library SafeMath {
    function add(uint256 x, uint256 y) internal pure returns (uint256 z) {
        require((z = x + y) >= x, "MATH:: ADD_OVERFLOW");
    }

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

    function mul(uint256 a, uint256 b) internal pure returns (uint256) {
        // 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;
        }

        uint256 c = a * b;
        require(c / a == b, "MATH:: MUL_OVERFLOW");

        return c;
    }

    function div(uint256 a, uint256 b) internal pure returns (uint256) {
        require(b > 0, "MATH:: DIVISION_BY_ZERO");
        uint256 c = a / b;
        // assert(a == b * c + a % b); // There is no case in which this doesn't hold

        return c;
    }

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

    // babylonian method (https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method)
    function sqrt(uint y) internal pure returns (uint z) {
        if (y > 3) {
            z = y;
            uint x = y / 2 + 1;
            while (x < z) {
                z = x;
                x = (y / x + x) / 2;
            }
        } else if (y != 0) {
            z = 1;
        }
    }
}
// A library for performing overflow-safe math, courtesy of DappHub: https://github.com/dapphub/ds-math/blob/d0ef6d6a5f/src/math.sol
// Modified to include only the essentials
function sqrt(uint y) internal pure returns (uint z) {
    if (y > 3) {
        z = y;
        uint x = y / 2 + 1;
        while (x < z) {
            z = x;
            x = (y / x + x) / 2;
        }
    } else if (y != 0) {
        z = 1;
    }
}

// babylonian method (https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method)