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]].
- <div style="color: red">See our [commentary](/hints/assignment_6_commentary) on your solutions.</div>
+ See our [commentary](/hints/assignment_6_commentary) on your solutions.
2. Prove that your monad satisfies the monad laws. First, give
and you'll get back an `int state` that when applied to a starting count of `0` yields the result `(6, 2)`---that is, the result of the computation was 6 and the number of operations was 2.
-However, there are several problems here. First off, you shouldn't name your function `lift`, because we're using that name for a function that's interdefinable with `bind` in a specific way. Our canonical `lift` function (also called `mapM` and `liftM` in Haskell) is:
+However, there are several problems here. First off, you shouldn't name your function `lift2`, because we're using that name for a function that's interdefinable with `bind` in a specific way. Our canonical `lift2` function is:
let lift2 (f : 'a -> 'b -> 'c) (u : 'a state) (v : 'b state) : 'c state =
bind u (fun x ->
bind v (fun y ->
unit (f x y)));;
+(Haskell calls this `liftM2`, and calls our `lift` either `liftM` or `mapM`.)
+
OK, so then you might call your function `loft2` instead. So what?
The remaining problem is more subtle. It's that your solution isn't very modular. You've crafted a tool `loft2` that fuses the operation of incrementing the count with the behavior of our `lift2`. What if we needed to deal with some unary functions as well? Then you'd need a `loft1`. What if we need to deal with some functions that are already monadic? Then you'd need a tool that fuses the count-incrementing with the behavior of `bind`. And so on.