```# fun f z -> z;;
- : 'a -> 'b -> 'b =
# fun f z -> f 1 z;;
- : (int -> 'a -> 'b) -> 'a -> 'b =
# fun f z -> f 2 (f 1 z);;
- : (int -> 'a -> 'a) -> 'a -> 'a =
# fun f z -> f 3 (f 2 (f 1 z))
- : (int -> 'a -> 'a) -> 'a -> 'a =
```
Finally, we're getting consistent principle types, so we can stop. These types should remind you of the simply-typed lambda calculus types for Church numerals (`(o -> o) -> o -> o`) with one extra bit thrown in (in this case, and int). So here's our type constructor for our hand-rolled lists: type 'a list' = (int -> 'a -> 'a) -> 'a -> 'a Generalizing to lists that contain any kind of element (not just ints), we have type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b 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 grasp the role of the `'b`'s in order to proceed to build a monad: l'_unit (x:'a):(('a, 'b) list) = fun x -> fun f z -> f x 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. l'_bind (u:('a,'b) list') (f:'a -> ('c, 'd) list'): ('c, 'd) list' = ... Unfortunately, we'll need to spell out the types: l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) (f: 'a -> ('c -> 'd -> 'd) -> 'd -> 'd) : ('c -> 'd -> 'd) -> 'd -> 'd = ... It's a rookie mistake to quail before complicated types. You should be no more intimiated by complex types than by a linguistic tree with deeply embedded branches: complex structure created by repeated application of simple rules. As usual, we need to unpack the `u` box. Examine the type of `u`. This time, `u` will only deliver up its contents if we give `u` as an argument a function expecting an `'a`. Once that argument is applied to an object of type `'a`, we'll have what we need. Thus: .... u (fun (x:'a) -> ... (f a) ... ) ... In order for `u` to have the kind of argument it needs, we have to adjust `(f a)` (which has type `('c -> 'd -> 'd) -> 'd -> 'd`) in order to deliver something of type `'b -> 'b`. The easiest way is to alias `'d` to `'b`, and provide `(f a)` with an argument of type `'c -> 'b -> 'b`. Thus: l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) : ('c -> 'b -> 'b) -> 'b -> 'b = .... u (fun (x:'a) -> f a k) ... [Excercise: can you arrive at a fully general bind for this type constructor, one that does not collapse `'d`'s with `'b`'s?] As usual, we have to abstract over `k`, but this time, no further adjustments are needed: l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) : ('c -> 'b -> 'b) -> 'b -> 'b = fun (k:'c -> 'b -> 'b) -> u (fun (x:'a) -> f a k) You should carefully check to make sure that this term is consistent with the typing. Our theory is that this monad should be capable of exactly replicating the behavior of the standard List monad. Let's test: l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3] l'_bind (fun f z -> f 1 (f 2 z)) (fun i -> fun f z -> f i (f (i+1) z)) ~~> Sigh. Ocaml won't show us our own list. So we have to choose an `f` and a `z` that will turn our hand-crafted lists into standard Ocaml lists, so that they will print out.
```# let cons h t = h :: t;;  (* Ocaml is stupid about :: *)
# l'_bind (fun f z -> f 1 (f 2 z))
(fun i -> fun f z -> f i (f (i+1) z)) cons [];;
- : int list = [1; 2; 2; 3]
```