70 lines
1.8 KiB
Plaintext
70 lines
1.8 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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import Mathlib.MeasureTheory.Integral.CircleIntegral
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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lemma b
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(S U : Set ℂ)
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(hS : S ∈ Filter.codiscreteWithin U) :
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DiscreteTopology ((S ∪ Uᶜ)ᶜ : Set ℂ) := by
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rw [mem_codiscreteWithin] at hS
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simp at hS
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have : (U \ S)ᶜ = S ∪ Uᶜ := by
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ext z
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simp
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tauto
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rw [discreteTopology_subtype_iff]
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intro x hx
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rw [← mem_iff_inf_principal_compl]
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simp at hx
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let A := hS x hx.2
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rw [← this]
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assumption
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lemma c
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(S U : Set ℂ)
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(hS : S ∈ Filter.codiscreteWithin U) :
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Countable ((S ∪ Uᶜ)ᶜ : Set ℂ) := by
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let A := b S U hS
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apply TopologicalSpace.separableSpace_iff_countable.1
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exact TopologicalSpace.SecondCountableTopology.to_separableSpace
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theorem integrability_congr_changeDiscrete
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{f₁ f₂ : ℂ → ℂ}
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{U : Set ℂ}
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{r : ℝ}
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(hf : f₁ =ᶠ[Filter.codiscreteWithin U] f₂) :
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IntervalIntegrable (f₁ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) → IntervalIntegrable (f₂ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) := by
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intro hf₁
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apply IntervalIntegrable.congr hf₁
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rw [Filter.eventuallyEq_iff_exists_mem]
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use (circleMap 0 r)⁻¹' ({z | f₁ z = f₂ z} ∩ U)
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constructor
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· apply Set.Countable.measure_zero
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have : (circleMap 0 r ⁻¹' ({z | f₁ z = f₂ z} ∩ U))ᶜ = (circleMap 0 r ⁻¹' ({z | f₁ z = f₂ z} ∩ U)ᶜ) := by
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exact rfl
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rw [this]
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apply Set.Countable.preimage_circleMap
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apply c
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sorry
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sorry
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· intro x hx
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simp at hx
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simp
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exact hx.1
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