library SafeMath {



  /**

  * @dev Multiplies two numbers, throws on overflow.

  */

  function mul(uint256 _a, uint256 _b) internal pure returns (uint256 c) {

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

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

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

    if (_a == 0) {

      return 0;

    }



    c = _a * _b;

    assert(c / _a == _b);

    return c;

  }



  /**

  * @dev Integer division of two numbers, truncating the quotient.

  */

  function div(uint256 _a, uint256 _b) internal pure returns (uint256) {

    // assert(_b > 0); // Solidity automatically throws 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 _a / _b;

  }



  /**

  * @dev Subtracts two numbers, throws on overflow (i.e. if subtrahend is greater than minuend).

  */

  function sub(uint256 _a, uint256 _b) internal pure returns (uint256) {

    assert(_b <= _a);

    return _a - _b;

  }



  /**

  * @dev Adds two numbers, throws on overflow.

  */

  function add(uint256 _a, uint256 _b) internal pure returns (uint256 c) {

    c = _a + _b;

    assert(c >= _a);

    return c;

  }

}
/**
 
 * @title SafeMath
 
 * @dev Math operations with safety checks that throw on error
 
 */
function div(uint256 _a, uint256 _b) internal pure returns (uint256) {

    // assert(_b > 0); // Solidity automatically throws 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 _a / _b;

  }
/**
 
 * @dev Integer division of two numbers, truncating the quotient.
 
 */