... f (u e) ...This subexpression types to `'b reader`, which is good. The only problem is that we made use of an environment `e` that we didn't already have, so we must abstract over that variable to balance the books: fun e -> f (u e) ... [To preview the discussion of the Curry-Howard correspondence, what we're doing here is constructing an intuitionistic proof of the type, and using the Curry-Howard labeling of the proof as our bind term.] This types to `env -> 'b reader`, but we want to end up with `env -> 'b`. Once again, the easiest way to turn a `'b reader` into a `'b` is to apply it to an environment. So we end up as follows:

r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = f (u e) eAnd we're done. This gives us a bind function of the right type. We can then check whether, in combination with the unit function we chose, it satisfies the monad laws, and behaves in the way we intend. And it does. [The bind we cite here is a condensed version of the careful `let a = u e in ...` constructions we provided in earlier lectures. We use the condensed version here in order to emphasize similarities of structure across monads.] The **State Monad** is similar. Once we've decided to use the following type constructor: type 'a state = store -> ('a, store) Then our unit is naturally: let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s) And we can reason our way to the bind function in a way similar to the reasoning given above. First, we need to apply `f` to the contents of the `u` box: let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = ... f (...) ... But unlocking the `u` box is a little more complicated. As before, we need to posit a state `s` that we can apply `u` to. Once we do so, however, we won't have an `'a`, we'll have a pair whose first element is an `'a`. So we have to unpack the pair: ... let (a, s') = u s in ... (f a) ... Abstracting over the `s` and adjusting the types gives the result: let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = fun (s : store) -> let (a, s') = u s in f a s' The **Option/Maybe Monad** doesn't follow the same pattern so closely, so we won't pause to explore it here, though conceptually its unit and bind follow just as naturally from its type constructor. Our other familiar monad is the **List Monad**, which we were told looks like this: type 'a list = ['a];; l_unit (a : 'a) = [a];; l_bind u f = List.concat (List.map f u);; Thinking through the list monad will take a little time, but doing so will provide a connection with continuations. Recall that `List.map` takes a function and a list and returns the result to applying the function to the elements of the list: List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]] and List.concat takes a list of lists and erases the embdded list boundaries: List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3] And sure enough, l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3] Now, why this unit, and why this bind? Well, ideally a unit should not throw away information, so we can rule out `fun x -> []` as an ideal unit. And units should not add more information than required, so there's no obvious reason to prefer `fun x -> [x,x]`. In other words, `fun x -> [x]` is a reasonable choice for a unit. As for bind, an `'a list` monadic object contains a lot of objects of type `'a`, and we want to make use of each of them (rather than arbitrarily throwing some of them away). The only thing we know for sure we can do with an object of type `'a` is apply the function of type `'a -> 'a list` to them. Once we've done so, we have a collection of lists, one for each of the `'a`'s. One possibility is that we could gather them all up in a list, so that `bind' [1;2] (fun i -> [i;i]) ~~> [[1;1];[2;2]]`. But that restricts the object returned by the second argument of `bind` to always be of type `'b list list`. We can elimiate that restriction by flattening the list of lists into a single list: this is just List.concat applied to the output of List.map. So there is some logic to the choice of unit and bind for the list monad. Yet we can still desire to go deeper, and see if the appropriate bind behavior emerges from the types, as it did for the previously considered monads. But we can't do that if we leave the list type as a primitive Ocaml type. However, we know several ways of implementing lists using just functions. In what follows, we're going to use type 3 lists, the right fold implementation (though it's important and intriguing to wonder how things would change if we used some other strategy for implementating lists). These were the lists that made lists look like Church numerals with extra bits embdded in them: empty list: fun f z -> z list with one element: fun f z -> f 1 z list with two elements: fun f z -> f 2 (f 1 z) list with three elements: fun f z -> f 3 (f 2 (f 1 z)) and so on. To save time, we'll let the OCaml interpreter infer the principle types of these functions (rather than inferring what the types should be ourselves): # fun f z -> z;; - : 'a -> 'b -> 'b =