do_core_upgrade

function do_core_upgrade( $reinstall = false ) {
	global $wp_filesystem;

	require_once ABSPATH . 'wp-admin/includes/class-wp-upgrader.php';

	if ( $reinstall ) {
		$url = 'update-core.php?action=do-core-reinstall';
	} else {
		$url = 'update-core.php?action=do-core-upgrade';
	}
	$url = wp_nonce_url( $url, 'upgrade-core' );

	$version = isset( $_POST['version'] ) ? $_POST['version'] : false;
	$locale  = isset( $_POST['locale'] ) ? $_POST['locale'] : 'en_US';
	$update  = find_core_update( $version, $locale );
	if ( ! $update ) {
		return;
	}

	// Allow relaxed file ownership writes for User-initiated upgrades when the API specifies
	// that it's safe to do so. This only happens when there are no new files to create.
	$allow_relaxed_file_ownership = ! $reinstall && isset( $update->new_files ) && ! $update->new_files;

	?>
	<div class="wrap">
	<h1><?php _e( 'Update WordPress' ); ?></h1>
	<?php

	$credentials = request_filesystem_credentials( $url, '', false, ABSPATH, array( 'version', 'locale' ), $allow_relaxed_file_ownership );
	if ( false === $credentials ) {
		echo '</div>';
		return;
	}

	if ( ! WP_Filesystem( $credentials, ABSPATH, $allow_relaxed_file_ownership ) ) {
		// Failed to connect. Error and request again.
		request_filesystem_credentials( $url, '', true, ABSPATH, array( 'version', 'locale' ), $allow_relaxed_file_ownership );
		echo '</div>';
		return;
	}

	if ( $wp_filesystem->errors->has_errors() ) {
		foreach ( $wp_filesystem->errors->get_error_messages() as $message ) {
			show_message( $message );
		}
		echo '</div>';
		return;
	}

	if ( $reinstall ) {
		$update->response = 'reinstall';
	}

	add_filter( 'update_feedback', 'show_message' );

	$upgrader = new Core_Upgrader();
	$result   = $upgrader->upgrade(
		$update,
		array(
			'allow_relaxed_file_ownership' => $allow_relaxed_file_ownership,
		)
	);

	if ( is_wp_error( $result ) ) {
		show_message( $result );
		if ( 'up_to_date' != $result->get_error_code() && 'locked' != $result->get_error_code() ) {
			show_message( __( 'Installation failed.' ) );
		}
		echo '</div>';
		return;
	}

	show_message( __( 'WordPress updated successfully.' ) );
	show_message(
		'<span class="hide-if-no-js">' . sprintf(
			/* translators: 1: WordPress version, 2: URL to About screen. */
			__( 'Welcome to WordPress %1$s. You will be redirected to the About WordPress screen. If not, click <a href="%2$s">here</a>.' ),
			$result,
			esc_url( self_admin_url( 'about.php?updated' ) )
		) . '</span>'
	);
	show_message(
		'<span class="hide-if-js">' . sprintf(
			/* translators: 1: WordPress version, 2: URL to About screen. */
			__( 'Welcome to WordPress %1$s. <a href="%2$s">Learn more</a>.' ),
			$result,
			esc_url( self_admin_url( 'about.php?updated' ) )
		) . '</span>'
	);
	?>
	</div>
	<script type="text/javascript">
	window.location = '<?php echo self_admin_url( 'about.php?updated' ); ?>';
	</script>
	<?php
}