pragma solidity ^0.6.0;


contract DSMath {
    function add(uint256 x, uint256 y) internal pure returns (uint256 z) {
        require((z = x + y) >= x);
    }

    function sub(uint256 x, uint256 y) internal pure returns (uint256 z) {
        require((z = x - y) <= x);
    }

    function mul(uint256 x, uint256 y) internal pure returns (uint256 z) {
        require(y == 0 || (z = x * y) / y == x);
    }

    function div(uint256 x, uint256 y) internal pure returns (uint256 z) {
        return x / y;
    }

    function min(uint256 x, uint256 y) internal pure returns (uint256 z) {
        return x <= y ? x : y;
    }

    function max(uint256 x, uint256 y) internal pure returns (uint256 z) {
        return x >= y ? x : y;
    }

    function imin(int256 x, int256 y) internal pure returns (int256 z) {
        return x <= y ? x : y;
    }

    function imax(int256 x, int256 y) internal pure returns (int256 z) {
        return x >= y ? x : y;
    }

    uint256 constant WAD = 10**18;
    uint256 constant RAY = 10**27;

    function wmul(uint256 x, uint256 y) internal pure returns (uint256 z) {
        z = add(mul(x, y), WAD / 2) / WAD;
    }

    function rmul(uint256 x, uint256 y) internal pure returns (uint256 z) {
        z = add(mul(x, y), RAY / 2) / RAY;
    }

    function wdiv(uint256 x, uint256 y) internal pure returns (uint256 z) {
        z = add(mul(x, WAD), y / 2) / y;
    }

    function rdiv(uint256 x, uint256 y) internal pure returns (uint256 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(uint256 x, uint256 n) internal pure returns (uint256 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);
            }
        }
    }
}