Version: 5.0

### 4Types in Typed Racket

Typed Racket provides a rich variety of types to describe data. This section introduces them.

#### 4.1Basic Types

The most basic types in Typed Racket are those for primitive data, such as True and False for booleans, String for strings, and Char for characters.

 > '"hello, world" - : String "hello, world" > #\f - : Char #\f > #t - : True #t > #f - : False #f

Each symbol is given a unique type containing only that symbol. The Symbol type includes all symbols.

 > 'foo - : 'foo 'foo > 'bar - : 'bar 'bar

Typed Racket also provides a rich hierarchy for describing particular kinds of numbers.

 > 0 - : 0 0 > -7 - : Integer -7 > 14 - : Exact-Positive-Integer 14 > 3.2 - : Float 3.2 > 7.0+2.8i - : Number 7.0+2.8i

Finally, any value is itself a type:

 > (ann 23 : 23) - : 23 23

#### 4.2Function Types

We have already seen some examples of function types. Function types are constructed using ->, with the argument types before the arrow and the result type after. Here are some example function types:

 (Number -> Number) (String String -> Boolean) (Char -> (values String Natural))

The first type requires a Number as input, and produces a Number. The second requires two arguments. The third takes one argument, and produces multiple values, of types String and Natural. Here are example functions for each of these types.

 > (lambda: ([x : Number]) x) - : (Number -> Number) # > (lambda: ([a : String] [b : String]) (equal? a b)) - : (String String -> Boolean) # > (lambda: ([c : Char]) (values (string c) (char->integer c))) - : (Char -> (values String Exact-Nonnegative-Integer)) #

#### 4.3Union Types

Sometimes a value can be one of several types. To specify this, we can use a union type, written with the type constructor U.

 > (let ([a-number 37]) (if (even? a-number) 'yes 'no))

- : (U 'yes 'no)

'no

Any number of types can be combined together in a union, and nested unions are flattened.

#### 4.4Recursive Types

Recursive types can refer to themselves. This allows a type to describe an infinite family of data. For example, this is the type of binary trees of numbers.

(Rec BT (U Number (Pair BT BT)))

The Rec type constructor specifies that the type BT refers to the whole binary tree type within the body of the Rec form.

#### 4.5Structure Types

Using define-struct: introduces new types, distinct from any previous type.

(define-struct: point ([x : Real] [y : Real]))

Instances of this structure, such as (make-point 7 12), have type point.

#### 4.6Subtyping

In Typed Racket, all types are placed in a hierarchy, based on what values are included in the type. When an element of a larger type is expected, an element of a smaller type may be provided. The smaller type is called a subtype of the larger type. The larger type is called a supertype. For example, Integer is a subtype of Real, since every integer is a real number. Therefore, the following code is acceptable to the type checker:

 (: f (Real -> Real)) (define (f x) (* x 0.75)) (: x Integer) (define x -125) (f x)

All types are subtypes of the Any type.

The elements of a union type are individually subtypes of the whole union, so String is a subtype of (U String Number). One function type is a subtype of another if they have the same number of arguments, the subtype’s arguments are more permissive (is a supertype), and the subtype’s result type is less permissive (is a subtype). For example, (Any -> String) is a subtype of (Number -> (U String #f)).

#### 4.7Polymorphism

Typed Racket offers abstraction over types as well as values.

##### 4.7.1Polymorphic Data Structures

Virtually every Racket program uses lists and sexpressions. Fortunately, Typed Racket can handle these as well. A simple list processing program can be written like this:

 #lang typed/racket (: sum-list ((Listof Number) -> Number)) (define (sum-list l) (cond [(null? l) 0] [else (+ (car l) (sum-list (cdr l)))]))

This looks similar to our earlier programs – except for the type of l, which looks like a function application. In fact, it’s a use of the type constructor Listof, which takes another type as its input, here Number. We can use Listof to construct the type of any kind of list we might want.

We can define our own type constructors as well. For example, here is an analog of the Maybe type constructor from Haskell:

 #lang typed/racket (define-struct: None ()) (define-struct: (a) Some ([v : a])) (define-type (Opt a) (U None (Some a))) (: find (Number (Listof Number) -> (Opt Number))) (define (find v l) (cond [(null? l) (make-None)] [(= v (car l)) (make-Some v)] [else (find v (cdr l))]))

The first define-struct: defines None to be a structure with no contents.

The second definition

(define-struct: (a) Some ([v : a]))

creates a parameterized type, Just, which is a structure with one element, whose type is that of the type argument to Just. Here the type parameters (only one, a, in this case) are written before the type name, and can be referred to in the types of the fields.

The type definiton
(define-type (Opt a) (U None (Some a)))
creates a parameterized type – Opt is a potential container for whatever type is supplied.

The find function takes a number v and list, and produces (make-Some v) when the number is found in the list, and (make-None) otherwise. Therefore, it produces a (Opt Number), just as the annotation specified.

##### 4.7.2Polymorphic Functions

Sometimes functions over polymorphic data structures only concern themselves with the form of the structure. For example, one might write a function that takes the length of a list of numbers:

 #lang typed/racket (: list-number-length ((Listof Number) -> Integer)) (define (list-number-length l) (if (null? l) 0 (add1 (list-number-length (cdr l)))))

and also a function that takes the length of a list of strings:

 #lang typed/racket (: list-string-length ((Listof String) -> Integer)) (define (list-string-length l) (if (null? l) 0 (add1 (list-string-length (cdr l)))))

Notice that both of these functions have almost exactly the same definition; the only difference is the name of the function. This is because neither function uses the type of the elements in the definition.

We can abstract over the type of the element as follows:

 #lang typed/racket (: list-length (All (A) ((Listof A) -> Integer))) (define (list-length l) (if (null? l) 0 (add1 (list-length (cdr l)))))

The new type constructor All takes a list of type variables and a body type. The type variables are allowed to appear free in the body of the All form.

#### 4.8Variable-Arity Functions: Programming with Rest Arguments

Typed Racket can handle some uses of rest arguments.

##### 4.8.1Uniform Variable-Arity Functions

In Racket, one can write a function that takes an arbitrary number of arguments as follows:

 #lang racket (define (sum . xs) (if (null? xs) 0 (+ (car xs) (apply sum (cdr xs))))) (sum) (sum 1 2 3 4) (sum 1 3)

The arguments to the function that are in excess to the non-rest arguments are converted to a list which is assigned to the rest parameter. So the examples above evaluate to 0, 10, and 4.

We can define such functions in Typed Racket as well:

 #lang typed/racket (: sum (Number * -> Number)) (define (sum . xs) (if (null? xs) 0 (+ (car xs) (apply sum (cdr xs)))))

This type can be assigned to the function when each element of the rest parameter is used at the same type.

##### 4.8.2Non-Uniform Variable-Arity Functions

However, the rest argument may be used as a heterogeneous list. Take this (simplified) definition of the Racket function map:

 #lang racket (define (map f as . bss) (if (or (null? as) (ormap null? bss)) null (cons (apply f (car as) (map car bss)) (apply map f (cdr as) (map cdr bss))))) (map add1 (list 1 2 3 4)) (map cons (list 1 2 3) (list (list 4) (list 5) (list 6))) (map + (list 1 2 3) (list 2 3 4) (list 3 4 5) (list 4 5 6))

Here the different lists that make up the rest argument bss can be of different types, but the type of each list in bss corresponds to the type of the corresponding argument of f. We also know that, in order to avoid arity errors, the length of bss must be one less than the arity of f (as as corresponds to the first argument of f).

The example uses of map evaluate to (list 2 3 4 5), (list (list 1 4) (list 2 5) (list 3 6)), and (list 10 14 18).

In Typed Racket, we can define map as follows:

 #lang typed/racket (: map (All (C A B ...) ((A B ... B -> C) (Listof A) (Listof B) ... B -> (Listof C)))) (define (map f as . bss) (if (or (null? as) (ormap null? bss)) null (cons (apply f (car as) (map car bss)) (apply map f (cdr as) (map cdr bss)))))

Note that the type variable B is followed by an ellipsis. This denotes that B is a dotted type variable which corresponds to a list of types, much as a rest argument corresponds to a list of values. When the type of map is instantiated at a list of types, then each type t which is bound by B (notated by the dotted pre-type t ... B) is expanded to a number of copies of t equal to the length of the sequence assigned to B. Then B in each copy is replaced with the corresponding type from the sequence.

So the type of (inst map Integer Boolean String Number) is