Update holomorphic.primitive.lean

Nevanlinna/holomorphic.primitive.lean

@@ -175,7 +175,7 @@ theorem primitive_lem1
 
 
 
-theorem primitive_fderivAtBasepoint
+theorem primitive_fderivAtBasepointZero
   {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
   (f : ℂ  → E)
   (hf : Continuous f) :
@@ -187,7 +187,7 @@ theorem primitive_fderivAtBasepoint
   rw [Asymptotics.isLittleO_iff]
   intro c hc
 
-  have {z : ℂ} {e : E} : z • e = (∫ (x : ℝ) in (0)..(z.re), e) + Complex.I • ∫ (x : ℝ) in (0)..(z.im), e:= by
+  have {z : ℂ} {e : E} : z • e = (∫ (_ : ℝ) in (0)..(z.re), e) + Complex.I • ∫ (_ : ℝ) in (0)..(z.im), e:= by
     simp
     rw [smul_comm]
     rw [← smul_assoc]
@@ -204,10 +204,46 @@ theorem primitive_fderivAtBasepoint
     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 sorry
-  have t₁ {r : ℝ} :IntervalIntegrable (fun x => f 0) MeasureTheory.volume 0 r := by sorry
-  have t₂ {a b : ℝ}: IntervalIntegrable (fun x_1 => f { re := a, im := x_1 }) MeasureTheory.volume 0 b := by sorry
-  have t₃ {a : ℝ} : IntervalIntegrable (fun x => f 0) MeasureTheory.volume 0 a := by sorry
+  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
+    exact hf
+    have : ((fun x => { re := a, im := x }) : ℝ → ℂ) = (fun x => { re := a, im := 0 } + { re := 0, im := x }) := by
+      funext x
+      apply Complex.ext
+      rw [Complex.add_re]
+      simp
+      rw [Complex.add_im]
+      simp
+    rw [this]
+    apply Continuous.add
+    fun_prop
+    have : (fun x => { re := 0, im := x } : ℝ → ℂ) = Complex.I • Complex.ofRealCLM := by
+      funext x
+      simp
+      have : (x : ℂ) = {re := x, im := 0} := by rfl
+      rw [this]
+      rw [Complex.I_mul]
+      simp
+    continuity
+
+  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
@@ -232,24 +268,25 @@ theorem primitive_fderivAtBasepoint
     intro y hy
     apply mem_ball_iff_norm.mp (h₂ε hy)
 
-  use Metric.ball 0 ε
+  use Metric.ball 0 (ε / (4 : ℝ))
   constructor
-  · exact Metric.ball_mem_nhds 0 h₁ε
+  · apply Metric.ball_mem_nhds 0
+    linarith
   · intro y hy
-    have h₁y : |y.re| < ε := by
+    have h₁y : |y.re| < ε / 4 := by
       calc |y.re|
       _ ≤ Complex.abs y := by apply Complex.abs_re_le_abs
-      _ < ε := by
+      _ < ε / 4 := by
         let A := mem_ball_iff_norm.1 hy
         simp at A
-        assumption
-    have h₂y : |y.im| < ε := by
+        linarith
+    have h₂y : |y.im| < ε / 4 := by
       calc |y.im|
       _ ≤ Complex.abs y := by apply Complex.abs_im_le_abs
-      _ < ε := by
+      _ < ε / 4 := by
         let A := mem_ball_iff_norm.1 hy
         simp at A
-        assumption
+        linarith
 
     have intervalComputation {x' y' : ℝ} (h : x' ∈ Ι 0 y') : |x'| ≤ |y'| := by
       let A := h.1
@@ -271,10 +308,10 @@ theorem primitive_fderivAtBasepoint
       apply intervalIntegral.norm_integral_le_of_norm_le_const
       intro x hx
 
-      have h₁x : |x| < ε := by
+      have h₁x : |x| < ε / 4 := by
         calc |x|
           _ ≤ |y.re| := intervalComputation hx
-          _ < ε := h₁y
+          _ < ε / 4 := h₁y
       apply le_of_lt
       apply h₃ε { re := x, im := 0 }
       rw [mem_ball_iff_norm]
@@ -282,17 +319,16 @@ theorem primitive_fderivAtBasepoint
       have : { re := x, im := 0 } = (x : ℂ) := by rfl
       rw [this]
       rw [Complex.abs_ofReal]
-      exact h₁x
-
+      linarith
 
     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
       intro x hx
 
-      have h₁x : |x| < ε := by
+      have h₁x : |x| < ε / 4 := by
         calc |x|
           _ ≤ |y.im| := intervalComputation hx
-          _ < ε := h₂y
+          _ < ε / 4 := h₂y
 
       apply le_of_lt
       apply h₃ε { re := y.re, im := x }
@@ -301,7 +337,7 @@ theorem primitive_fderivAtBasepoint
       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 }
-        _ ≤ 2 * ε := by
+        _ < ε := by
           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‖
@@ -318,12 +354,30 @@ theorem primitive_fderivAtBasepoint
       _ ≤ (c / (4 : ℝ)) * (|y.re| + |y.im|) := by
         simp
         rw [mul_add]
-      _ ≤ c * ‖y‖ := by
+      _ ≤ (c / (4 : ℝ)) * (4 * ‖y‖) := by
+        have : |y.re| + |y.im| ≤ 4 * ‖y‖ := by
+          calc |y.re| + |y.im|
+            _ ≤ ‖y‖ + ‖y‖ := by
+              apply add_le_add
+              apply Complex.abs_re_le_abs
+              apply Complex.abs_im_le_abs
+            _ ≤ 4 * ‖y‖ := by
+              rw [← two_mul]
+              apply mul_le_mul
+              linarith
+              rfl
+              exact norm_nonneg y
+              linarith
+
         apply mul_le_mul
-        apply div_le_self
-        exact le_of_lt hc
+        rfl
+        exact this
+        apply add_nonneg
+        apply abs_nonneg
+        apply abs_nonneg
+        linarith
+      _ ≤ c * ‖y‖ := by
         linarith
-        sorry
 
 
 theorem primitive_additivity