library SafeMathLib {
    
  /**
  * @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;
    require(c / a == b, "SafeMath mul failed");
    return c;
  }

  /**
  * @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) 
  {
    require(b <= a, "SafeMath sub failed");
    return a - b;
  }

  /**
  * @dev Adds two numbers, throws on overflow.
  */
  function add(uint256 a, uint256 b)
      internal
      pure
      returns (uint256 c) 
  {
    c = a + b;
    require(c >= a, "SafeMath add failed");
    return c;
  }
  
  /**
    * @dev gives square root of given x.
    */
  function sqrt(uint256 x)
      internal
      pure
      returns (uint256 y) 
  {
    uint256 z = ((add(x,1)) / 2);
    y = x;
    while (z < y) 
    {
      y = z;
      z = ((add((x / z),z)) / 2);
    }
  }
  
  /**
    * @dev gives square. multiplies x by x
    */
  function sq(uint256 x)
      internal
      pure
      returns (uint256)
  {
    return (mul(x,x));
  }
  
  /**
    * @dev x to the power of y 
    */
  function pwr(uint256 x, uint256 y)
      internal 
      pure 
      returns (uint256)
  {
    if (x==0)
        return (0);
    else if (y==0)
        return (1);
    else 
    {
      uint256 z = x;
      for (uint256 i = 1; i < y; i++)
        z = mul(z,x);
      return (z);
    }
  }
}
function add(uint256 a, uint256 b)
    internal
    pure
    returns (uint256 c) 
{
  c = a + b;
  require(c >= a, "SafeMath add failed");
  return c;
}

/**
 * @dev Adds two numbers, throws on overflow.
 */