Write these up and submit them to Canvas, the day before the lecture.

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You do not need to write a summary/commentary for the papers:
* Abstract Interpretation
* Abstract Interpretation: a unified lattice model for static analysis of
  programs by construction or approximation of fixpoints

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Monotonicity of the transfer and lub functions

A transfer function is not required to be monotonic, but the lub function must
be monotonic.
* Why must the lub function be monotonic?
* What might go wrong if lub is were somehow not monotonic?

Give an example of using a non-monotonic function in place of the lub function
and show how it causes the problem you specified above.  Using the same
lattice and a lub function that is monotonic, show that the problem does not
occur.

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Reasoning about programs

Stein's algorithm (https://en.wikipedia.org/wiki/Binary_GCD_algorithm)
computes the greatest common divisor (GCD) of two integers.  On some CPU
architectures, Stein's algorithm admits a faster implementation than the far
more famous Euclidean algorithm (https://en.wikipedia.org/wiki/Euclidean_algorithm).

Here is one possible implementation of Stein's algorithm:

    def gcd(int a, int b):
        int d = 0
        while a is even and b is even:
            a = a / 2
            b = b / 2
            d = d + 1
        while a != b:
            if a is even:   a = a / 2
            elif b is even: b = b / 2
            elif a > b:     a = (a - b) / 2
            else:           b = (b - a) / 2
        return a * 2^d

A program will crash if it ever divides by zero.  Does this program ever
crash due to division by zero?  Carefully explain how you figured this out.

Stein's algorithm is intended to compute a value; it should always
terminate.  Is it possible for this program to run forever?  Carefully
explain how you figured this out.

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end.