From f0b84fcbffbef0ab9c6d0a6d7737ea4b764b4c0e Mon Sep 17 00:00:00 2001
From: Stefan Kebekus <kebekus@users.noreply.github.com>
Date: Wed, 7 Aug 2024 10:46:08 +0200
Subject: [PATCH] Update holomorphic_primitive2.lean

---
 Nevanlinna/holomorphic_primitive2.lean | 68 +++++++++++++++++---------
 1 file changed, 46 insertions(+), 22 deletions(-)

diff --git a/Nevanlinna/holomorphic_primitive2.lean b/Nevanlinna/holomorphic_primitive2.lean
index 1e4c877..a202aef 100644
--- a/Nevanlinna/holomorphic_primitive2.lean
+++ b/Nevanlinna/holomorphic_primitive2.lean
@@ -89,18 +89,45 @@ theorem primitive_fderivAtBasepointZero
     rw [← intervalIntegral.integral_sub t₂ t₃]
   rw [Filter.eventually_iff_exists_mem]
 
-  let s := f⁻¹' Metric.ball (f 0) (c / (4 : ℝ))
-  have h₁s : IsOpen s := IsOpen.preimage hf Metric.isOpen_ball
-  have h₂s : 0 ∈ s := by
-    apply Set.mem_preimage.mpr
-    apply Metric.mem_ball_self
-    linarith
+  obtain ⟨s, h₁s, h₂s⟩ : ∃ s ⊆ f⁻¹' Metric.ball (f 0) (c / (4 : ℝ)), IsOpen s ∧ 0 ∈ s := by
+    have B : Metric.ball (f 0) (c / 4) ∈ nhds (f 0) := by
+      apply Metric.ball_mem_nhds (f 0)
+      linarith
+    exact eventually_nhds_iff.mp (continuousAt_def.1 hf (Metric.ball (f 0) (c / (4 : ℝ))) B)
 
-  obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 h₁s 0 h₂s
+  obtain ⟨ε, h₁ε, h₂ε⟩ : ∃ ε > 0, (Metric.ball 0 ε) ×ℂ (Metric.ball 0 ε) ⊆ s := by
+    obtain ⟨ε', h₁ε', h₂ε'⟩ : ∃ ε' > 0, Metric.ball 0 ε' ⊆ s := by
+      apply Metric.mem_nhds_iff.mp
+      apply IsOpen.mem_nhds
+      exact h₂s.1
+      exact h₂s.2
+    use (2 : ℝ)⁻¹ * ε'
+    constructor
+    · simpa
+    · intro x hx
+      apply h₂ε'
+      simp
+      calc Complex.abs x
+      _ ≤ |x.re| + |x.im| := Complex.abs_le_abs_re_add_abs_im x
+      _ < (2 : ℝ)⁻¹ * ε' + |x.im| := by
+        apply (add_lt_add_iff_right |x.im|).mpr
+        have : x.re ∈ Metric.ball 0 (2⁻¹ * ε') := (Complex.mem_reProdIm.1 hx).1
+        simp at this
+        exact this
+      _ < (2 : ℝ)⁻¹ * ε' + (2 : ℝ)⁻¹ * ε' := by
+        apply (add_lt_add_iff_left ((2 : ℝ)⁻¹ * ε')).mpr
+        have : x.im ∈ Metric.ball 0 (2⁻¹ * ε') := (Complex.mem_reProdIm.1 hx).2
+        simp at this
+        exact this
+      _ = ε' := by
+        rw [← add_mul]
+        abel_nf
+        simp
 
-  have h₃ε : ∀ y ∈ Metric.ball 0 ε, ‖(f y) - (f 0)‖ < (c / (4 : ℝ)) := by
+  have h₃ε : ∀ y ∈ (Metric.ball 0 ε) ×ℂ (Metric.ball 0 ε), ‖(f y) - (f 0)‖ < (c / (4 : ℝ)) := by
     intro y hy
-    apply mem_ball_iff_norm.mp (h₂ε hy)
+    apply mem_ball_iff_norm.mp
+    exact h₁s (h₂ε hy)
 
   use Metric.ball 0 (ε / (4 : ℝ))
   constructor
@@ -148,12 +175,11 @@ theorem primitive_fderivAtBasepointZero
           _ < ε / 4 := h₁y
       apply le_of_lt
       apply h₃ε { re := x, im := 0 }
-      rw [mem_ball_iff_norm]
-      simp
-      have : { re := x, im := 0 } = (x : ℂ) := by rfl
-      rw [this]
-      rw [Complex.abs_ofReal]
-      linarith
+      constructor
+      · simp
+        linarith
+      · simp
+        exact h₁ε
 
     have t₂ : ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ ≤ (c / (4 : ℝ)) * |y.im - 0| := by
       apply intervalIntegral.norm_integral_le_of_norm_le_const
@@ -166,13 +192,11 @@ theorem primitive_fderivAtBasepointZero
 
       apply le_of_lt
       apply h₃ε { re := y.re, im := x }
-      simp
-
-      calc Complex.abs { re := y.re, im := x }
-        _ ≤ |y.re| + |x| := by
-          apply Complex.abs_le_abs_re_add_abs_im { re := y.re, im := x }
-        _ < ε := by
-          linarith
+      constructor
+      · simp
+        linarith
+      · simp
+        linarith
 
     calc ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0) + Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖
       _ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by