contract DSMath {

    function add(uint x, uint y) internal pure returns (uint z) {

        require((z = x + y) >= x);

    }

    function sub(uint x, uint y) internal pure returns (uint z) {

        require((z = x - y) <= x);

    }

    function mul(uint x, uint y) internal pure returns (uint z) {

        require(y == 0 || (z = x * y) / y == x);

    }



    function min(uint x, uint y) internal pure returns (uint z) {

        return x <= y ? x : y;

    }

    function max(uint x, uint y) internal pure returns (uint z) {

        return x >= y ? x : y;

    }

    function imin(int x, int y) internal pure returns (int z) {

        return x <= y ? x : y;

    }

    function imax(int x, int y) internal pure returns (int z) {

        return x >= y ? x : y;

    }



    uint constant WAD = 10 ** 18;

    uint constant RAY = 10 ** 27;



    function wmul(uint x, uint y) internal pure returns (uint z) {

        z = add(mul(x, y), WAD / 2) / WAD;

    }

    function rmul(uint x, uint y) internal pure returns (uint z) {

        z = add(mul(x, y), RAY / 2) / RAY;

    }

    function wdiv(uint x, uint y) internal pure returns (uint z) {

        z = add(mul(x, WAD), y / 2) / y;

    }

    function rdiv(uint x, uint y) internal pure returns (uint z) {

        z = add(mul(x, RAY), y / 2) / y;

    }



    // This famous algorithm is called "exponentiation by squaring"

    // and calculates x^n with x as fixed-point and n as regular unsigned.

    //

    // It's O(log n), instead of O(n) for naive repeated multiplication.

    //

    // These facts are why it works:

    //

    //  If n is even, then x^n = (x^2)^(n/2).

    //  If n is odd,  then x^n = x * x^(n-1),

    //   and applying the equation for even x gives

    //    x^n = x * (x^2)^((n-1) / 2).

    //

    //  Also, EVM division is flooring and

    //    floor[(n-1) / 2] = floor[n / 2].

    //

    function rpow(uint x, uint n) internal pure returns (uint z) {

        z = n % 2 != 0 ? x : RAY;



        for (n /= 2; n != 0; n /= 2) {

            x = rmul(x, x);



            if (n % 2 != 0) {

                z = rmul(z, x);

            }

        }

    }

}
function rpow(uint x, uint n) internal pure returns (uint z) {

    z = n % 2 != 0 ? x : RAY;



    for (n /= 2; n != 0; n /= 2) {

        x = rmul(x, x);



        if (n % 2 != 0) {

            z = rmul(z, x);

        }

    }

}

// This famous algorithm is called "exponentiation by squaring"
// and calculates x^n with x as fixed-point and n as regular unsigned.
//
// It's O(log n), instead of O(n) for naive repeated multiplication.
//
// These facts are why it works:
//
//  If n is even, then x^n = (x^2)^(n/2).
//  If n is odd,  then x^n = x * x^(n-1),
//   and applying the equation for even x gives
//    x^n = x * (x^2)^((n-1) / 2).
//
//  Also, EVM division is flooring and
//    floor[(n-1) / 2] = floor[n / 2].
//