working…

Nevanlinna/firstMain.lean

@@ -16,6 +16,15 @@ theorem MeromorphicOn.restrict
   MeromorphicOn f (Metric.closedBall 0 r) := by
   exact fun x a => h₁f x trivial
 
+theorem MeromorphicOn.restrict_inv
+  {f : ℂ → ℂ}
+  (h₁f : MeromorphicOn f ⊤)
+  (r : ℝ) :
+  h₁f.inv.restrict r = (h₁f.restrict r).inv := by
+  funext x
+  simp
+
+
 noncomputable def MeromorphicOn.N_zero
   {f : ℂ → ℂ}
   (hf : MeromorphicOn f ⊤) :
@@ -28,6 +37,43 @@ noncomputable def MeromorphicOn.N_infty
   ℝ → ℝ :=
   fun r ↦ ∑ᶠ z, (max 0 (-((hf.restrict r).divisor z))) * log (r * ‖z‖⁻¹)
 
+theorem Nevanlinna_counting₀
+  {f : ℂ → ℂ}
+  (hf : MeromorphicOn f ⊤) :
+  hf.inv.N_infty = hf.N_zero := by
+  funext r
+  unfold MeromorphicOn.N_zero MeromorphicOn.N_infty
+  let A := (hf.restrict r).divisor.finiteSupport (isCompact_closedBall 0 r)
+  repeat
+    rw [finsum_eq_sum_of_support_subset (s := A.toFinset)]
+  apply Finset.sum_congr rfl
+  intro x hx
+  congr
+  rw [hf.restrict_inv r]
+  rw [MeromorphicOn.divisor_inv]
+  simp
+  --
+  exact fun x a => hf x trivial
+  --
+  intro x
+  contrapose
+  simp
+  intro hx
+  rw [hx]
+  tauto
+  --
+  intro x
+  contrapose
+  simp
+  intro hx h₁x
+  rw [hf.restrict_inv r] at h₁x
+  have hh : MeromorphicOn f (Metric.closedBall 0 r) := hf.restrict r
+  rw [hh.divisor_inv] at h₁x
+  simp at h₁x
+  rw [hx] at h₁x
+  tauto
+
+
 theorem Nevanlinna_counting
   {f : ℂ → ℂ}
   (hf : MeromorphicOn f ⊤) :
@@ -73,27 +119,6 @@ noncomputable def MeromorphicOn.m_infty
   ℝ → ℝ :=
   fun r ↦ (2 * π)⁻¹ * ∫ x in (0)..(2 * π), logpos ‖f (circleMap 0 r x)‖
 
-theorem Nevanlinna_proximity₀
-  {f : ℂ → ℂ}
-  {r : ℝ}
-  (h₁f : MeromorphicOn f ⊤)
-  (hr : 0 < r) :
-  (2 * π)⁻¹ * ∫ x in (0)..(2 * π), log ‖f (circleMap 0 r x)‖ = (h₁f.m_infty r) - (h₁f.inv.m_infty r) := by
-
-  unfold MeromorphicOn.m_infty
-  rw [← mul_sub]; congr
-  rw [← intervalIntegral.integral_sub]; congr
-  funext x
-  simp_rw [loglogpos]; congr
-  exact Eq.symm (IsAbsoluteValue.abv_inv Norm.norm (f (circleMap 0 r x)))
-  --
-  apply MeromorphicOn.integrable_logpos_abs_f hr
-  intro z hx
-  exact h₁f z trivial
-  --
-  apply MeromorphicOn.integrable_logpos_abs_f hr
-  exact MeromorphicOn.inv_iff.mpr fun x a => h₁f x trivial
-
 
 theorem Nevanlinna_proximity
   {f : ℂ → ℂ}
@@ -101,16 +126,6 @@ theorem Nevanlinna_proximity
   (h₁f : MeromorphicOn f ⊤) :
   (2 * π)⁻¹ * ∫ x in (0)..(2 * π), log ‖f (circleMap 0 r x)‖ = (h₁f.m_infty r) - (h₁f.inv.m_infty r) := by
 
-  by_cases h₁r : r = 0
-  · unfold MeromorphicOn.m_infty
-    rw [← mul_sub]; congr
-    simp only [h₁r, circleMap_zero_radius, Function.const_apply, intervalIntegral.integral_const, sub_zero, smul_eq_mul, Pi.inv_apply, norm_inv]
-    rw [← mul_sub]; congr
-    exact loglogpos
-
-  by_cases h₂r : 0 < r
-  · exact Nevanlinna_proximity₀ h₁f h₂r
-
   unfold MeromorphicOn.m_infty
   rw [← mul_sub]; congr
   rw [← intervalIntegral.integral_sub]; congr
@@ -119,30 +134,44 @@ theorem Nevanlinna_proximity
   exact Eq.symm (IsAbsoluteValue.abv_inv Norm.norm (f (circleMap 0 r x)))
   --
   apply MeromorphicOn.integrable_logpos_abs_f
+  intro z hx
+  exact h₁f z trivial
+  --
+  apply MeromorphicOn.integrable_logpos_abs_f
+  exact MeromorphicOn.inv_iff.mpr fun x a => h₁f x trivial
 
 
-  sorry
-  sorry
-
 noncomputable def MeromorphicOn.T_infty
   {f : ℂ → ℂ}
   (hf : MeromorphicOn f ⊤) :
   ℝ → ℝ :=
   hf.m_infty + hf.N_infty
 
+
 theorem Nevanlinna_firstMain₁
   {f : ℂ → ℂ}
   (h₁f : MeromorphicOn f ⊤)
   (h₂f : StronglyMeromorphicAt f 0)
   (h₃f : f 0 ≠ 0) :
   (fun r ↦ log ‖f 0‖) + h₁f.inv.T_infty = h₁f.T_infty := by
+
+  rw [add_eq_of_eq_sub]
+  unfold MeromorphicOn.T_infty
+
+  have {A B C D : ℝ → ℝ} : A + B - (C + D) = A - C + (B - D) := by
+    ring
+  rw [this]
+  clear this
+
   funext r
   simp
-  unfold MeromorphicOn.T_infty
+  rw [← Nevanlinna_proximity h₁f]
+
+
+
   unfold MeromorphicOn.N_infty
   unfold MeromorphicOn.m_infty
   simp
-
   sorry
 
 theorem Nevanlinna_firstMain₂

Nevanlinna/meromorphicAt.lean

@@ -147,4 +147,74 @@ theorem MeromorphicAt.order_mul
             · exact Set.mem_inter h₃t₁ h₃t₂
 
 
+theorem MeromorphicAt.order_neg_zero_iff
+  {f : ℂ → ℂ}
+  {z₀ : ℂ}
+  (hf : MeromorphicAt f z₀) :
+  hf.order ≠ ⊤ ↔ ∃ (g : ℂ → ℂ), AnalyticAt ℂ g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z = (z - z₀) ^ (hf.order.untop' 0) • g z := by
+
+  constructor
+  · intro h
+    exact (hf.order_eq_int_iff (hf.order.untop' 0)).1 (Eq.symm (untop'_of_ne_top h))
+  · rw [← hf.order_eq_int_iff]
+    intro h
+    exact Option.ne_none_iff_exists'.mpr ⟨hf.order.untop' 0, h⟩
+
+
+theorem MeromorphicAt.order_inv
+  {f : ℂ → ℂ}
+  {z₀ : ℂ}
+  (hf : MeromorphicAt f z₀) :
+  hf.order = -hf.inv.order := by
+
+  by_cases h₂f : hf.order = ⊤
+  · simp [h₂f]
+    have : hf.inv.order = ⊤ := by
+      rw [hf.inv.order_eq_top_iff]
+      rw [eventually_nhdsWithin_iff]
+      rw [eventually_nhds_iff]
+
+      rw [hf.order_eq_top_iff] at h₂f
+      rw [eventually_nhdsWithin_iff] at h₂f
+      rw [eventually_nhds_iff] at h₂f
+      obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f
+      use t
+      constructor
+      · intro y h₁y h₂y
+        simp
+        rw [h₁t y h₁y h₂y]
+      · exact ⟨h₂t, h₃t⟩
+    rw [this]
+    simp
+
+  · have : hf.order = hf.order.untop' 0 := by
+      simp [h₂f, untop'_of_ne_top]
+    rw [this]
+    rw [eq_comm]
+    rw [neg_eq_iff_eq_neg]
+    apply (hf.inv.order_eq_int_iff (-hf.order.untop' 0)).2
+    rw [hf.order_eq_int_iff] at this
+    obtain ⟨g, h₁g, h₂g, h₃g⟩ := this
+    use g⁻¹
+    constructor
+    · exact AnalyticAt.inv h₁g h₂g
+    · constructor
+      · simp [h₂g]
+      · rw [eventually_nhdsWithin_iff]
+        rw [eventually_nhds_iff]
+        rw [eventually_nhdsWithin_iff] at h₃g
+        rw [eventually_nhds_iff] at h₃g
+        obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₃g
+        use t
+        constructor
+        · intro y h₁y h₂y
+          simp
+          let A := h₁t y h₁y h₂y
+          rw [A]
+          simp
+          rw [mul_comm]
+        · exact ⟨h₂t, h₃t⟩
+
+
+
 -- might want theorem MeromorphicAt.order_zpow

Nevanlinna/meromorphicOn_divisor.lean

@@ -146,6 +146,29 @@ theorem MeromorphicOn.divisor_mul
     simp
 
 
+theorem MeromorphicOn.divisor_inv
+  {f: ℂ → ℂ}
+  {U : Set ℂ}
+  (h₁f : MeromorphicOn f U) :
+  h₁f.inv.divisor.toFun = -h₁f.divisor.toFun := by
+  funext z
+
+  by_cases hz : z ∈ U
+  · rw [MeromorphicOn.divisor_def₁]
+    simp
+    rw [MeromorphicOn.divisor_def₁]
+    rw [MeromorphicAt.order_inv]
+    simp
+    by_cases h₂f : (h₁f z hz).order = ⊤
+    · simp [h₂f]
+    · let A := untop'_of_ne_top (d := 0) h₂f
+      rw [← A]
+      exact rfl
+    repeat exact hz
+  · unfold MeromorphicOn.divisor
+    simp [hz]
+
+
 theorem MeromorphicOn.divisor_of_makeStronglyMeromorphicOn
   {f : ℂ → ℂ}
   {U : Set ℂ}

Nevanlinna/meromorphicOn_integrability.lean

@@ -144,7 +144,11 @@ theorem MeromorphicOn.integrable_log_abs_f
       rw [this] at h₁f
       exact MeromorphicOn.integrable_log_abs_f₀ h₂r h₁f
     · have t₀ : 0 < -r := by
-        sorry
+        have hr_neg : r < 0 := by
+          apply lt_of_le_of_ne
+          exact le_of_not_lt h₂r
+          exact h₁r
+        linarith
       have : |r| = -r := by
         apply abs_of_neg
         exact Left.neg_pos_iff.mp t₀
@@ -157,13 +161,11 @@ theorem MeromorphicOn.integrable_log_abs_f
       simpa
 
 
-
 theorem MeromorphicOn.integrable_logpos_abs_f
   {f : ℂ → ℂ}
   {r : ℝ}
-  (hr : 0 < r)
   -- WARNING: Not optimal. It suffices to be meromorphic on the Sphere
-  (h₁f : MeromorphicOn f (Metric.closedBall (0 : ℂ) r)) :
+  (h₁f : MeromorphicOn f (Metric.closedBall (0 : ℂ) |r|)) :
   IntervalIntegrable (fun z ↦ logpos ‖f (circleMap 0 r z)‖) MeasureTheory.volume 0 (2 * π) := by
 
   simp_rw [logpos_norm]
@@ -171,8 +173,8 @@ theorem MeromorphicOn.integrable_logpos_abs_f
 
   apply IntervalIntegrable.add
   apply IntervalIntegrable.const_mul
-  exact MeromorphicOn.integrable_log_abs_f hr h₁f
+  exact MeromorphicOn.integrable_log_abs_f h₁f
 
   apply IntervalIntegrable.const_mul
   apply IntervalIntegrable.norm
-  exact MeromorphicOn.integrable_log_abs_f hr h₁f
+  exact MeromorphicOn.integrable_log_abs_f h₁f