From f4655ef1d3e1e9e843dddf84a62046ee348be642 Mon Sep 17 00:00:00 2001
From: Stefan Kebekus <kebekus@users.noreply.github.com>
Date: Mon, 5 Aug 2024 14:53:26 +0200
Subject: [PATCH] =?UTF-8?q?working=E2=80=A6?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 Nevanlinna/holomorphic_primitive2.lean | 61 +++++++++++++++++++-------
 1 file changed, 45 insertions(+), 16 deletions(-)

diff --git a/Nevanlinna/holomorphic_primitive2.lean b/Nevanlinna/holomorphic_primitive2.lean
index 976ce6a..766218d 100644
--- a/Nevanlinna/holomorphic_primitive2.lean
+++ b/Nevanlinna/holomorphic_primitive2.lean
@@ -291,18 +291,28 @@ theorem primitive_hasDerivAtBasepoint
 
 theorem primitive_additivity
   {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
-  (f : ℂ  → E)
-  (z₀ : ℂ)
-  (rx ry : ℝ)
-  (hry : 0 < ry)
+  {f : ℂ  → E}
+  {z₀ : ℂ}
+  {rx ry : ℝ}
   (hf : DifferentiableOn ℂ f (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry))
-  (z₁ : ℂ)
+  (hry : 0 < ry)
+  {z₁ : ℂ}
   (hz₁ : z₁ ∈ (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry))
   :
-  ∃ εx εy : ℝ, ∀ z ∈ (Metric.ball z₁.re εx ×ℂ Metric.ball z₁.im εy), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by
+  ∃ εx > 0, ∃ εy > 0, ∀ z ∈ (Metric.ball z₁.re εx ×ℂ Metric.ball z₁.im εy), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by
+
+  let εx := rx - dist z₀.re z₁.re
+  have hεx : εx > 0 := by
+    sorry
+  let εy := ry - dist z₀.im z₁.im
+  have hεy : εy > 0 := by
+    sorry
+
+  use εx
+  use hεx
+  use εy
+  use hεy
 
-  use rx - dist z₀.re z₁.re
-  use ry - dist z₀.im z₁.im
   intro z hz
 
   unfold primitive
@@ -506,20 +516,41 @@ theorem primitive_additivity'
   primitive z₀ f =ᶠ[nhds z₁] fun z ↦ (primitive z₁ f z) + (primitive z₀ f z₁) := by
 
   let ε := R - dist z₀ z₁
+
+  have hε : 0 < ε := by
+    dsimp [ε]
+    simp
+    exact Metric.mem_ball'.mp hz₁
+
   let rx := dist z₀.re z₁.re + ε/(2 : ℝ)
   let ry := dist z₀.im z₁.im + ε/(2 : ℝ)
 
   have h'ry : 0 < ry := by
-    sorry
+    dsimp [ry]
+    apply add_pos_of_nonneg_of_pos
+    exact dist_nonneg
+    simpa
 
   have h'f : DifferentiableOn ℂ f (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry) := by
+    apply hf.mono
+    intro x hx
+    simp
+    let A := hx.1
+    simp at A
+    let B := hx.2
+    simp at B
+    calc dist x z₀
+    _ = √((x.re - z₀.re) ^ 2 + (x.im - z₀.im) ^ 2) := by exact Complex.dist_eq_re_im x z₀
+    _ =
     sorry
 
   have h'z₁ : z₁ ∈ (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry) := by
-    sorry
+    dsimp [rx, ry]
+    constructor
+    · rw [dist_comm]; simp; exact hε
+    · rw [dist_comm]; simp; exact hε
 
-  let A := primitive_additivity f z₀ rx ry h'ry h'f z₁ h'z₁
-  obtain ⟨εx, εy, hε⟩ := A
+  obtain ⟨εx, hεx, εy, hεy, hε⟩ := primitive_additivity h'f h'ry h'z₁
 
   apply Filter.eventuallyEq_iff_exists_mem.2
   use (Metric.ball z₁.re εx ×ℂ Metric.ball z₁.im εy)
@@ -529,10 +560,8 @@ theorem primitive_additivity'
     exact Metric.isOpen_ball
     exact Metric.isOpen_ball
     constructor
-    · simp
-      sorry
-    · simp
-      sorry
+    · simpa
+    · simpa
   · intro x hx
     simp
     rw [← sub_zero (primitive z₀ f x), ← hε x hx]