1. **Build a state monad.** Based on the division by zero monad,
build a system that will evaluate arithmetic expressions. Instead of
returning a simple integer as a result, it will deliver the correct
answer along with a count of the number of operations performed during
the calculation. That is, the desired behavior should be like this:
# lift ( + ) (lift ( / ) (unit 20) (unit 2))
(lift ( * ) (unit 2) (unit 3)) 0;;
- : int * int = (16, 3)
Here, `lift` is the function that uses `bind` to prepare an ordinary
arithmetic operator (such as addition `( + )`, division `( / )`, or
multiplication `( * )`) to recieve objects from the counting monad as
arguments. The response of the interpreter says two things: that
(20/2) + (2*3) = 16, and that the computation took three arithmetic
steps. By the way, that zero at the end provides the monadic object
with a starting point (0 relevant computations have occurred previous
to the current computation).
Assume for the purposes of this excercise that no one ever tries to
divide by zero (so there should be no int option types anywhere in
your solution).
You'll need to define a computation monad type, unit, bind, and lift.
We encourage you to consider this hint: [[Assignment 6 Hint 1]].
2. Prove that your monad satisfies the monad laws. First, give
examples illustrating specific cases in which the monad laws are
obeyed, then explain (briefly, not exhaustively) why the laws hold in
general for your unit and bind operators.
3. How would you extend your strategy if you wanted to count
arithmetic operations, but you also wanted to be safe from division by
zero? This is a deep question: how should you combine two monads into
a single system? If you don't arrive at working code, you can still
discuss the issues and design choices.