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Nevanlinna/holomorphic_primitive2.lean

@@ -337,13 +337,107 @@ theorem primitive_additivity
   (R : ℝ)
   (hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
   (z₁ : ℂ)
-  (hz₁ : z₁ ∈ Metric.ball z₀ R) :
-  ∀ z ∈ Metric.ball z₁ (R - ‖z₁‖), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by
-  intro z _
+  (hz₁ : z₁ ∈ Metric.ball z₀ R)
+  :
+  ∀ z ∈ Metric.ball z₁ (R - dist z₁ z₀), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by
+  intro z hz
+
+  have H : (Set.uIcc z₁.re z.re ×ℂ Set.uIcc z₀.im z₁.im) ⊆ Metric.ball z₀ R := by
+    intro x hx
+
+    have a₀ : x.re ∈ Set.uIcc z₁.re z.re := by exact (Complex.mem_reProdIm.1 hx).1
+    have a₁ : x.im ∈ Set.uIcc z₀.im z₁.im := by exact (Complex.mem_reProdIm.1 hx).2
+
+
+    have A₀ : dist x.re z₀.re ≤ dist x.re z₁.re + dist z₁.re z₀.re := by apply dist_triangle
+
+    have A₁ : dist x.im z₀.im ≤ dist z₁.im z₀.im := by
+      apply Real.dist_right_le_of_mem_uIcc
+      rwa [Set.uIcc_comm]
+
+    have A₂ : dist x.re z₁.re ≤ dist z.re z₁.re := by
+      apply Real.dist_right_le_of_mem_uIcc
+      rwa [Set.uIcc_comm]
+
+    have A₃ : dist z.re z₁.re < R - dist z₁ z₀ := by
+      have : ∀ x₀ x₁ : ℂ, dist x₀.re x₁.re ≤ dist x₀ x₁ := by
+        intro x₀ x₁
+        rw [Complex.dist_eq_re_im, Real.dist_eq]
+        apply Real.le_sqrt_of_sq_le
+        simp
+        exact sq_nonneg (x₀.im - x₁.im)
+      calc dist z.re z₁.re
+        _ ≤ dist z z₁ := by apply this z z₁
+        _ < R - dist z₁ z₀ := by exact hz
+
+    simp
+
+    have B₀ : dist x z₀ ≤ dist x ⟨z₁.re, x.im⟩ + dist ⟨z₁.re, x.im⟩ z₀ := by apply dist_triangle
+
+    have B₁ : dist ⟨z₁.re, x.im⟩ z₀ ≤ dist z₁ z₀ := by
+      rw [Complex.dist_eq_re_im]
+      rw [Complex.dist_eq_re_im]
+      simp
+      apply Real.sqrt_le_sqrt
+      simp
+      exact sq_le_sq.mpr A₁
+
+    calc dist x z₀
+      _ ≤ dist x ⟨z₁.re, x.im⟩ + dist ⟨z₁.re, x.im⟩ z₀ := by apply dist_triangle
+      _ = dist x.re z₁.re + dist ⟨z₁.re, x.im⟩ z₀ := by rw [Complex.dist_of_im_eq]; rfl
+      _ ≤ dist z.re z₁.re + dist ⟨z₁.re, x.im⟩ z₀ := by apply add_le_add_right A₂
+      _ < (R - dist z₁ z₀) + dist ⟨z₁.re, x.im⟩ z₀ := by apply add_lt_add_right A₃
+      _ ≤ (R - dist z₁ z₀) + dist z₁ z₀ := by exact (add_le_add_iff_left (R - dist z₁ z₀)).mpr B₁
+      _ = R := by exact sub_add_cancel R (dist z₁ z₀)
+
 
   unfold primitive
+
+
+  have integrability₁
+    (a₁ a₂ b : ℝ) :
+    IntervalIntegrable (fun x => f { re := x, im := b }) MeasureTheory.volume a₁ a₂ := by
+    apply ContinuousOn.intervalIntegrable
+    apply ContinuousOn.comp
+    exact hf.continuousOn
+
+    have : ((fun x => { re := x, im := b }) : ℝ → ℂ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
+      funext x
+      apply Complex.ext
+      rw [Complex.add_re]
+      simp
+      rw [Complex.add_im]
+      simp
+    apply Continuous.continuousOn
+    rw [this]
+    continuity
+    --
+    intro w hw
+    simp
+    rw [Complex.dist_eq_re_im]
+
   have : (∫ (x : ℝ) in z₀.re..z.re, f { re := x, im := z₀.im }) = (∫ (x : ℝ) in z₀.re..z₁.re, f { re := x, im := z₀.im }) + (∫ (x : ℝ) in z₁.re..z.re, f { re := x, im := z₀.im }) := by
     rw [intervalIntegral.integral_add_adjacent_intervals]
+    -- IntervalIntegrable (fun x => f { re := x, im := z₀.im }) MeasureTheory.volume z₀.re z₁.re
+    apply ContinuousOn.intervalIntegrable
+    apply ContinuousOn.comp
+    exact hf.continuousOn
+    have {b : ℝ} : ((fun x => { re := x, im := b }) : ℝ → ℂ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
+      funext x
+      apply Complex.ext
+      rw [Complex.add_re]
+      simp
+      rw [Complex.add_im]
+      simp
+    apply Continuous.continuousOn
+    rw [this]
+    continuity
+    -- Remains: Set.MapsTo (fun x => { re := x, im := z₀.im }) (Set.uIcc z₀.re z₁.re) (Metric.ball z₀ R)
+    intro w hw
+    simp
+
+
+
     sorry --apply integrability₁ f hf
     sorry --apply integrability₁ f hf
   rw [this]
@@ -365,6 +459,7 @@ theorem primitive_additivity
     simp
 
     let A₀ : dist x.re z₀.re ≤ dist x.re z₁.re + dist z₁.re z₀.re := by apply dist_triangle
+    let A₁ : dist x.im z₀.im ≤ dist z₁.im z₀.im := by sorry