library SignedSafeMath {
  int256 constant private _INT256_MIN = -2**255;

  /**
   * @dev Multiplies two signed integers, reverts on overflow.
   */
  function mul(int256 a, int256 b) internal pure returns (int256) {
    // Gas optimization: this is cheaper than requiring 'a' not being zero, but the
    // benefit is lost if 'b' is also tested.
    // See: https://github.com/OpenZeppelin/openzeppelin-contracts/pull/522
    if (a == 0) {
      return 0;
    }

    require(!(a == -1 && b == _INT256_MIN), "SignedSafeMath: multiplication overflow");

    int256 c = a * b;
    require(c / a == b, "SignedSafeMath: multiplication overflow");

    return c;
  }

  /**
   * @dev Integer division of two signed integers truncating the quotient, reverts on division by zero.
   */
  function div(int256 a, int256 b) internal pure returns (int256) {
    require(b != 0, "SignedSafeMath: division by zero");
    require(!(b == -1 && a == _INT256_MIN), "SignedSafeMath: division overflow");

    int256 c = a / b;

    return c;
  }

  /**
   * @dev Subtracts two signed integers, reverts on overflow.
   */
  function sub(int256 a, int256 b) internal pure returns (int256) {
    int256 c = a - b;
    require((b >= 0 && c <= a) || (b < 0 && c > a), "SignedSafeMath: subtraction overflow");

    return c;
  }

  /**
   * @dev Adds two signed integers, reverts on overflow.
   */
  function add(int256 a, int256 b) internal pure returns (int256) {
    int256 c = a + b;
    require((b >= 0 && c >= a) || (b < 0 && c < a), "SignedSafeMath: addition overflow");

    return c;
  }

  /**
   * @notice Computes average of two signed integers, ensuring that the computation
   * doesn't overflow.
   * @dev If the result is not an integer, it is rounded towards zero. For example,
   * avg(-3, -4) = -3
   */
  function avg(int256 _a, int256 _b)
    internal
    pure
    returns (int256)
  {
    if ((_a < 0 && _b > 0) || (_a > 0 && _b < 0)) {
      return add(_a, _b) / 2;
    }
    int256 remainder = (_a % 2 + _b % 2) / 2;
    return add(add(_a / 2, _b / 2), remainder);
  }
}
function div(int256 a, int256 b) internal pure returns (int256) {
  require(b != 0, "SignedSafeMath: division by zero");
  require(!(b == -1 && a == _INT256_MIN), "SignedSafeMath: division overflow");

  int256 c = a / b;

  return c;
}

/**
 * @dev Integer division of two signed integers truncating the quotient, reverts on division by zero.
 */