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-1. Test all three monad laws for the intensionality monad. To do
-this, download the code and load it into your Ocaml evaluator (`# #use
-"intensionality-monad.ml";;`). For instance, what does evaluating
-`bind (unit 'c') (swap left) 2 == swap left 'c' 2;;` show? Please
-explain briefly but clearly what you are doing in your discussion.
+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:
+
+ # lift2 ( + ) (lift2 ( / ) (unit 20) (unit 2))
+ (lift2 ( * ) (unit 2) (unit 3)) 0;;
+ - : int * int = (16, 3)
+
+ Here, `lift2` 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 lift2.
+We encourage you to consider this hint: [[hints/Assignment 6 Hint 1]].
+
+ See our [commentary](/hints/assignment_6_commentary) on your solutions.
+
+
+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.