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;
    require(c / a == b, "SafeMath mul failed");
    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 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);
    }
  }
}
/**
 * @title SafeMath v0.1.9
 * @dev Math operations with safety checks that throw on error
 * change notes:  original SafeMath library from OpenZeppelin modified by Inventor
 * - added sqrt
 * - added sq
 * - added pwr 
 * - changed asserts to requires with error log outputs
 * - removed div, its useless
 */
function sub(uint256 a, uint256 b)
internal
pure
returns(uint256) {
  require(b <= a, "SafeMath sub failed");
  return a - b;
}

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