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Nevanlinna/analyticOnNhd_divisor.lean

@@ -1,6 +1,3 @@
-import Mathlib.Analysis.SpecialFunctions.Integrals
-import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
-import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
 import Nevanlinna.analyticAt
 import Nevanlinna.divisor
 
@@ -8,8 +5,6 @@ open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral
 
 
-
-
 noncomputable def AnalyticOnNhd.zeroDivisor
   {f : ℂ → ℂ}
   {U : Set ℂ}
@@ -25,99 +20,40 @@ noncomputable def AnalyticOnNhd.zeroDivisor
 
   supportInU := by
     intro z hz
-    simp only [Function.mem_support] at hz
+    simp at hz
-    simp only [Function.mem_support, ne_eq, dite_eq_else, Nat.cast_eq_zero, ENat.toNat_eq_zero, not_forall, not_or] at hz
+    by_contra h₂z
+    simp [h₂z] at hz
 
+  locallyFiniteInU := by
+    intro z hz
 
-    discreteSupport := by
+    apply eventually_nhdsWithin_iff.2
-      simp_rw [← singletons_open_iff_discrete]
+    rw [eventually_nhds_iff]
-      simp_rw [Metric.isOpen_singleton_iff]
-      simp
 
-      --  simp only [dite_eq_ite, gt_iff_lt, Subtype.forall, Function.mem_support, ne_eq, ite_eq_else, Classical.not_imp, not_or, Subtype.mk.injEq]
+    rcases AnalyticAt.eventually_eq_zero_or_eventually_ne_zero (hf z hz) with h|h
-
+    · rw [eventually_nhds_iff] at h
-      intro a ha ⟨h₁a, h₂a⟩
+      obtain ⟨N, h₁N, h₂N, h₃N⟩ := h
-      obtain ⟨g, h₁g, h₂g, h₃g⟩ := (AnalyticAt.order_neq_top_iff (hf a ha)).1 h₂a
+      use N
-      rw [Metric.eventually_nhds_iff_ball] at h₃g
-
-      have : ∃ ε > 0, ∀ y ∈ Metric.ball (↑a) ε, g y ≠ 0 := by
-        have h₄g : ContinuousAt g a :=
-          AnalyticAt.continuousAt h₁g
-        have : {0}ᶜ ∈ nhds (g a) := by
-          exact compl_singleton_mem_nhds_iff.mpr h₂g
-        let F := h₄g.preimage_mem_nhds this
-        rw [Metric.mem_nhds_iff] at F
-        obtain ⟨ε, h₁ε, h₂ε⟩ := F
-        use ε
-        constructor; exact h₁ε
-        intro y hy
-        let G := h₂ε hy
-        simp at G
-        exact G
-
-      obtain ⟨ε₁, h₁ε₁⟩ := this
-      obtain ⟨ε₂, h₁ε₂, h₂ε₂⟩ := h₃g
-      use min ε₁ ε₂
       constructor
-      · have : 0 < min ε₁ ε₂ := by
+      · intro y h₁y _
-          rw [lt_min_iff]
+        by_cases h₃y : y ∈ U
-          exact And.imp_right (fun _ => h₁ε₂) h₁ε₁
+        · simp [h₃y]
-        exact this
+          right
+          rw [AnalyticAt.order_eq_top_iff (hf y h₃y)]
+          rw [eventually_nhds_iff]
+          use N
+        · simp [h₃y]
+      · tauto
 
-      intro y hy ⟨h₁y, h₂y⟩ h₃
+    · rw [eventually_nhdsWithin_iff, eventually_nhds_iff] at h
-
+      obtain ⟨N, h₁N, h₂N, h₃N⟩ := h
-      have h'₂y : ↑y ∈ Metric.ball (↑a) ε₂ := by
+      use N
-        simp
+      constructor
-        calc dist y a
+      · intro y h₁y h₂y
-        _ < min ε₁ ε₂ := by assumption
+        by_cases h₃y : y ∈ U
-        _ ≤ ε₂ := by exact min_le_right ε₁ ε₂
+        · simp [h₃y]
-
+          left
-      have h₃y : ↑y ∈ Metric.ball (↑a) ε₁ := by
+          rw [AnalyticAt.order_eq_zero_iff (hf y h₃y)]
-        simp
+          exact h₁N y h₁y h₂y
-        calc dist y a
+        · simp [h₃y]
-        _ < min ε₁ ε₂ := by assumption
+      · tauto
-        _ ≤ ε₁ := by exact min_le_left ε₁ ε₂
-
-      have F := h₂ε₂ y h'₂y
-      have : f y = 0 := by
-        rw [AnalyticAt.order_eq_zero_iff (hf y hy)] at h₁y
-        tauto
-      rw [this] at F
-      simp at F
-
-      have : g y ≠ 0 := by
-        exact h₁ε₁.2 y h₃y
-      simp [this] at F
-      rw [sub_eq_zero] at F
-      tauto
-
-
-theorem AnalyticOnNhd.support_of_zeroDivisor
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  (hf : AnalyticOnNhd ℂ f U) :
-  Function.support hf.zeroDivisor ⊆ U := by
-
-  intro z
-  contrapose
-  intro hz
-  dsimp [AnalyticOnNhd.zeroDivisor]
-  simp
-  tauto
-
-
-theorem AnalyticOnNhd.support_of_zeroDivisor₂
-  {f : ℂ → ℂ}
-  {U : Set ℂ}
-  (hf : AnalyticOnNhd ℂ f U) :
-  Function.support hf.zeroDivisor ⊆ f⁻¹' {0} := by
-
-  intro z hz
-  dsimp [AnalyticOnNhd.zeroDivisor] at hz
-  have t₀ := hf.support_of_zeroDivisor hz
-  simp [hf.support_of_zeroDivisor hz] at hz
-  let A := hz.1
-  let C := (hf z t₀).order_eq_zero_iff
-  simp
-  rw [C] at A
-  tauto

Nevanlinna/divisor.lean

@@ -12,7 +12,7 @@ structure Divisor
   where
   toFun : ℂ → ℤ
   supportInU : toFun.support ⊆ U
-  locallyFiniteInU : ∀ x ∈ U, ∃  N ∈ 𝓝 x, (N ∩ toFun.support).Finite
+  locallyFiniteInU : ∀ x ∈ U, toFun =ᶠ[𝓝[≠] x] 0
 
 instance
   (U : Set ℂ) :