library SafeMath {

  /**
  * @dev Multiplies two numbers, reverts on overflow.
  */
  function mul(uint256 _a, uint256 _b) internal pure returns (uint256) {
    if (_a == 0) {
      return 0;
    }

    uint256 c = _a * _b;
    require(c / _a == _b);

    return c;
  }

  /**
  * @dev Integer division of two numbers truncating the quotient, reverts on division by zero.
  */
  function div(uint256 _a, uint256 _b) internal pure returns (uint256) {
    require(_b > 0); // Solidity only automatically asserts when dividing by 0
    uint256 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(uint256 _a, uint256 _b) internal pure returns (uint256) {
    require(_b <= _a);
    uint256 c = _a - _b;

    return c;
  }

  /**
  * @dev Adds two numbers, reverts on overflow.
  */
  function add(uint256 _a, uint256 _b) internal pure returns (uint256) {
    uint256 c = _a + _b;
    require(c >= _a);

    return c;
  }

  /**
  * @dev Divides two numbers and returns the remainder (unsigned integer modulo),
  * reverts when dividing by zero.
  */
  function mod(uint256 a, uint256 b) internal pure returns (uint256) {
    require(b != 0);
    return a % b;
  }
}
/**
 * @title SafeMath
 * @dev Math operations with safety checks that revert on error
 */
function mul(uint256 _a, uint256 _b) internal pure returns (uint256) {
    if (_a == 0) {
      return 0;
    }

    uint256 c = _a * _b;
    require(c / _a == _b);

    return c;
  }
/**
 * @dev Multiplies two numbers, reverts on overflow.
 */