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Nevanlinna/holomorphic_primitive2.lean

@@ -335,13 +335,19 @@ theorem primitive_additivity
   (f : ℂ  → E)
   (z₀ : ℂ)
   (rx ry : ℝ)
+  (hrx : 0 < rx)
+  (hry : 0 < ry)
   (hf : DifferentiableOn ℂ f (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry))
   (z₁ : ℂ)
   (hz₁ : z₁ ∈ (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry))
   :
-  ∀ z ∈ Metric.ball z₁ (R - dist z₁ z₀), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by
+  ∃ 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
+
+  use rx - dist z₀.re z₁.re
+  use ry - dist z₀.im z₁.im
   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
 
@@ -383,7 +389,7 @@ theorem primitive_additivity
       _ < (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
 
@@ -405,14 +411,16 @@ theorem primitive_additivity
     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)
+    -- Remains: Set.MapsTo (fun x => { re := x, im := z₀.im }) (Set.uIcc z₀.re z₁.re) (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry)
     intro w hw
     simp
-    simp_rw [Complex.dist_of_im_eq]
+    apply Complex.mem_reProdIm.mpr
+    constructor
+    · simp
       calc dist w z₀.re
       _ ≤ dist z₁.re z₀.re := by apply Real.dist_right_le_of_mem_uIcc; rwa [Set.uIcc_comm] at hw
-      _ ≤ dist z₁ z₀ := by sorry
+      _ < rx := by apply Metric.mem_ball.mp (Complex.mem_reProdIm.1 hz₁).1
-      _ < R := by simp at hz₁; assumption
+    · simpa
 
     -- IntervalIntegrable (fun x => f { re := x, im := z₀.im }) MeasureTheory.volume z₁.re z.re
     apply ContinuousOn.intervalIntegrable
@@ -428,27 +436,29 @@ theorem primitive_additivity
     apply Continuous.continuousOn
     rw [this]
     continuity
-    -- -- Remains: Set.MapsTo (fun x => { re := x, im := z₀.im }) (Set.uIcc z₁.re z.re) (Metric.ball z₀ R)
+    -- Remains: Set.MapsTo (fun x => { re := x, im := z₀.im }) (Set.uIcc z₁.re z.re) (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry)
     intro w hw
     simp
-    simp_rw [Complex.dist_of_im_eq]
+    constructor
+    · simp
       calc dist w z₀.re
       _ ≤ dist w z₁.re + dist z₁.re z₀.re := by exact dist_triangle w z₁.re z₀.re
-      _ ≤ dist z₁.re z.re + dist z₁.re z₀.re := by
+      _ ≤ dist z.re z₁.re + dist z₁.re z₀.re := by
-        apply add_le_add_right
+        apply (add_le_add_iff_right (dist z₁.re z₀.re)).mpr
-        apply Real.dist_le_of_mem_uIcc hw
+        rw [Set.uIcc_comm] at hw
+        exact Real.dist_right_le_of_mem_uIcc hw
+      _ < (rx - dist z₀.re z₁.re) + dist z₁.re z₀.re := by
+        apply (add_lt_add_iff_right (dist z₁.re z₀.re)).mpr
+        apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).1
+      _ = rx := by
+        rw [dist_comm]
         simp
-      _ ≤ dist z₁ z + dist z₁ z₀ := by
+    · simpa
-        sorry
-      _ < (R - dist z₁ z₀) + dist z₁ z₀ := by
-        apply add_lt_add_right
-        simp at hz
-        rwa [dist_comm]
-      _ = R := by simp
   rw [this]
 
   have : (∫ (x : ℝ) in z₀.im..z.im, f { re := z.re, im := x }) = (∫ (x : ℝ) in z₀.im..z₁.im, f { re := z.re, im := x }) + (∫ (x : ℝ) in z₁.im..z.im, f { re := z.re, im := x }) := by
     rw [intervalIntegral.integral_add_adjacent_intervals]
+
     -- IntervalIntegrable (fun x => f { re := z.re, im := x }) MeasureTheory.volume z₀.im z₁.im
     apply ContinuousOn.intervalIntegrable
     apply ContinuousOn.comp
@@ -464,15 +474,67 @@ theorem primitive_additivity
     apply Continuous.add
     fun_prop
     fun_prop
-    -- Set.MapsTo (Complex.mk z.re) (Set.uIcc z₀.im z₁.im) (Metric.ball z₀ R)
+    -- Set.MapsTo (Complex.mk z.re) (Set.uIcc z₀.im z₁.im) (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry)
     intro w hw
+    constructor
+    · simp
+      calc dist z.re z₀.re
+      _ ≤ dist z.re z₁.re + dist z₁.re z₀.re := by exact dist_triangle z.re z₁.re z₀.re
+      _ < (rx - dist z₀.re z₁.re) + dist z₁.re z₀.re := by
+        apply (add_lt_add_iff_right (dist z₁.re z₀.re)).mpr
+        apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).1
+      _ = rx := by
+        rw [dist_comm]
         simp
+    · simp
+      calc dist w z₀.im
+      _ ≤ dist z₁.im z₀.im := by rw [Set.uIcc_comm] at hw; exact Real.dist_right_le_of_mem_uIcc hw
+      _ < ry := by
+        rw [← Metric.mem_ball]
+        exact hz₁.2
 
-
+    -- IntervalIntegrable (fun x => f { re := z.re, im := x }) MeasureTheory.volume z₁.im z.im
-    sorry --apply integrability₂ f hf
+    apply ContinuousOn.intervalIntegrable
-
+    apply ContinuousOn.comp
-    sorry --apply integrability₂ f hf
+    exact hf.continuousOn
+    apply Continuous.continuousOn
+    have {b : ℝ}:  (Complex.mk b) = (fun x => Complex.I • Complex.ofRealCLM x + { re := b, im := 0 }) := by
+      funext x
+      apply Complex.ext
+      rw [Complex.add_re]
+      simp
+      simp
     rw [this]
+    apply Continuous.add
+    fun_prop
+    fun_prop
+    -- Set.MapsTo (Complex.mk z.re) (Set.uIcc z₁.im z.im) (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry)
+    intro w hw
+    constructor
+    · simp
+      calc dist z.re z₀.re
+      _ ≤ dist z.re z₁.re + dist z₁.re z₀.re := by exact dist_triangle z.re z₁.re z₀.re
+      _ < (rx - dist z₀.re z₁.re) + dist z₁.re z₀.re := by
+        apply (add_lt_add_iff_right (dist z₁.re z₀.re)).mpr
+        apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).1
+      _ = rx := by
+        rw [dist_comm]
+        simp
+    · simp
+      calc dist w z₀.im
+      _ ≤ dist w z₁.im + dist z₁.im z₀.im := by exact dist_triangle w z₁.im z₀.im
+      _ ≤ dist z.im z₁.im + dist z₁.im z₀.im := by
+        apply (add_le_add_iff_right (dist z₁.im z₀.im)).mpr
+        rw [Set.uIcc_comm] at hw
+        exact Real.dist_right_le_of_mem_uIcc hw
+      _ < (ry - dist z₀.im z₁.im) + dist z₁.im z₀.im := by
+        apply (add_lt_add_iff_right (dist z₁.im z₀.im)).mpr
+        apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).2
+      _ = ry := by
+        rw [dist_comm]
+        simp
+  rw [this]
+
   simp
 
   have {a b c d e f g h : E} : (a + b) + (c + d) - (e + f) - (g + h) = b + (a - g) - e - f + d  - h + (c) := by
@@ -483,7 +545,6 @@ theorem primitive_additivity
   have H' : DifferentiableOn ℂ f (Set.uIcc z₁.re z.re ×ℂ Set.uIcc z₀.im z₁.im) := by
     apply DifferentiableOn.mono hf
     intro x hx
-    simp
     exact H hx
 
 

lake-manifest.json

@@ -5,7 +5,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "0f3e143dffdc3a591662f3401ce1d7a3405227c0",
+   "rev": "41bc768e2224d6c75128a877f1d6e198859b3178",
    "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -25,7 +25,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "209712c78b16c795453b6da7f7adbda4589a8f21",
+   "rev": "6058ab8d938c5104eace7d0fb5ac17b21cb067b1",
    "name": "aesop",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -65,7 +65,7 @@
    "type": "git",
    "subDir": null,
    "scope": "",
-   "rev": "c96401c7496392a2200dc78c0b334df0561466dc",
+   "rev": "d56e389960165aa122e7c97c098d67e4def09470",
    "name": "mathlib",
    "manifestFile": "lake-manifest.json",
    "inputRev": null,