Working

Nevanlinna/meromorphicOn_integrability.lean

@@ -12,39 +12,58 @@ open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral
 
 
-lemma a
-  (S : Set ℂ)
-  (hS : S ∈ Filter.codiscreteWithin ⊤) :
-  DiscreteTopology (Sᶜ : Set ℂ) := by
+lemma b
+  (S U : Set ℂ)
+  (hS : S ∈ Filter.codiscreteWithin U) :
+  DiscreteTopology ((S ∪ Uᶜ)ᶜ : Set ℂ) := by
 
   rw [mem_codiscreteWithin] at hS
   simp at hS
-  have : (Set.univ \ S)ᶜ = S := by ext z; simp
-  rw [this] at hS
+  have : (U \ S)ᶜ = S ∪ Uᶜ := by
+    ext z
+    simp
+    tauto
 
   rw [discreteTopology_subtype_iff]
   intro x hx
   rw [← mem_iff_inf_principal_compl]
-  exact (hS x)
+
+  simp at hx
+  let A := hS x hx.2
+  rw [← this]
+  assumption
+
+
+lemma c
+  (S U : Set ℂ)
+  (hS : S ∈ Filter.codiscreteWithin U) :
+  Countable ((S ∪ Uᶜ)ᶜ : Set ℂ) := by
+  let A := b S U hS
+  apply TopologicalSpace.separableSpace_iff_countable.1
+  exact TopologicalSpace.SecondCountableTopology.to_separableSpace
 
 
 theorem integrability_congr_changeDiscrete
   {f₁ f₂ : ℂ → ℂ}
+  {U : Set ℂ}
   {r : ℝ}
-  (hf : f₁ =ᶠ[Filter.codiscreteWithin ⊤] f₂) :
+  (hf : f₁ =ᶠ[Filter.codiscreteWithin U] f₂) :
   IntervalIntegrable (f₁ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) → IntervalIntegrable (f₂ ∘ (circleMap 0 r)) MeasureTheory.volume 0 (2 * π) := by
   intro hf₁
 
   apply IntervalIntegrable.congr hf₁
   rw [Filter.eventuallyEq_iff_exists_mem]
-  use (circleMap 0 r)⁻¹' { z | f₁ z = f₂ z}
+  use (circleMap 0 r)⁻¹' ({z | f₁ z = f₂ z} ∩ U)
   constructor
   · apply Set.Countable.measure_zero
-    have : (circleMap 0 r ⁻¹' {z | f₁ z = f₂ z})ᶜ = (circleMap 0 r ⁻¹' {z | f₁ z = f₂ z}ᶜ) := by
+    have : (circleMap 0 r ⁻¹' ({z | f₁ z = f₂ z} ∩ U))ᶜ = (circleMap 0 r ⁻¹' ({z | f₁ z = f₂ z} ∩ U)ᶜ) := by
       exact rfl
     rw [this]
     apply Set.Countable.preimage_circleMap
-
+    apply c
     sorry
     sorry
-  · sorry
+  · intro x hx
+    simp at hx
+    simp
+    exact hx.1