Autocomplete Combobox in WPF

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C#

WPF

combobox

Guy's after trying to autocomplete a textbox i move towards now to an autocomplete combobox.Because i cannot add all controls to a textbox.But in combobox,i did that.But a problem also stands there.When i type in combobox it's popup filters the items according to typed text nicely,but i need to select a option by mouse,cannot by keyboard arrow's.I tried to solve it with keydown event but somehow it cannot focus combobox i think.Someone can suggest me here what wrong i am doing or how can i enable keyboard arrowkey events?Below are my codes:

XML:

<ComboBox Height="23" HorizontalAlignment="Left" Margin="160,236,0,0" Name="comboBox1" VerticalAlignment="Top" Width="120" IsEditable="True" ItemsSource="{Binding}" KeyDown="comboBox1_KeyDown">
    <ComboBox.Resources>
        <sys:Double x:Key="{x:Static SystemParameters.VerticalScrollBarWidthKey}">0</sys:Double>
    </ComboBox.Resources>
</ComboBox>

C#:

    public partial class MainWindow : Window
    {
        List<string> nameList;
        
        public MainWindow()
        {
            InitializeComponent();
            nameList = new List<string> 
            {
                "A0-Word","B0-Word","C0-Word",
                "A1-Word","A111","A11122","B1-Word","C1-Word",
                "B2-Word","C2-Word",
                "C3-Word",
            };

           comboBox1.DataContext = nameList;

            comboBox1.Loaded += delegate
            {
                System.Windows.Controls.TextBox textBox = comboBox1.Template.FindName("PART_EditableTextBox", comboBox1) as System.Windows.Controls.TextBox;
                Popup popup = comboBox1.Template.FindName("PART_Popup", comboBox1) as Popup;
                if (textBox != null)
                {
                    textBox.TextChanged += delegate
                    {
                            comboBox1.Items.Filter += (item) =>
                            {
                                if (item.ToString().StartsWith(textBox.Text))
                                {
                                    popup.IsOpen = true;
                                    return true;
                                    
                                }
                                else
                                {
                                   // popup.IsOpen = false;
                                    return false;
                                }
                            };

                    };
                }
            };

            comboBox1.KeyDown+=new System.Windows.Input.KeyEventHandler(comboBox1_KeyDown);
            
        }

private void comboBox1_KeyDown(object sender, System.Windows.Input.KeyEventArgs e)
{

    if (e.Key == Key.Down)
    {
        e.Handled = true;
        comboBox1.Items.MoveCurrentToNext();
    }
}