Migrate layers example to material_ui (#190237)

Migrates the layers example to material_ui. Maybe in the future we want
to move this example to material_ui, or refactor it to not use
material_ui, but this PR is not considering that for now.

I did a `flutter run` for each of the affected examples here, and they
all ran fine.

Part of https://github.com/flutter/flutter/issues/190093.
diff --git a/dev/bots/check_examples_cross_imports.dart b/dev/bots/check_examples_cross_imports.dart
index 699c150..8887a3c 100644
--- a/dev/bots/check_examples_cross_imports.dart
+++ b/dev/bots/check_examples_cross_imports.dart
@@ -570,14 +570,6 @@
     'examples/api/test/widgets/text_magnifier/text_magnifier.0_test.dart',
     'examples/flutter_view/lib/main.dart',
     'examples/image_list/lib/main.dart',
-    'examples/layers/rendering/touch_input.dart',
-    'examples/layers/services/isolate.dart',
-    'examples/layers/widgets/gestures.dart',
-    'examples/layers/widgets/spinning_mixed.dart',
-    'examples/layers/widgets/media_query.dart',
-    'examples/layers/widgets/sectors.dart',
-    'examples/layers/widgets/styled_text.dart',
-    'examples/layers/test/gestures_test.dart',
     'examples/multiple_windows/lib/app/main_window.dart',
     'examples/multiple_windows/lib/app/tooltip_button.dart',
     'examples/multiple_windows/lib/app/tooltip_window_edit_dialog.dart',
diff --git a/examples/layers/pubspec.yaml b/examples/layers/pubspec.yaml
index 204a644..c3529ed 100644
--- a/examples/layers/pubspec.yaml
+++ b/examples/layers/pubspec.yaml
@@ -8,16 +8,15 @@
 dependencies:
   flutter:
     sdk: flutter
-
+  material_ui: ^0.0.2
 
 dev_dependencies:
   flutter_test:
     sdk: flutter
 
-
 flutter:
   assets:
     - services/data.json
   uses-material-design: true
 
-# PUBSPEC CHECKSUM: 60tfp7
+# PUBSPEC CHECKSUM: oescuq
diff --git a/examples/layers/rendering/touch_input.dart b/examples/layers/rendering/touch_input.dart
index dcb46b4..2d7030b 100644
--- a/examples/layers/rendering/touch_input.dart
+++ b/examples/layers/rendering/touch_input.dart
@@ -5,8 +5,8 @@
 // This example shows how to use process input events in the underlying render
 // tree.
 
-import 'package:flutter/material.dart'; // Imported just for its color palette.
 import 'package:flutter/rendering.dart';
+import 'package:material_ui/material_ui.dart'; // Imported just for its color palette.
 
 import 'src/binding.dart';
 
diff --git a/examples/layers/services/isolate.dart b/examples/layers/services/isolate.dart
index 228be44..12d9a63 100644
--- a/examples/layers/services/isolate.dart
+++ b/examples/layers/services/isolate.dart
@@ -5,8 +5,8 @@
 import 'dart:convert';
 import 'dart:isolate';
 
-import 'package:flutter/material.dart';
 import 'package:flutter/services.dart';
+import 'package:material_ui/material_ui.dart';
 
 typedef OnProgressListener = void Function(double completed, double total);
 typedef OnResultListener = void Function(String result);
diff --git a/examples/layers/test/gestures_test.dart b/examples/layers/test/gestures_test.dart
index 47cd843..044c8b1 100644
--- a/examples/layers/test/gestures_test.dart
+++ b/examples/layers/test/gestures_test.dart
@@ -2,8 +2,8 @@
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
-import 'package:flutter/material.dart';
 import 'package:flutter_test/flutter_test.dart';
+import 'package:material_ui/material_ui.dart';
 
 import '../widgets/gestures.dart';
 
diff --git a/examples/layers/widgets/gestures.dart b/examples/layers/widgets/gestures.dart
index ae622f3..e45b39e 100644
--- a/examples/layers/widgets/gestures.dart
+++ b/examples/layers/widgets/gestures.dart
@@ -2,7 +2,7 @@
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
-import 'package:flutter/material.dart';
+import 'package:material_ui/material_ui.dart';
 
 class _GesturePainter extends CustomPainter {
   const _GesturePainter({
diff --git a/examples/layers/widgets/media_query.dart b/examples/layers/widgets/media_query.dart
index 2811709..4c67e9e 100644
--- a/examples/layers/widgets/media_query.dart
+++ b/examples/layers/widgets/media_query.dart
@@ -2,7 +2,7 @@
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
-import 'package:flutter/material.dart';
+import 'package:material_ui/material_ui.dart';
 
 class AdaptedListItem extends StatelessWidget {
   const AdaptedListItem({super.key, required this.name});
diff --git a/examples/layers/widgets/sectors.dart b/examples/layers/widgets/sectors.dart
index 3a48868..9871ca9 100644
--- a/examples/layers/widgets/sectors.dart
+++ b/examples/layers/widgets/sectors.dart
@@ -4,7 +4,7 @@
 
 import 'dart:math' as math;
 
-import 'package:flutter/material.dart';
+import 'package:material_ui/material_ui.dart';
 
 import '../rendering/src/sector_layout.dart';
 
diff --git a/examples/layers/widgets/spinning_mixed.dart b/examples/layers/widgets/spinning_mixed.dart
index c30e8da..184d219 100644
--- a/examples/layers/widgets/spinning_mixed.dart
+++ b/examples/layers/widgets/spinning_mixed.dart
@@ -2,8 +2,8 @@
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
-import 'package:flutter/material.dart';
 import 'package:flutter/rendering.dart';
+import 'package:material_ui/material_ui.dart';
 
 import '../rendering/src/solid_color_box.dart';
 
diff --git a/examples/layers/widgets/styled_text.dart b/examples/layers/widgets/styled_text.dart
index 4ef75dd..f8c221d 100644
--- a/examples/layers/widgets/styled_text.dart
+++ b/examples/layers/widgets/styled_text.dart
@@ -2,7 +2,7 @@
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
-import 'package:flutter/material.dart';
+import 'package:material_ui/material_ui.dart';
 
 typedef _TextTransformer = Widget Function(String name, String text);
 
diff --git a/pubspec.lock b/pubspec.lock
index 8f82714..1345c36 100644
--- a/pubspec.lock
+++ b/pubspec.lock
@@ -218,6 +218,14 @@
       url: "https://pub.dev"
     source: hosted
     version: "1.0.9"
+  cupertino_ui:
+    dependency: transitive
+    description:
+      name: cupertino_ui
+      sha256: "427360789a03b76eaf659c37131570d2fc98e381f80d55f252bd9e30f37e7126"
+      url: "https://pub.dev"
+    source: hosted
+    version: "0.0.2"
   dap:
     dependency: transitive
     description:
@@ -658,6 +666,14 @@
       url: "https://pub.dev"
     source: hosted
     version: "0.13.0"
+  material_ui:
+    dependency: transitive
+    description:
+      name: material_ui
+      sha256: e1fa67393d807bd1841e5ece1ec260ed84f440ae7351e40b6406f733e4fe31cf
+      url: "https://pub.dev"
+    source: hosted
+    version: "0.0.2"
   meta:
     dependency: "direct main"
     description: