library SafeMath {



  /**

  * @dev Multiplies two numbers, throws on overflow.

  */

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

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

  assert(b <= a);

  return a - b;

}

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