library SafeMath64 {
    string private constant ERROR_ADD_OVERFLOW = "MATH64_ADD_OVERFLOW";
    string private constant ERROR_SUB_UNDERFLOW = "MATH64_SUB_UNDERFLOW";
    string private constant ERROR_MUL_OVERFLOW = "MATH64_MUL_OVERFLOW";
    string private constant ERROR_DIV_ZERO = "MATH64_DIV_ZERO";

    /**
    * @dev Multiplies two numbers, reverts on overflow.
    */
    function mul(uint64 _a, uint64 _b) internal pure returns (uint64) {
        uint256 c = uint256(_a) * uint256(_b);
        require(c < 0x010000000000000000, ERROR_MUL_OVERFLOW); // 2**64 (less gas this way)

        return uint64(c);
    }

    /**
    * @dev Integer division of two numbers truncating the quotient, reverts on division by zero.
    */
    function div(uint64 _a, uint64 _b) internal pure returns (uint64) {
        require(_b > 0, ERROR_DIV_ZERO); // Solidity only automatically asserts when dividing by 0
        uint64 c = _a / _b;
        // assert(_a == _b * c + _a % _b); // There is no case in which this doesn't hold

        return c;
    }

    /**
    * @dev Subtracts two numbers, reverts on overflow (i.e. if subtrahend is greater than minuend).
    */
    function sub(uint64 _a, uint64 _b) internal pure returns (uint64) {
        require(_b <= _a, ERROR_SUB_UNDERFLOW);
        uint64 c = _a - _b;

        return c;
    }

    /**
    * @dev Adds two numbers, reverts on overflow.
    */
    function add(uint64 _a, uint64 _b) internal pure returns (uint64) {
        uint64 c = _a + _b;
        require(c >= _a, ERROR_ADD_OVERFLOW);

        return c;
    }

    /**
    * @dev Divides two numbers and returns the remainder (unsigned integer modulo),
    * reverts when dividing by zero.
    */
    function mod(uint64 a, uint64 b) internal pure returns (uint64) {
        require(b != 0, ERROR_DIV_ZERO);
        return a % b;
    }
}
/**
 * @title SafeMath64
 * @dev Math operations for uint64 with safety checks that revert on error
 */
function div(uint64 _a, uint64 _b) internal pure returns (uint64) {
    require(_b > 0, ERROR_DIV_ZERO); // Solidity only automatically asserts when dividing by 0
    uint64 c = _a / _b;
    // assert(_a == _b * c + _a % _b); // There is no case in which this doesn't hold

    return c;
}

/**
 * @dev Integer division of two numbers truncating the quotient, reverts on division by zero.
 */