Update holomorphic_JensenFormula2.lean

Nevanlinna/holomorphic_JensenFormula2.lean

@@ -7,7 +7,7 @@ import Nevanlinna.specialFunctions_CircleIntegral_affine
 
 open Real
 
-
+/-
 noncomputable def Zeroset
   {f : ℂ → ℂ}
   {s : Set ℂ}
@@ -72,7 +72,7 @@ noncomputable def order
   let A := ((ZeroFinset_mem_iff h₁f i).1 i.2).1
   let B := (h₁f i.1 A).analyticAt
   exact B.order.toNat
-
+-/
 
 theorem jensen_case_R_eq_one
   (f : ℂ → ℂ)
@@ -180,29 +180,28 @@ theorem jensen_case_R_eq_one
     exact Ne.symm (zero_ne_one' ℝ)
 
 
-  have h₁Gi : ∀ i ∈ (ZeroFinset h₁f h₂f).attach, IntervalIntegrable (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π) := by
+  have h₁Gi : ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, IntervalIntegrable (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π) := by
     -- This is hard. Need to invoke specialFunctions_CircleIntegral_affine.
     sorry
 
-  have : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) = (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x)))) + ∑ x ∈ (ZeroFinset h₁f h₂f).attach, ↑(order x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
+  have : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
+    = (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
+        + ∑ x ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (h'₁f.order x).toNat * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
     dsimp [G]
     rw [intervalIntegral.integral_add]
     rw [intervalIntegral.integral_finset_sum]
     simp_rw [intervalIntegral.integral_const_mul]
 
-    --  ∀ i ∈ (ZeroFinset h₁f).attach, IntervalIntegrable (fun x => ↑(order i) *
-    --  log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
+    --  ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset,
+    --  IntervalIntegrable (fun x => (h'₁f.order i).toNat *
+    --    log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
     intro i hi
     apply IntervalIntegrable.const_mul
-    have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := by exact ((ZeroFinset_mem_iff h₁f i).1 i.2).1
-    simp at this
+    --simp at this
     by_cases h₂i : ‖i.1‖ = 1
     -- case pos
     exact int'₂ h₂i
     -- case neg
-    have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
-
-
     apply Continuous.intervalIntegrable
     apply continuous_iff_continuousAt.2
     intro x
@@ -234,32 +233,32 @@ theorem jensen_case_R_eq_one
     apply ContinuousAt.comp
     apply Real.continuousAt_log
     simp [h₂F]
-    --
+    -- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
     apply ContinuousAt.comp
     apply Complex.continuous_abs.continuousAt
     apply ContinuousAt.comp
     apply DifferentiableAt.continuousAt (𝕜 := ℂ )
     apply HolomorphicAt.differentiableAt
-    simp [h₁F]
-    --
+    simp [h'₁F]
+    -- ContinuousAt (fun x => circleMap 0 1 x) x
     apply Continuous.continuousAt
     apply continuous_circleMap
-    --
-    have : (fun x => ∑ s ∈ (ZeroFinset h₁f h₂f).attach, ↑(order s) * log (Complex.abs (circleMap 0 1 x - ↑s)))
-      = ∑ s ∈ (ZeroFinset h₁f h₂f).attach, (fun x => ↑(order s) * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
+
+    have : (fun x => ∑ s ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (h'₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s)))
+      = ∑ s ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (fun x => (h'₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
       funext x
       simp
     rw [this]
     apply IntervalIntegrable.sum
     intro i h₂i
     apply IntervalIntegrable.const_mul
-    have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := by exact ((ZeroFinset_mem_iff h₁f i).1 i.2).1
-    simp at this
+    --have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := i.2
+    --simp at this
     by_cases h₂i : ‖i.1‖ = 1
     -- case pos
     exact int'₂ h₂i
     -- case neg
-    have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
+    --have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
     apply Continuous.intervalIntegrable
     apply continuous_iff_continuousAt.2
     intro x
@@ -287,16 +286,13 @@ theorem jensen_case_R_eq_one
     have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
       intro z hz
       apply logabs_of_holomorphicAt_is_harmonic
-      apply h₁F z hz
+      apply h'₁F z hz
       exact h₂F z hz
 
     apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
   simp_rw [← Complex.norm_eq_abs] at this
   rw [t₁] at this
 
-  --let Z₁ := (ZeroFinset h₁f h₂f) ∩ (Metric.ball 0 1)
-
-  let Z₂ := { x : ZeroFinset h₁f h₂f | ‖x.1‖ = 1 }
 
 
   sorry