Add tests for restorable_value.0.dart API example. (#148676)

This PR contributes to https://github.com/flutter/flutter/issues/130459

### Description
- Updates `examples/api/lib/widgets/restoration_properties/restorable_value.0.dart` to meet the latest API examples structure
- Adds tests for `examples/api/lib/widgets/restoration_properties/restorable_value.0.dart`

dev/bots/check_code_samples.dart

@@ -388,7 +388,6 @@ final Set<String> _knownMissingTests = <String>{
   'examples/api/test/widgets/scroll_position/scroll_metrics_notification.0_test.dart',
   'examples/api/test/widgets/media_query/media_query_data.system_gesture_insets.0_test.dart',
   'examples/api/test/widgets/async/future_builder.0_test.dart',
-  'examples/api/test/widgets/restoration_properties/restorable_value.0_test.dart',
   'examples/api/test/widgets/animated_switcher/animated_switcher.0_test.dart',
   'examples/api/test/widgets/transitions/relative_positioned_transition.0_test.dart',
   'examples/api/test/widgets/transitions/positioned_transition.0_test.dart',

examples/api/lib/widgets/restoration_properties/restorable_value.0.dart

@@ -13,14 +13,13 @@ class RestorableValueExampleApp extends StatelessWidget {
 
   @override
   Widget build(BuildContext context) {
-    return WidgetsApp(
-      title: 'RestorableValue Sample',
-      color: const Color(0xffffffff),
-      builder: (BuildContext context, Widget? child) {
-        return const Center(
-          child: RestorableValueExample(restorationId: 'main'),
-        );
-      },
+    return MaterialApp(
+      home: Scaffold(
+        appBar: AppBar(
+          title: const Text('RestorableValue Sample'),
+        ),
+        body: const RestorableValueExample(restorationId: 'main'),
+      ),
     );
   }
 }
@@ -72,9 +71,11 @@ class _RestorableValueExampleState extends State<RestorableValueExample> with Re
 
   @override
   Widget build(BuildContext context) {
-    return OutlinedButton(
-      onPressed: _incrementAnswer,
-      child: Text('${_answer.value}'),
+    return Center(
+      child: OutlinedButton(
+        onPressed: _incrementAnswer,
+        child: Text('${_answer.value}'),
+      ),
     );
   }
 }

examples/api/test/widgets/actions/actions.0_test.dart

@@ -21,7 +21,7 @@ void main() {
     return button.style?.foregroundColor?.resolve(<WidgetState>{});
   }
 
-  testWidgets('increments and decrements value', (WidgetTester tester) async {
+  testWidgets('Increments and decrements value', (WidgetTester tester) async {
     await tester.pumpWidget(
       const example.ActionsExampleApp(),
     );

examples/api/test/widgets/restoration_properties/restorable_value.0_test.dart

@@ -0,0 +1,56 @@
+// Copyright 2014 The Flutter Authors. All rights reserved.
+// 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_api_samples/widgets/restoration_properties/restorable_value.0.dart'
+    as example;
+import 'package:flutter_test/flutter_test.dart';
+
+void main() {
+  testWidgets('Increments answer on OutlinedButton tap', (WidgetTester tester) async {
+    await tester.pumpWidget(
+      const example.RestorableValueExampleApp(),
+    );
+
+    // Verify that the initial answer value in the example equals 42.
+    expect(find.text('42'), findsOneWidget);
+
+    // Tap the button to increment the answer value by 1.
+    await tester.tap(find.byType(OutlinedButton));
+    await tester.pump();
+
+    // Verify that the answer value increased by 1.
+    expect(find.text('43'), findsOneWidget);
+  });
+
+  testWidgets('Restores answer value after restart', (WidgetTester tester) async {
+    await tester.pumpWidget(
+      const MaterialApp(
+        home: RootRestorationScope(
+          restorationId: 'root',
+          child: example.RestorableValueExample(restorationId: 'child'),
+        ),
+      ),
+    );
+
+    // The initial answer value in the example equals 42.
+    expect(find.text('42'), findsOneWidget);
+
+    // Tap the button 10 times to change the answer value.
+    for (int i = 0; i < 10; i++) {
+      await tester.tap(find.byType(OutlinedButton));
+      await tester.pump();
+    }
+
+    // Verify that the answer value increased by 10.
+    expect(find.text('52'), findsOneWidget);
+
+    // Simulate restoring the state of the widget tree after the application
+    // is restarted.
+    await tester.restartAndRestore();
+
+    // Verify that the answer value is restored correctly.
+    expect(find.text('52'), findsOneWidget);
+  });
+}