Update holomorphic_primitive2.lean

Nevanlinna/holomorphic_primitive2.lean

@@ -47,51 +47,6 @@ theorem primitive_fderivAtBasepointZero
     arg 1
     arg 2
     rw [this]
-  have {A B C D :E} : (A + B) - (C + D) = (A - C) + (B - D) := by
-    abel
-
-  have t₀ {r : ℝ} : IntervalIntegrable (fun x => f { re := x, im := 0 }) MeasureTheory.volume 0 r := by
-    apply Continuous.intervalIntegrable
-    apply Continuous.comp
-    exact hf
-    have : (fun x => ({ re := x, im := 0 } : ℂ)) = Complex.ofRealLI := by rfl
-    rw [this]
-    continuity
-
-  have t₁ {r : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 r := by
-    apply Continuous.intervalIntegrable
-    apply Continuous.comp
-    exact hf
-    fun_prop
-  have t₂ {a b : ℝ} : IntervalIntegrable (fun x_1 => f { re := a, im := x_1 }) MeasureTheory.volume 0 b := by
-    apply Continuous.intervalIntegrable
-    apply Continuous.comp hf
-    have : (Complex.mk a) = (fun x => Complex.I • Complex.ofRealCLM x + { re := a, im := 0 }) := by
-      funext x
-      apply Complex.ext
-      rw [Complex.add_re]
-      simp
-      simp
-    rw [this]
-    apply Continuous.add
-    continuity
-    fun_prop
-
-  have t₃ {a : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 a := by
-    apply Continuous.intervalIntegrable
-    apply Continuous.comp
-    exact hf
-    fun_prop
-  conv =>
-    left
-    intro x
-    left
-    arg 1
-    rw [this]
-    rw [← smul_sub]
-    rw [← intervalIntegral.integral_sub t₀ t₁]
-    rw [← intervalIntegral.integral_sub t₂ t₃]
-  rw [Filter.eventually_iff_exists_mem]
 
   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
@@ -108,7 +63,7 @@ theorem primitive_fderivAtBasepointZero
     apply Metric.ball_mem_nhds (f 0)
     simpa
 
-  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) ×ℂ (Metric.ball 0 ε) ⊆ s ∧ (Metric.ball 0 ε) ×ℂ (Metric.ball 0 ε) ⊆ Metric.ball 0 R := by
     obtain ⟨ε', h₁ε', h₂ε'⟩ : ∃ ε' > 0, Metric.ball 0 ε' ⊆ s ∩ Metric.ball 0 R := by
       apply Metric.mem_nhds_iff.mp
       apply IsOpen.mem_nhds
@@ -118,12 +73,33 @@ theorem primitive_fderivAtBasepointZero
       constructor
       · exact h₂s.2
       · simpa
+
     use (2 : ℝ)⁻¹ * ε'
+
     constructor
     · simpa
+    · constructor
       · intro x hx
-      apply Set.inter_subset_left
-      apply h₂ε'
+        apply (h₂ε' _).1
+        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
+      · intro x hx
+        apply (h₂ε' _).2
         simp
         calc Complex.abs x
         _ ≤ |x.re| + |x.im| := Complex.abs_le_abs_re_add_abs_im x
@@ -145,7 +121,71 @@ theorem primitive_fderivAtBasepointZero
   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
-    exact h₁s (h₂ε hy)
+    apply h₁s
+    exact h₂ε.1 hy
+
+  have t₀ {r : ℝ} (hr : r ∈ Metric.ball 0 ε) : IntervalIntegrable (fun x => f { re := x, im := 0 }) MeasureTheory.volume 0 r := by
+    apply ContinuousOn.intervalIntegrable
+    apply ContinuousOn.comp
+    exact hf
+    have : (fun x => ({ re := x, im := 0 } : ℂ)) = Complex.ofRealLI := by rfl
+    rw [this]
+    apply Continuous.continuousOn
+    continuity
+    intro x hx
+    apply h₂ε.2
+    simp
+    constructor
+    · simp
+      calc |x|
+      _ < ε := by
+        sorry
+    · simpa
+
+  have t₁ {r : ℝ}  (hr : r ∈ Metric.ball 0 ε) : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 r := by
+    apply ContinuousOn.intervalIntegrable
+    apply ContinuousOn.comp
+    apply hf
+    fun_prop
+    intro x hx
+    simpa
+
+
+  have t₂ {a b : ℝ} : IntervalIntegrable (fun x_1 => f { re := a, im := x_1 }) MeasureTheory.volume 0 b := by
+    apply Continuous.intervalIntegrable
+    apply Continuous.comp hf
+    have : (Complex.mk a) = (fun x => Complex.I • Complex.ofRealCLM x + { re := a, im := 0 }) := by
+      funext x
+      apply Complex.ext
+      rw [Complex.add_re]
+      simp
+      simp
+    rw [this]
+    apply Continuous.add
+    continuity
+    fun_prop
+
+  have t₃ {a : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 a := by
+    apply Continuous.intervalIntegrable
+    apply Continuous.comp
+    exact hf
+    fun_prop
+
+  have {A B C D :E} : (A + B) - (C + D) = (A - C) + (B - D) := by
+    abel
+  conv =>
+    left
+    intro x
+    left
+    arg 1
+    rw [this]
+    rw [← smul_sub]
+
+    rw [← intervalIntegral.integral_sub t₀ t₁]
+    rw [← intervalIntegral.integral_sub t₂ t₃]
+
+  rw [Filter.eventually_iff_exists_mem]
+
 
   use Metric.ball 0 (ε / (4 : ℝ))
   constructor