jQuery Validate のリモート メソッドを使用してユーザー名の可用性を確認するにはどうすればよいですか?

$("#signupForm").validate({
    rules: {
        username: {
            required: true,
            minlength: 3,
            remote: "check-username.php"
        }
    },
    messages: {
        username: {
            remote: "This username is already taken! Try another."
        }
    }
});

<?php
require_once "./source/includes/data.php";
header('Content-type: application/json');

$name = mysql_real_escape_string($_POST['username']);

$query = mysql_query("SELECT * FROM mmh_user_info WHERE username ='$username'");
$result = mysql_num_rows($query);
if ($result == 0){
$valid = 'true';}
else{
$valid = 'false';
}
echo $valid;
?>

$("#signupForm").submit(function(e) {
    e.preventDefault();
    $(this).validate();
});

// Initialize the database connection

// Query to check if the username exists
$query = "SELECT * FROM mmh_user_info WHERE username='$name'";
$result = mysql_query($query);

// Check the result and return a JSON response
if (mysql_num_rows($result) > 0) {
    echo json_encode(array('status' => 'error', 'message' => 'Username already exists'));
} else {
    echo json_encode(array('status' => 'success', 'message' => 'Username available'));
}