
Plaintext
114 lines
2.9 KiB
Plaintext
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_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.open_of_order_eq_top
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁f : MeromorphicOn f U) :
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IsOpen { u : U | (h₁f u.1 u.2).order = ⊤ } := by
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apply isOpen_iff_forall_mem_open.mpr
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intro z hz
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simp at hz
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have h₁z := hz
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rw [MeromorphicAt.order_eq_top_iff] at hz
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rw [eventually_nhdsWithin_iff] at hz
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rw [eventually_nhds_iff] at hz
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obtain ⟨t', h₁t', h₂t', h₃t'⟩ := hz
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let t : Set U := Subtype.val ⁻¹' t'
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use t
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constructor
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· intro w hw
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simp
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by_cases h₁w : w = z
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· rwa [h₁w]
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· --
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rw [MeromorphicAt.order_eq_top_iff]
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rw [eventually_nhdsWithin_iff]
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rw [eventually_nhds_iff]
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use t' \ {z.1}
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constructor
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· intro y h₁y h₂y
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apply h₁t'
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exact Set.mem_of_mem_diff h₁y
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exact Set.mem_of_mem_inter_right h₁y
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· constructor
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· apply IsOpen.sdiff h₂t'
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exact isClosed_singleton
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· rw [Set.mem_diff]
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constructor
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· exact hw
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· simp
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exact Subtype.coe_ne_coe.mpr h₁w
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· constructor
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· exact isOpen_induced h₂t'
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· exact h₃t'
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theorem MeromorphicOn.open_of_order_neq_top
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁f : MeromorphicOn f U) :
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IsOpen { u : U | (h₁f u.1 u.2).order ≠ ⊤ } := by
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apply isOpen_iff_forall_mem_open.mpr
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intro z hz
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simp at hz
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let A := (h₁f z.1 z.2).eventually_eq_zero_or_eventually_ne_zero
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rcases A with h|h
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· rw [← (h₁f z.1 z.2).order_eq_top_iff] at h
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tauto
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· let A := (h₁f z.1 z.2).eventually_analyticAt
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let B := Filter.Eventually.and h A
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rw [eventually_nhdsWithin_iff] at B
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rw [eventually_nhds_iff] at B
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obtain ⟨t', h₁t', h₂t', h₃t'⟩ := B
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let t : Set U := Subtype.val ⁻¹' t'
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use t
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constructor
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· intro w hw
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simp
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by_cases h₁w : w = z
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· rwa [h₁w]
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· let B := h₁t' w hw
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simp at B
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have : (w : ℂ) ≠ (z : ℂ) := by exact Subtype.coe_ne_coe.mpr h₁w
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let C := B this
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let D := C.2.order_eq_zero_iff.2 C.1
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rw [C.2.meromorphicAt_order, D]
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simp
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· constructor
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· exact isOpen_induced h₂t'
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· exact h₃t'
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theorem MeromorphicOn.order_ne_top
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁U : IsConnected U)
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(h₁f : MeromorphicOn f U) :
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(∃ z₀ : U, (h₁f z₀.1 z₀.2).order = ⊤) ↔ (∀ z : U, (h₁f z.1 z.2).order = ⊤) := by
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constructor
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· intro h
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obtain ⟨h₁z₀, h₂z₀⟩ := h
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intro hz
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sorry
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· intro h
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obtain ⟨w, hw⟩ := h₁U.nonempty
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use ⟨w, hw⟩
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exact h ⟨w, hw⟩
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