library SafeMath {

    /**

     * @dev Returns the addition of two unsigned integers, reverting on

     * overflow.

     *

     * Counterpart to Solidity's `+` operator.

     *

     * Requirements:

     *

     * - Addition cannot overflow.

     */

    function add(uint256 a, uint256 b) internal pure returns (uint256) {

        uint256 c = a + b;

        require(c >= a, "SafeMath: addition overflow");



        return c;

    }



    /**

     * @dev Returns the subtraction of two unsigned integers, reverting on

     * overflow (when the result is negative).

     *

     * Counterpart to Solidity's `-` operator.

     *

     * Requirements:

     *

     * - Subtraction cannot overflow.

     */

    function sub(uint256 a, uint256 b) internal pure returns (uint256) {

        return sub(a, b, "SafeMath: subtraction overflow");

    }



    /**

     * @dev Returns the subtraction of two unsigned integers, reverting with custom message on

     * overflow (when the result is negative).

     *

     * Counterpart to Solidity's `-` operator.

     *

     * Requirements:

     *

     * - Subtraction cannot overflow.

     */

    function sub(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {

        require(b <= a, errorMessage);

        uint256 c = a - b;



        return c;

    }



    /**

     * @dev Returns the multiplication of two unsigned integers, reverting on

     * overflow.

     *

     * Counterpart to Solidity's `*` operator.

     *

     * Requirements:

     *

     * - Multiplication cannot overflow.

     */

    function mul(uint256 a, uint256 b) internal pure returns (uint256) {

        // Gas optimization: this is cheaper than requiring 'a' not being zero, but the

        // benefit is lost if 'b' is also tested.

        // See: https://github.com/OpenZeppelin/openzeppelin-contracts/pull/522

        if (a == 0) {

            return 0;

        }



        uint256 c = a * b;

        require(c / a == b, "SafeMath: multiplication overflow");



        return c;

    }



    /**

     * @dev Returns the integer division of two unsigned integers. Reverts on

     * division by zero. The result is rounded towards zero.

     *

     * Counterpart to Solidity's `/` operator. Note: this function uses a

     * `revert` opcode (which leaves remaining gas untouched) while Solidity

     * uses an invalid opcode to revert (consuming all remaining gas).

     *

     * Requirements:

     *

     * - The divisor cannot be zero.

     */

    function div(uint256 a, uint256 b) internal pure returns (uint256) {

        return div(a, b, "SafeMath: division by zero");

    }



    /**

     * @dev Returns the integer division of two unsigned integers. Reverts with custom message on

     * division by zero. The result is rounded towards zero.

     *

     * Counterpart to Solidity's `/` operator. Note: this function uses a

     * `revert` opcode (which leaves remaining gas untouched) while Solidity

     * uses an invalid opcode to revert (consuming all remaining gas).

     *

     * Requirements:

     *

     * - The divisor cannot be zero.

     */

    function div(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {

        require(b > 0, errorMessage);

        uint256 c = a / b;

        // assert(a == b * c + a % b); // There is no case in which this doesn't hold



        return c;

    }



    /**

     * @dev Returns the remainder of dividing two unsigned integers. (unsigned integer modulo),

     * Reverts when dividing by zero.

     *

     * Counterpart to Solidity's `%` operator. This function uses a `revert`

     * opcode (which leaves remaining gas untouched) while Solidity uses an

     * invalid opcode to revert (consuming all remaining gas).

     *

     * Requirements:

     *

     * - The divisor cannot be zero.

     */

    function mod(uint256 a, uint256 b) internal pure returns (uint256) {

        return mod(a, b, "SafeMath: modulo by zero");

    }



    /**

     * @dev Returns the remainder of dividing two unsigned integers. (unsigned integer modulo),

     * Reverts with custom message when dividing by zero.

     *

     * Counterpart to Solidity's `%` operator. This function uses a `revert`

     * opcode (which leaves remaining gas untouched) while Solidity uses an

     * invalid opcode to revert (consuming all remaining gas).

     *

     * Requirements:

     *

     * - The divisor cannot be zero.

     */

    function mod(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {

        require(b != 0, errorMessage);

        return a % b;

    }

}
function sub(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {

    require(b <= a, errorMessage);

    uint256 c = a - b;



    return c;

}

/**
 
 * @dev Returns the subtraction of two unsigned integers, reverting with custom message on
 
 * overflow (when the result is negative).
 
 *
 
 * Counterpart to Solidity's `-` operator.
 
 *
 
 * Requirements:
 
 *
 
 * - Subtraction cannot overflow.
 
 */