package higher_kinded

README.mdx

"Higher kinded types"
=====================

OCaml natively supports parameterized type constructors, such as
`option`. The parameters of a type constructor may only be types, not
arbitrary type constructors.

The following is not legal syntax:

```
type 'a person =
  { name : string 'a
  ; age  : int 'a
  }
```

It is not possible to define such a type where `'a` can be replaced
with something like `option` or `ref`, because you can't apply `'a` to
other types like `string` or `int`. In other words, although `int
option` is a valid type expression, `int 'a` is not.

The `Higher_kinded` library makes something similar possible. The
above example would be defined like this:

```ocaml
type 'a person =
  { name : (string -> 'a) Higher_kinded.t
  ; age : (int -> 'a) Higher_kinded.t
  }
```

The fundamental concept of `Higher_kinded` is that a value of type `(a
-> ... -> z -> C.higher_kinded) Higher_kinded.t` is equivalent to a
value of type `(a, ..., z) C.t`. The only reason it is rendered as a
function is that `->` is the only right associative type operator,
which is useful for reasons that will be explained later.

A signature defining a type constructor can include one of the
`Higher_kinded.S` signatures, and its implementation should use one of
the `Higher_kinded.Make` functors. For example, `Option` could look
something like this:

```ocaml
# module Option : sig
    type 'a t = 'a option

    include Higher_kinded.S with type 'a t := 'a t
  end = struct
    type 'a t = 'a option

    include Higher_kinded.Make (Base.Option)
  end
module Option :
  sig
    type 'a t = 'a option
    type higher_kinded
    val inject : 'a t -> ('a -> higher_kinded) Higher_kinded.t
    val project : ('a -> higher_kinded) Higher_kinded.t -> 'a t
  end
```

Now it is possible to define values of type `(int ->
Option.higher_kinded) Higher_kinded.t`:

```ocaml
# let a = Option.inject (None : int option)
val a : (int -> Option.higher_kinded) Higher_kinded.t = <abstr>
# let b = Option.inject (Some 42)
val b : (int -> Option.higher_kinded) Higher_kinded.t = <abstr>
```

Here is how to observe them:

```ocaml
# Option.project b
- : int option = Some 42
```

Now that `Option` can be used this way, we can express the `person`
example from earlier:

```ocaml
# let alice = { name = Option.inject (Some "alice doe"); age = Option.inject None }
val alice : Option.higher_kinded person = {name = <abstr>; age = <abstr>}
```

If we did the same thing with refs:

```ocaml
module Ref : sig
  type 'a t = 'a ref

  include Higher_kinded.S with type 'a t := 'a t
end = struct
  type 'a t = 'a ref

  include Higher_kinded.Make (Base.Ref)
end
```

we could write:

```ocaml
# let secret_agent =
    { name = Ref.inject (ref "alice"); age = Ref.inject (ref 55) }
val secret_agent : Ref.higher_kinded person = {name = <abstr>; age = <abstr>}
```

Here's how we could modify the references:

```ocaml
Ref.project secret_agent.name := "Austin Powers";
Ref.project secret_agent.age := 35
```

The `inject` and `project` functions have no runtime cost; they only
change the type.

You can also use `Higher_kinded` for types that have multiple type
parameters. Here is an example using `Result`:

```ocaml
# module Result : sig
    type ('a, 'e) t = ('a, 'e) result

    include Higher_kinded.S2 with type ('a, 'e) t := ('a, 'e) t
  end = struct
    type ('a, 'e) t = ('a, 'e) result

    include Higher_kinded.Make2 (Base.Result)
  end
module Result :
  sig
    type ('a, 'e) t = ('a, 'e) result
    type higher_kinded
    val inject : ('a, 'z) t -> ('a -> 'z -> higher_kinded) Higher_kinded.t
    val project : ('a -> 'z -> higher_kinded) Higher_kinded.t -> ('a, 'z) t
  end
```

You can even use multi-parameter higher kinded witnesses in positions
expecting lower arity types.  For example, suppose that you had an
error type defined:

```ocaml
type error =
  | Integer_overflow
```

Then you could write:

```ocaml
# let immortal =
    { name = Result.inject (Ok "Keanu")
    ; age = Result.inject (Error Integer_overflow)
    }
val immortal : (error -> Result.higher_kinded) person =
  {name = <abstr>; age = <abstr>}
```

The resulting type uses a partially applied witness (in the above
example, `error -> Result.higher_kinded`) that explains how to fill in
the remaining arguments.

This library is a variation on Jeremy Yallop and Leo White's
"Lightweight higher-kinded polymorphism".