From ebeba74314e71ae51b3afc87656def8638d7829a Mon Sep 17 00:00:00 2001
From: Stefan Kebekus <kebekus@users.noreply.github.com>
Date: Thu, 14 Nov 2024 11:24:44 +0100
Subject: [PATCH] =?UTF-8?q?Working=E2=80=A6?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 Nevanlinna/analyticAt.lean | 21 +++++++++++++++++++--
 lake-manifest.json         |  8 ++++----
 2 files changed, 23 insertions(+), 6 deletions(-)

diff --git a/Nevanlinna/analyticAt.lean b/Nevanlinna/analyticAt.lean
index 2b4ffc3..87731af 100644
--- a/Nevanlinna/analyticAt.lean
+++ b/Nevanlinna/analyticAt.lean
@@ -282,6 +282,17 @@ theorem analyticAt_finprod
     exact analyticAt_const
 
 
+lemma AnalyticAt.zpow_nonneg
+  {f : ℂ → ℂ}
+  {z₀ : ℂ}
+  {n : ℤ}
+  (hf : AnalyticAt ℂ f z₀)
+  (hn : 0 ≤ n) :
+  AnalyticAt ℂ (fun x ↦ (f x) ^ n) z₀ := by
+  simp_rw [(Eq.symm (Int.toNat_of_nonneg hn) : n = OfNat.ofNat n.toNat), zpow_ofNat]
+  apply AnalyticAt.pow hf
+
+
 theorem AnalyticAt.zpow
   {f : ℂ → ℂ}
   {z₀ : ℂ}
@@ -289,5 +300,11 @@ theorem AnalyticAt.zpow
   (h₁f : AnalyticAt ℂ f z₀)
   (h₂f : f z₀ ≠ 0) :
   AnalyticAt ℂ (fun x ↦ (f x) ^ n) z₀ := by
-
-  sorry
+  by_cases hn : 0 ≤ n
+  · exact zpow_nonneg h₁f hn
+  · rw [(Int.eq_neg_comm.mp rfl : n = - (- n))]
+    conv =>
+      arg 2
+      intro x
+      rw [zpow_neg]
+    exact AnalyticAt.inv (zpow_nonneg h₁f (by linarith)) (zpow_ne_zero (-n) h₂f)
diff --git a/lake-manifest.json b/lake-manifest.json
index b0837ec..ae6cb22 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -5,7 +5,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "325d06c3ef6bcb16c43c4ac6614976731ce6a63e",
+   "rev": "b100ff2565805e9f30a482788b3fc66937a7f38a",
    "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -25,7 +25,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "5dfdc66e0d503631527ad90c1b5d7815c86a8662",
+   "rev": "de91b59101763419997026c35a41432ac8691f15",
    "name": "aesop",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -55,7 +55,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "ac7b989cbf99169509433124ae484318e953d201",
+   "rev": "b0b73e5bc33f1bc4d3c0f254630dd0e262cecc08",
    "name": "importGraph",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -85,7 +85,7 @@
    "type": "git",
    "subDir": null,
    "scope": "",
-   "rev": "b8570f16d4170663edb8a12b8eae9f0d84e96ab4",
+   "rev": "e3d7d7f053ea862ffb7d1613eff6de0f2e3e8e14",
    "name": "mathlib",
    "manifestFile": "lake-manifest.json",
    "inputRev": null,