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Editing: class-wp-widget-search.php

<?php
/**
 * Widget API: WP_Widget_Search class
 *
 * @package WordPress
 * @subpackage Widgets
 * @since 4.4.0
 */

/**
 * Core class used to implement a Search widget.
 *
 * @since 2.8.0
 *
 * @see WP_Widget
 */
class WP_Widget_Search extends WP_Widget {

/**
	 * Sets up a new Search widget instance.
	 *
	 * @since 2.8.0
	 */
	public function __construct() {
		$widget_ops = array(
			'classname'                   => 'widget_search',
			'description'                 => __( 'A search form for your site.' ),
			'customize_selective_refresh' => true,
			'show_instance_in_rest'       => true,
		);
		parent::__construct( 'search', _x( 'Search', 'Search widget' ), $widget_ops );
	}

/**
	 * Outputs the content for the current Search widget instance.
	 *
	 * @since 2.8.0
	 *
	 * @param array $args     Display arguments including 'before_title', 'after_title',
	 *                        'before_widget', and 'after_widget'.
	 * @param array $instance Settings for the current Search widget instance.
	 */
	public function widget( $args, $instance ) {
		$title = ! empty( $instance['title'] ) ? $instance['title'] : '';

/** This filter is documented in wp-includes/widgets/class-wp-widget-pages.php */
		$title = apply_filters( 'widget_title', $title, $instance, $this->id_base );

echo $args['before_widget'];
		if ( $title ) {
			echo $args['before_title'] . $title . $args['after_title'];
		}

// Use active theme search form if it exists.
		get_search_form();

echo $args['after_widget'];
	}

/**
	 * Outputs the settings form for the Search widget.
	 *
	 * @since 2.8.0
	 *
	 * @param array $instance Current settings.
	 */
	public function form( $instance ) {
		$instance = wp_parse_args( (array) $instance, array( 'title' => '' ) );
		$title    = $instance['title'];
		?>
		<p>
			<label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?></label>
			<input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo esc_attr( $title ); ?>" />
		</p>
		<?php
	}

/**
	 * Handles updating settings for the current Search widget instance.
	 *
	 * @since 2.8.0
	 *
	 * @param array $new_instance New settings for this instance as input by the user via
	 *                            WP_Widget::form().
	 * @param array $old_instance Old settings for this instance.
	 * @return array Updated settings.
	 */
	public function update( $new_instance, $old_instance ) {
		$instance          = $old_instance;
		$new_instance      = wp_parse_args( (array) $new_instance, array( 'title' => '' ) );
		$instance['title'] = sanitize_text_field( $new_instance['title'] );
		return $instance;
	}
}