working...

Nevanlinna/meromorphicOn.lean

@@ -93,7 +93,6 @@ theorem MeromorphicOn.open_of_order_neq_top
       · exact h₃t'
 
 
-
 theorem MeromorphicOn.clopen_of_order_eq_top
   {f : ℂ → ℂ}
   {U : Set ℂ}
@@ -104,3 +103,29 @@ theorem MeromorphicOn.clopen_of_order_eq_top
   · rw [← isOpen_compl_iff]
     exact open_of_order_neq_top h₁f
   · exact open_of_order_eq_top h₁f
+
+
+theorem MeromorphicOn.order_ne_top
+  {U : Set ℂ}
+  (hU : IsConnected U)
+  (h₁f : StronglyMeromorphicOn f U)
+  (h₂f : ∃ u : U, f u ≠ 0) :
+  ∀ u : U, (h₁f u u.2).meromorphicAt.order ≠ ⊤ := by
+
+  let A := h₁f.meromorphicOn.clopen_of_order_eq_top
+  have : PreconnectedSpace U := by
+    apply isPreconnected_iff_preconnectedSpace.mp
+    exact IsConnected.isPreconnected hU
+  rw [isClopen_iff] at A
+  rcases A with h|h
+  · intro u
+    have : u ∉ (∅ : Set U) := by exact fun a => a
+    rw [← h] at this
+    simp at this
+    tauto
+  · obtain ⟨u, hu⟩ := h₂f
+    have : u ∈ (Set.univ : Set U) := by trivial
+    rw [← h] at this
+    simp at this
+    rw [(h₁f u u.2).order_eq_zero_iff.2 hu] at this
+    tauto

Nevanlinna/stronglyMeromorphicOn.lean

@@ -1,3 +1,4 @@
+import Nevanlinna.meromorphicOn_divisor
 import Nevanlinna.stronglyMeromorphicAt
 import Mathlib.Algebra.BigOperators.Finprod
 
@@ -79,79 +80,74 @@ theorem makeStronglyMeromorphicOn_changeDiscrete
     · simp [h₂v]
 
 
-theorem stronglyMeromorphicOn_ratlPolynomial
+theorem analyticAt_ratlPolynomial₁
+  {z : ℂ}
+  (d : ℂ → ℤ)
+  (P : Finset ℂ) :
+  z ∉ P → AnalyticAt ℂ (∏ u ∈ P, fun z ↦ (z - u) ^ d u) z := by
+  intro hz
+  rw [Finset.prod_fn]
+  apply Finset.analyticAt_prod
+  intro u hu
+  apply AnalyticAt.zpow
+  apply AnalyticAt.sub
+  apply analyticAt_id
+  apply analyticAt_const
+  rw [sub_ne_zero, ne_comm]
+  exact ne_of_mem_of_not_mem hu hz
+
+
+theorem stronglyMeromorphicOn_ratlPolynomial₂
+  {U : Set ℂ}
+  (d : ℂ → ℤ)
+  (P : Finset ℂ) :
+  StronglyMeromorphicOn (∏ u ∈ P, fun z ↦ (z - u) ^ d u) U := by
+
+  intro z hz
+  by_cases h₂z : z ∈ P
+  · rw [← Finset.mul_prod_erase P _ h₂z]
+    right
+    use d z
+    use ∏ x ∈ P.erase z, fun z => (z - x) ^ d x
+    constructor
+    · have : z ∉ P.erase z := Finset.not_mem_erase z P
+      apply analyticAt_ratlPolynomial₁ d (P.erase z) this
+    · constructor
+      · simp only [Finset.prod_apply]
+        rw [Finset.prod_ne_zero_iff]
+        intro u hu
+        apply zpow_ne_zero
+        rw [sub_ne_zero]
+        by_contra hCon
+        rw [hCon] at hu
+        let A := Finset.not_mem_erase u P
+        tauto
+      · exact Filter.Eventually.of_forall (congrFun rfl)
+  · apply AnalyticAt.stronglyMeromorphicAt
+    exact analyticAt_ratlPolynomial₁ d P (z := z) h₂z
+
+
+
+theorem stronglyMeromorphicOn_ratlPolynomial₃
   {U : Set ℂ}
   (d : ℂ → ℤ) :
   StronglyMeromorphicOn (∏ᶠ u, fun z ↦ (z - u) ^ d u) U := by
   by_cases hd : (Function.mulSupport fun u z => (z - u) ^ d u).Finite
   · rw [finprod_eq_prod _ hd]
-    intro z h₁z
-    by_cases h₂z : d z = 0
-    · apply AnalyticAt.stronglyMeromorphicAt
-      rw [Finset.prod_fn]
-      apply Finset.analyticAt_prod
-      intro u hu
-      by_cases huz : u = z
-      · rw [← huz] at h₂z
-        rw [h₂z]
-        simp
-        exact analyticAt_const
-      · apply AnalyticAt.zpow
-        apply AnalyticAt.sub
-        apply analyticAt_id
-        apply analyticAt_const
-        rwa [sub_ne_zero, ne_comm]
-    · have : z ∈ hd.toFinset := by
-        simp
-        by_contra hCon
-        have A : 0 ≠ (1 : ℂ → ℂ) z := by simp
-        rw [← hCon] at A
-        simp only [sub_self] at A
-        rw [ne_comm] at A
-        rw [zpow_ne_zero_iff] at A
-        tauto
-        exact h₂z
-      rw [← Finset.mul_prod_erase hd.toFinset _ this]
-      right
-      use d z
-      use ∏ x ∈ hd.toFinset.erase z, fun z => (z - x) ^ d x
-      constructor
-      · rw [Finset.prod_fn]
-        apply Finset.analyticAt_prod
-        intro u hu
-        apply AnalyticAt.zpow
-        apply AnalyticAt.sub
-        apply analyticAt_id
-        apply analyticAt_const
-        rw [sub_ne_zero, ne_comm]
-        by_contra hCon
-        simp at hu
-        tauto
-      · constructor
-        · simp only [Finset.prod_apply]
-          rw [Finset.prod_ne_zero_iff]
-          intro u hu
-          rw [zpow_ne_zero_iff]
-          rw [sub_ne_zero]
-          by_contra hCon
-          rw [hCon] at hu
-          let A := Finset.not_mem_erase u hd.toFinset
-          tauto
-          --
-          have : u ∈ hd.toFinset := by
-            exact Finset.mem_of_mem_erase hu
-          simp at this
-          by_contra hCon
-          rw [hCon] at this
-          simp at this
-          tauto
-        · exact Filter.Eventually.of_forall (congrFun rfl)
-
+    apply stronglyMeromorphicOn_ratlPolynomial₂ (U := U) d hd.toFinset
   · rw [finprod_of_infinite_mulSupport hd]
     apply AnalyticOn.stronglyMeromorphicOn
     apply analyticOnNhd_const
 
 
+theorem stronglyMeromorphicOn_divisor_ratlPolynomial
+  {U : Set ℂ}
+  (d : ℂ → ℤ)
+  (hd : Set.Finite d.support) :
+  (stronglyMeromorphicOn_ratlPolynomial₃ d (U := U)).meromorphicOn.divisor = d := by
+  sorry
+
+
 theorem makeStronglyMeromorphicOn_changeDiscrete'
   {f : ℂ → ℂ}
   {U : Set ℂ}

Nevanlinna/stronglyMeromorphicOn_eliminate.lean

@@ -8,7 +8,6 @@ import Nevanlinna.meromorphicOn_divisor
 import Nevanlinna.stronglyMeromorphicOn
 import Nevanlinna.mathlibAddOn
 
-
 open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral
 
@@ -304,27 +303,10 @@ theorem MeromorphicOn.decompose₃
   ∃ g : ℂ → ℂ, (MeromorphicOn g U)
     ∧ (AnalyticOn ℂ g U)
     ∧ (∀ u : U, g u ≠ 0)
-    ∧ (f = g * ∏ᶠ u : U, fun z ↦ (z - u.1) ^ (h₁f.meromorphicOn.divisor u.1)) := by
+    ∧ (f = g * ∏ᶠ u, fun z ↦ (z - u) ^ (h₁f.meromorphicOn.divisor u)) := by
 
   have h₃f : ∀ u : U, (h₁f u u.2).meromorphicAt.order ≠ ⊤ := by
-    let A := h₁f.meromorphicOn.clopen_of_order_eq_top
-    have : PreconnectedSpace U := by
-      apply isPreconnected_iff_preconnectedSpace.mp
-      exact IsConnected.isPreconnected h₂U
-    rw [isClopen_iff] at A
-    rcases A with h|h
-    · intro u
-      have : u ∉ (∅ : Set U) := by exact fun a => a
-      rw [← h] at this
-      simp at this
-      tauto
-    · obtain ⟨u, hu⟩ := h₂f
-      let A := (h₁f u u.2).order_eq_zero_iff.2 hu
-      have : u ∈ (Set.univ : Set U) := by trivial
-      rw [← h] at this
-      simp at this
-      rw [A] at this
-      tauto
+    apply MeromorphicOn.order_ne_top h₂U h₁f h₂f
 
   have h₄f : Set.Finite (Function.support h₁f.meromorphicOn.divisor) := by
     exact h₁f.meromorphicOn.divisor.finiteSupport h₁U
@@ -336,34 +318,14 @@ theorem MeromorphicOn.decompose₃
     apply h₃f
 
   obtain ⟨g, h₁g, h₂g, h₃g, h₄g⟩ := MeromorphicOn.decompose₂ h₁f (P := P) hP
+
   let h := ∏ p ∈ P, fun z => (z - p.1) ^ h₁f.meromorphicOn.divisor p.1
+
   have h₂h : StronglyMeromorphicOn h U := by
-    -- WARNING: This is a general lemma we should add!
-    intro u hu
-    right
-    by_cases h₁u : ⟨u, hu⟩ ∈ P
-    · sorry
-    · use 0
-      use h
-      simp only [zpow_zero, smul_eq_mul, one_mul, eventually_true, and_true]
-      constructor
-      ·
-        sorry
-      · unfold h
-        simp only [Finset.prod_apply]
-        rw [Finset.prod_ne_zero_iff]
-        intro p hp
-        apply zpow_ne_zero
-        rw [sub_ne_zero]
-        by_contra hCon
-        have : ⟨u, hu⟩ = p := by
-          exact SetCoe.ext hCon
-        rw [← this] at hp
-        tauto
-
-
+    unfold h
+    apply stronglyMeromorphicOn_ratlPolynomial₂
   have h₁h : MeromorphicOn h U := by
-    sorry
+    exact StronglyMeromorphicOn.meromorphicOn h₂h
   have h₃h : h₁h.divisor = h₁f.meromorphicOn.divisor := by
     sorry