# 5.2. Modules¶

We begin with a couple of examples of the OCaml module system before diving into the details.

A structure is simply a collection of definitions, such as:

struct
let inc x = x + 1
type primary_color = Red | Green | Blue
exception Oops
end


In a way, the structure is like a record: the structure has some distinct components with names. But unlike a record, it can define new types, exceptions, and so forth.

By itself the code above won’t compile, because structures do not have the same first-class status as values like integers or functions. You can’t just enter that code in utop, or pass that structure to a function, etc. What you can do is bind the structure to a name:

module MyModule = struct
let inc x = x + 1
type primary_color = Red | Green | Blue
exception Oops
end

module MyModule :
sig
val inc : int -> int
type primary_color = Red | Green | Blue
exception Oops
end


The output from OCaml has the form:

module MyModule : sig ... end


This indicates that MyModule has been defined, and that it has been inferred to have the module type that appears to the right of the colon. That module type is written as signature:

sig
val inc : int -> int
type primary_color = Red | Green | Blue
exception Oops
end


The signature itself is a collection of specifications. The specifications for variant types and exceptions are simply their original definitions, so primary_color and Oops are no different than they were in the original structure. The specification for inc though is written with the val keyword, exactly as the toplevel would respond if we defined inc in it.

Note

This use of the word “specification” is perhaps confusing, since many programmers would use that word to mean “the comments specifying the behavior of a function.” But if we broaden our sight a little, we could allow that the type of a function is part of its specification. So it’s at least a related sense of the word.

The definitions in a module are usually more closely related than those in MyModule. Often a module will implement some data structure. For example, here is a module for stacks implemented as linked lists:

module ListStack = struct
(** [empty] is the empty stack. *)
let empty = []

(** [is_empty s] is whether [s] is empty. *)
let is_empty = function [] -> true | _ -> false

(** [push x s] pushes [x] onto the top of [s]. *)
let push x s = x :: s

(** [Empty] is raised when an operation cannot be applied
to an empty stack. *)
exception Empty

(** [peek s] is the top element of [s].
Raises [Empty] if [s] is empty. *)
let peek = function
| [] -> raise Empty
| x :: _ -> x

(** [pop s] is all but the top element of [s].
Raises [Empty] if [s] is empty. *)
let pop = function
| [] -> raise Empty
| _ :: s -> s
end

module ListStack :
sig
val empty : 'a list
val is_empty : 'a list -> bool
val push : 'a -> 'a list -> 'a list
exception Empty
val peek : 'a list -> 'a
val pop : 'a list -> 'a list
end


Important

The specification of pop might surprise you. Note that it does not return the top element. That’s the job of peek. Instead, pop returns all but the top element.

We can then use that module to manipulate a stack:

ListStack.push 2 (ListStack.push 1 ListStack.empty)

- : int list = [2; 1]


Warning

There’s a common confusion lurking here for those programmers coming from object-oriented languages. It’s tempting to think of ListStack as being an object on which you invoke methods. Indeed ListStack.push vaguely looks like we’re invoking a push method on a ListStack object. But that’s not what is happening. In an OO language you could instantiate many stack objects. But here, there is only one ListStack. Moreover it is not an object, in large part because it has no notion of a this or self keyword to denote the receiving object of the method call.

That’s admittedly rather verbose code. Soon we’ll see several solutions to that problem, but for now here’s one:

ListStack.(push 2 (push 1 empty))

- : int list = [2; 1]


By writing ListStack.(e), all the names from ListStack become usable in e without needing to write the prefix ListStack. each time. Another improvement could be using the pipeline operator:

ListStack.(empty |> push 1 |> push 2)

- : int list = [2; 1]


Now we can read the code left-to-right without having to parse parentheses. Nice.

Warning

There’s another common OO confusion lurking here. It’s tempting to think of ListStack as being a class from which objects are instantiated. That’s not the case though. Notice how there is no new operator used to create a stack above, nor any constructors (in the OO sense of that word).

Modules are considerably more basic than classes. A module is just a collection of definitions in its own namespace. In ListStack, we have some definitions of functions—push, pop, etc.—and one value, empty.

So whereas in Java we might create a couple of stacks using code like this:

Stack s1 = new Stack();
s1.push(1);
s1.push(2);
Stack s2 = new Stack();
s2.push(3);


In OCaml the same stacks could be created as follows:

let s1 = ListStack.(empty |> push 1 |> push 2)
let s2 = ListStack.(empty |> push 3)

val s1 : int list = [2; 1]

val s2 : int list = [3]


## 5.2.1. Module Definitions¶

The module definition keyword is much like the let definition keyword that we learned before. (The OCaml designers hypothetically could have chosen to use let_module instead of module to emphasize the similarity.) The difference is just that:

• let binds a value to a name, whereas

• module binds a module value to a name.

Syntax.

The most common syntax for a module definition is simply:

module ModuleName = struct
module_items
end


where module_items inside a structure can include let definitions, type definitions, and exception definitions, as well as nested module definitions. Module names must begin with an uppercase letter, and idiomatically they use CamelCase rather than Snake_case.

But a more accurate version of the syntax would be:

module ModuleName = module_expression


where a struct is just one sort of module_expression. Here’s another: the name of an already defined module. For example, you can write module L = List if you’d like a short alias for the List module. We’ll see other sorts of module expressions later in this section and chapter.

The definitions inside a structure can optionally be terminated by ;; as in the toplevel:

module M = struct
let x = 0;;
type t = int;;
end

module M : sig val x : int type t = int end


Sometimes that can be useful to add temporarily if you are trying to diagnose a syntax error. It will help OCaml understand that you want two definitions to be syntactically separate. After fixing whatever the underlying error is, though, you can remove the ;;.

One use case for ;; is if you want to evaluate an expression as part of a module:

module M = struct
let x = 0;;
assert (x = 0);;
end

module M : sig val x : int end


But that can be rewritten without ;; as:

module M = struct
let x = 0
let _ = assert (x = 0)
end

module M : sig val x : int end


Structures can also be written on a single line, with optional ;; between items for readability:

module N = struct let x = 0 let y = 1 end
module O = struct let x = 0;; let y = 1 end

module N : sig val x : int val y : int end

module O : sig val x : int val y : int end


An empty structure is permitted:

module E = struct end

module E : sig end


Dynamic semantics.

We already know that expressions are evaluated to values. Similarly, a module expression is evaluated to a module value or just “module” for short. The only interesting kind of module expression we have so far, from the perspective of evaluation anyway, is the structure. Evaluation of structures is easy: just evaluate each definition in it, in the order they occur. Because of that, earlier definitions are therefore in scope in later definitions, but not vice versa. So this module is fine:

module M = struct
let x = 0
let y = x
end

module M : sig val x : int val y : int end


But this module is not, because at the time the let definition of x is being evaluated, y has not yet been bound:

module M = struct
let x = y
let y = 0
end

File "[13]", line 2, characters 10-11:
2 |   let x = y
^
Error: Unbound value y


Of course, mutual recursion can be used if desired:

module M = struct
let rec even = function 0 -> true | n -> odd (n - 1)
and odd = function 1 -> true | n -> even (n - 1)
end

module M : sig val even : int -> bool val odd : int -> bool end


Static semantics.

A structure is well typed if all the definitions in it are themselves well-typed, according to all the typing rules we have already learned.

As we’ve seen in toplevel output, the module type of a structure is a signature. There’s more to module types than that, though. Let’s put that off for a moment to first talk about scope.

## 5.2.2. Scope and Open¶

After a module M has been defined, you can access the names within it using the dot operator. For example:

module M = struct let x = 42 end

module M : sig val x : int end

M.x

- : int = 42


Of course from outside the module the name x by itself is not meaningful:

x

File "[17]", line 1, characters 0-1:
1 | x
^
Error: Unbound value x


But you can bring all of the definitions of a module into the current scope using open:

open M

x

- : int = 42


Opening a module is like writing a local definition for each name defined in the module. For example, open String brings all the definitions from the String module into scope, and has an effect similar to the following on the local namespace:

let length = String.length
let get = String.get
let lowercase_ascii = String.lowercase_ascii
...


If there are types, exceptions, or modules defined in a module, those also are brought into scope with open.

The Always-Open Module. There is a special module called Stdlib that is automatically opened in every OCaml program. It contains the “built-in” functions and operators. You therefore never need to prefix any of the names it defines with Stdlib., though you could do so if you ever needed to unambiguously identify a name from it. In earlier days, this module was named Pervasives, and you might still see that name in some code bases.

Open as a Module Item. An open is another sort of module_item. So we can open one module inside another:

module M = struct
open List

(** [uppercase_all lst] upper-cases all the elements of [lst]. *)
let uppercase_all = map String.uppercase_ascii
end

module M : sig val uppercase_all : string list -> string list end


Since List is open, the name map from it is in scope. But what if we wanted to get rid of the String. as well?

module M = struct
open List
open String

(** [uppercase_all lst] upper-cases all the elements of [lst]. *)
let uppercase_all = map uppercase_ascii
end

File "[21]", line 6, characters 26-41:
6 |   let uppercase_all = map uppercase_ascii
^^^^^^^^^^^^^^^
Error: This expression has type string -> string
but an expression was expected of type char -> char
Type string is not compatible with type char


Now we have a problem, because String also defines the name map, but with a different type than List. As usual a later definition shadows an earlier one, so it’s String.map that gets chosen instead of List.map as we intended.

If you’re using many modules inside your code, chances are you’ll have at least one collision like this. Often it will be with a standard higher-order function like map that is defined in many library modules.

Tip

It is therefore generally good practice not to open all the modules you’re going to use at the top of a .ml file or structure. This is perhaps different than how you’re used to working with languages like Java, where you might import many packages with *. Instead, it’s good to restrict the scope in which you open modules.

Limiting the Scope of Open. We’ve already seen one way of limiting the scope of an open: M.(e). Inside e all the names from module M are in scope. This is useful for briefly using M in a short expression:

(* remove surrounding whitespace from [s] and convert it to lower case *)
let s = "BigRed "
let s' = s |> String.trim |> String.lowercase_ascii (* long way *)
let s'' = String.(s |> trim |> lowercase_ascii) (* short way *)

val s : string = "BigRed "

val s' : string = "bigred"

val s'' : string = "bigred"


But what if you want to bring a module into scope for an entire function, or some other large block of code? The (admittedly strange) syntax for that is let open M in e. It makes all the names from M be in scope in e. For example:

(** [lower_trim s] is [s] in lower case with whitespace removed. *)
let lower_trim s =
let open String in
s |> trim |> lowercase_ascii

val lower_trim : string -> string = <fun>


Going back to our uppercase_all example, it might be best to eschew any kind of opening and simply to be explicit about which module we are using where:

module M = struct
(** [uppercase_all lst] upper-cases all the elements of [lst]. *)
let uppercase_all = List.map String.uppercase_ascii
end

module M : sig val uppercase_all : string list -> string list end


## 5.2.3. Module Type Definitions¶

We’ve already seen that OCaml will infer a signature as the type of a module. Let’s now see how to write those modules types ourselves. As an example, here is a module type for our list-based stacks:

module type LIST_STACK = sig
exception Empty
val empty : 'a list
val is_empty : 'a list -> bool
val push : 'a -> 'a list -> 'a list
val peek : 'a list -> 'a
val pop : 'a list -> 'a list
end

module type LIST_STACK =
sig
exception Empty
val empty : 'a list
val is_empty : 'a list -> bool
val push : 'a -> 'a list -> 'a list
val peek : 'a list -> 'a
val pop : 'a list -> 'a list
end


Now that we have both a module and a module type for list-based stacks, we should move the specification comments from the structure into the signature. Those comments are properly part of the specification of the names in the signature. They specify behavior, thus augmenting the specification of types provided by the val declarations.

module type LIST_STACK = sig
(** [Empty] is raised when an operation cannot be applied
to an empty stack. *)
exception Empty

(** [empty] is the empty stack. *)
val empty : 'a list

(** [is_empty s] is whether [s] is empty. *)
val is_empty : 'a list -> bool

(** [push x s] pushes [x] onto the top of [s]. *)
val push : 'a -> 'a list -> 'a list

(** [peek s] is the top element of [s].
Raises [Empty] if [s] is empty. *)
val peek : 'a list -> 'a

(** [pop s] is all but the top element of [s].
Raises [Empty] if [s] is empty. *)
val pop : 'a list -> 'a list
end

module ListStack = struct
let empty = []

let is_empty = function [] -> true | _ -> false

let push x s = x :: s

exception Empty

let peek = function
| [] -> raise Empty
| x :: _ -> x

let pop = function
| [] -> raise Empty
| _ :: s -> s
end

module type LIST_STACK =
sig
exception Empty
val empty : 'a list
val is_empty : 'a list -> bool
val push : 'a -> 'a list -> 'a list
val peek : 'a list -> 'a
val pop : 'a list -> 'a list
end

module ListStack :
sig
val empty : 'a list
val is_empty : 'a list -> bool
val push : 'a -> 'a list -> 'a list
exception Empty
val peek : 'a list -> 'a
val pop : 'a list -> 'a list
end


Nothing so far, however, tells OCaml that there is a relationship between LIST_STACK and ListStack. If we want OCaml to ensure that ListStack really does have the module type specified by LIST_STACK, we can add a type annotation in the first line of the module definition:

module ListStack : LIST_STACK = struct
let empty = []

let is_empty = function [] -> true | _ -> false

let push x s = x :: s

exception Empty

let peek = function
| [] -> raise Empty
| x :: _ -> x

let pop = function
| [] -> raise Empty
| _ :: s -> s
end

module ListStack : LIST_STACK


The compiler agrees that the module ListStack does define all the items specified by LIST_STACK with appropriate types. If we had accidentally omitted some item, the type annotation would have been rejected:

module ListStack : LIST_STACK = struct
let empty = []

let is_empty = function [] -> true | _ -> false

let push x s = x :: s

exception Empty

let peek = function
| [] -> raise Empty
| x :: _ -> x

(* [pop] is missing *)
end

File "[28]", lines 1-15, characters 32-3:
1 | ................................struct
2 |   let empty = []
3 |
4 |   let is_empty = function [] -> true | _ -> false
5 |
...
12 |     | x :: _ -> x
13 |
14 |   (* [pop] is missing *)
15 | end
Error: Signature mismatch:
...
The value pop' is required but not provided
File "[26]", line 21, characters 2-30: Expected declaration


Syntax.

The most common syntax for a module type is simply:

module type ModuleTypeName = sig
specifications
end


where specifications inside a signature can include val declarations, type definitions, exception definitions, and nested module type definitions. Like structures, a signature can be written on many lines or just one line, and the empty signature sig end is allowed.

But, as we saw with module definitions, a more accurate version of the syntax would be:

module type ModuleTypeName = module_type


where a signature is just one sort of module_type. Another would be the name of an already defined module type—e.g., module type LS = LIST_STACK. We’ll see other module types later in this section and chapter.

By convention, module type names are usually CamelCase, like module names. So why did we use ALL_CAPS above for LIST_STACK? It was to avoid a possible point of confusion in that example, which we now illustrate. We could instead have used ListStack as the name of both the module and the module type:

module type ListStack = sig ... end
module ListStack : ListStack = struct ... end


In OCaml the namespaces for modules and module types are distinct, so it’s perfectly valid to have a module named ListStack and and module type named ListStack. The compiler will not get confused about which you mean, because they occur in distinct syntactic contexts. But as a human you might well get confused by those seemingly overloaded names.

Note

The use of ALL_CAPS for module types was at one point common, and you might see it still. It’s an older convention from Standard ML. But the social conventions of all caps have changed since those days. To modern readers, a name like LIST_STACK might feel like your code is impolitely shouting at you. That is a connotation that evolved in the 1980s. Older programming languages (e.g., Pascal, COBOL, FORTRAN) commonly used all caps for keywords and even their own names. Modern languages still idiomatically use all caps for constants—see, for example, Java’s Math.PI or Python’s style guide.

More Syntax.

We should also add syntax now for module type annotations. Module definitions may include an optional type annotation:

module ModuleName : module_type = module_expression


And module expressions may include manual type annotations:

(module_expression : module_type)


That syntax is analogous to how we can write (e : t) to manually specify the type t of an expression e.

Here are a few examples to show how that syntax can be used:

module ListStackAlias : LIST_STACK = ListStack
(* equivalently *)
module ListStackAlias = (ListStack : LIST_STACK)

module M : sig val x : int end = struct let x = 42 end
(* equivalently *)
module M = (struct let x = 42 end : sig val x : int end)

module ListStackAlias : LIST_STACK

module ListStackAlias : LIST_STACK

module M : sig val x : int end

module M : sig val x : int end


And, module types can include nested module specifications:

module type X = sig
val x : int
end

module type T = sig
module Inner : X
end

module M : T = struct
module Inner : X = struct
let x = 42
end
end

module type X = sig val x : int end

module type T = sig module Inner : X end

module M : T


In the example above, T specifies that there must be an inner module named Inner whose module type is X. Here, the type annotation is mandatory, because otherwise nothing would be known about Inner. In implementing T, module M therefore has to provide a module (i) with that name, which also (ii) meets the specifications of module type X.

Dynamic semantics.

Since module types are in fact types, they are not evaluated. They have no dynamic semantics.

Static semantics.

Earlier in this section we delayed discussing the static semantics of module expressions. Now that we have learned about module types, we can return to that discussion. We do so, next, in its own section, because the discussion will be lengthy.

## 5.2.4. Module Type Semantics¶

If M is just a struct block, its module type is whatever signature the compiler infers for it. But that can be changed by module type annotations. The key question we have to answer is: what does a type annotation mean for modules? That is, what does it mean when we write the : T in module M : T = ...?

There are two properties the compiler guarantees:

1. Signature matching: every name declared in T is defined in M at the same or a more general type.

2. Opacity: any name defined in M that does not appear in T is not visible to code outside of M.

But a more complete answer turns out to involve subtyping, which is a concept you’ve probably seen before in an object-oriented language. We’re going to take a brief detour into that realm now, then come back to OCaml and modules.

In Java, the extends keyword creates subtype relationships between classes:

class C { }
class D extends C { }

D d = new D();
C c = d;


Subtyping is what permits the assignment of d to c on the last line of that example. Because D extends C, Java considers D to be a subtype of C, and therefore permits an object instantiated from D to be used any place where an object instantiated from C is expected. It’s up to the programmer of D to ensure that doesn’t lead to any run-time errors, of course. The methods of D have to ensure that class invariants of C hold, for example. So by writing D extends C, the programmer is taking on some responsibility, and in turn gaining some flexibility by being able to write such assignment statements.

So what is a “subtype”? That notion is in many ways dependent on the language. For a language-independent notion, we turn to Barbara Liskov. She won the Turing Award in 2008 in part for her work on object-oriented language design. Twenty years before that, she invented what is now called the Liskov Substitution Principle to explain subtyping. It says that if S is a subtype of T, then substituting an object of type S for an object of type T should not change any desirable behaviors of a program. You can see that at work in the Java example above, both in terms of what the language allows and what the programmer must guarantee.

The particular flavor of subtyping in Java is called nominal subtyping, which is to say, it is based on names. In our example, D is a subtype of C just because of the way the names were declared. The programmer decreed that subtype relationship, and the language accepted the decree without question. Indeed the only subtype relationships that exist are those that have been decreed by name through such uses of extends and implements.

Now it’s time to return to OCaml. Its module system also uses subtyping, with the same underlying intuition about the Liskov Substitution Principle. But OCaml uses a different flavor called structural subtyping. That is, it is based on the structure of modules rather than their names. “Structure” here simply means the definitions contained in the module. Those definitions are used to determine whether (M : T) is acceptable as a type annotation, where M is a module and T is a module type.

Let’s play with this idea of structure through several examples, starting with this module:

module M = struct
let x = 0
let z = 2
end

module M : sig val x : int val z : int end


Module M contains two definitions. You can see those in the signature for the module that OCaml outputs: it contains x : int and z : int. Because of the former, the module type annotation below is accepted:

module type X = sig
val x : int
end

module MX = (M : X)

module type X = sig val x : int end

module MX : X


Module type X requires a module item named x with type int. Module M does contain such an item. So (M : X) is valid. The same would work for z:

module type Z = sig
val z : int
end

module MZ = (M : Z)

module type Z = sig val z : int end

module MZ : Z


Or for both x and z:

module type XZ = sig
val x : int
val z : int
end

module MXZ = (M : XZ)

module type XZ = sig val x : int val z : int end

module MXZ : XZ


But not for y, because M contains no such item:

module type Y = sig
val y : int
end

module MY = (M : Y)

module type Y = sig val y : int end

File "[35]", line 5, characters 13-14:
5 | module MY = (M : Y)
^
Error: Signature mismatch:
Modules do not match:
sig val x : int val z : int end
is not included in
Y
The value y' is required but not provided
File "[35]", line 2, characters 2-13: Expected declaration


Take a close look at that error message. Learning to read such errors on small examples will help you when they appear in large bodies of code. OCaml is comparing two signatures, corresponding to the two expressions on either side of the colon in (M : Y). The line

sig val x : int val z : int end


is the signature that OCaml is using for M. Since M is a module, that signature is just the names and types as they were defined in M. OCaml compares that signature to Y, and discovers a mismatch:

The value y' is required but not provided


That’s because Y requires y but M provides no such definition.

Here’s another error message to practice reading:

module type Xstring = sig
val x : string
end

module MXstring = (M : Xstring)

module type Xstring = sig val x : string end

File "[36]", line 5, characters 19-20:
5 | module MXstring = (M : Xstring)
^
Error: Signature mismatch:
Modules do not match:
sig val x : int val z : int end
is not included in
Xstring
Values do not match: val x : int is not included in val x : string
File "[36]", line 2, characters 2-16: Expected declaration
File "[31]", line 2, characters 6-7: Actual declaration


This time the error is

Values do not match: val x : int is not included in val x : string


The error changed, because M does provide a definition of x, but at a different type than Xstring requires. That’s what “is not included in” means here. So why doesn’t OCaml say something a little more straightforward, like “is not the same as”? It’s because the types do not have to be exactly the same. If the provided value’s type is polymorphic, it suffices for the required value’s type to be an instantiation of that polymorphic type.

For example, if a signature requires a type int -> int, it suffices for a structure to provide a value of type 'a -> 'a:

module type IntFun = sig
val f : int -> int
end

module IdFun = struct
let f x = x
end

module Iid = (IdFun : IntFun)

module type IntFun = sig val f : int -> int end

module IdFun : sig val f : 'a -> 'a end

module Iid : IntFun


So far all these examples were just a matter of comparing the definitions required by a signature to the definitions provided by a structure. But here’s an example that might be surprising:

module MXZ' = ((M : X) : Z)

File "[38]", line 1, characters 15-22:
1 | module MXZ' = ((M : X) : Z)
^^^^^^^
Error: Signature mismatch:
Modules do not match: X is not included in Z
The value z' is required but not provided
File "[33]", line 2, characters 2-13: Expected declaration


Why does OCaml complain that z is required but not provided? We know from the definition of M that it indeed does have a value z : int. Yet the error message perhaps strangely claims:

The value z' is required but not provided.


The reason for this error is that we’ve already supplied the type annotation X in the module expression (M : X). That causes the module expression to be known only at the module type X. In other words, we’ve forgotten irrevocably about the existence of z after that annotation. All that is known is that the module has items required by X.

After all those examples, here are the static semantics of module type annotations:

• Module type annotation (M : T) is valid if the module type of M is a subtype of T. The module type of (M : T) is then T in any further type checking.

• Module type S is a subtype of T if the set of definitions in S is a superset of those in T. Definitions in T are permitted to instantiate type variables from S.

The “sub” vs. “super” in the second rule is not a typo. Consider these module types and modules:

module type T = sig
val a : int
end

module type S = sig
val a : int
val b : bool
end

module A = struct
let a = 0
end

module AB = struct
let a = 0
let b = true
end

module AC = struct
let a = 0
let c = 'c'
end

module type T = sig val a : int end

module type S = sig val a : int val b : bool end

module A : sig val a : int end

module AB : sig val a : int val b : bool end

module AC : sig val a : int val c : char end


Module type S provides a superset of the definitions in T, because it adds a definition of b. So why is S called a subtype of T? Think about the set $$\mathit{Type}(T)$$ of all module values M such that M : T. That set contains A, AB, AC, and many others. Also think about the set $$\mathit{Type}(S)$$ of all module values M such that M : S. That set contains AB but not A nor AC. So $$\mathit{Type}(S) \subset \mathit{Type}(T)$$, because there are some module values that are in $$\mathit{Type}(T)$$ but not in $$\mathit{Type}(S)$$.

As another example, a module type StackHistory for stacks might customize our usual Stack signature by adding an operation history : 'a t -> int to return how many items have ever been pushed on the stack in its history. That history operation makes the set of definitions in StackHistory bigger than the set in Stack, hence the use of “superset” in the rule above. But the set of module values that implement StackHistory is smaller than the set of module values that implement Stack, hence the use of “subset”.

## 5.2.5. Module Types are Static¶

Decisions about validity of module type annotations are made at compile time rather than run time.

Important

Module type annotations therefore offer potential confusion to programmers accustomed to object-oriented languages, in which subtyping works differently.

Python programmers, for example, are accustomed to so-called “duck typing”. They might expect ((M : X) : Z) to be valid, because z does exist at run-time in M. But in OCaml, the compile-time type of (M : X) has hidden z from view irrevocably.

Java programmers, on the other hand, might expect that module type annotations work like type casts. So it might seem valid to first “cast” M to X then to Z. In Java such type casts are checked, as needed, at run time. But OCaml module type annotations are static. Once an annotation of X is made, there is no way to check at compile time what other items might exist in the module—that would require a run-time check, which OCaml does not permit.

In both cases it might feel as though OCaml is being too restrictive. Maybe. But in return for that restrictiveness, OCaml is guaranteeing an absence of run-time errors of the kind that would occur in Java or Python, whether because of a run-time error from a cast, or a run-time error from a missing method.

## 5.2.6. First-Class Modules¶

Modules are not as first-class in OCaml as functions. But it is possible to package modules as first-class values. Briefly:

• (module M : T) packages module M with module type T into a value.

• (val e : T) un-packages e into a module with type T`.

We won’t cover this much further, but if you’re curious you can have a look at the manual.