Then our unit is naturally:
- let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s)
+ 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:
List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
-and List.concat takes a list of lists and erases the embedded list
+and `List.concat` takes a list of lists and erases the embedded list
boundaries:
List.concat [[1; 2]; [2; 3]] ~~> [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
+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
the object returned by the second argument of `bind` to always be of
type `'b list list`. We can eliminate 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
+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
+lists using just functions. In what follows, we're going to use version
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 implementing lists). These were the lists that made
So an `('a, 'b) list'` is a list containing elements of type `'a`,
where `'b` is the type of some part of the plumbing. This is more
general than an ordinary OCaml list, but we'll see how to map them
-into OCaml lists soon. We don't need to fully grasp the role of the `'b`'s
+into OCaml lists soon. We don't need to fully grasp the role of the `'b`s
in order to proceed to build a monad:
l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun k z -> k a z
-No problem. Arriving at bind is a little more complicated, but
-exactly the same principles apply, you just have to be careful and
-systematic about it.
+Take an `'a` and return its v3-style singleton. No problem. Arriving at bind
+is a little more complicated, but exactly the same principles apply, you just
+have to be careful and systematic about it.
l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ...