1 Starting with Typed Scheme
1.1 A First Function
1.2 Adding more complexity
1.3 Defining New Datatypes
1.4 Recursive Datatypes and Unions
2 Polymorphism
3 Type Reference
4 Special Form Reference
4.1 Binding Forms
4.2 Structure Definitions
4.3 Type Aliases
4.4 Type Annotation and Instantiation
4.5 Require
require/ typed
require/ opaque-type
Version: 4.0


Typed Scheme: Scheme with Static Types

 #lang typed-scheme

Typed Scheme is a Scheme-like language, with a type system that supports common Scheme programming idioms. Explicit type declarations are required – that is, there is no type inference. The language supports a number of features from previous work on type systems that make it easier to type Scheme programs, as well as a novel idea dubbed occurrence typing for case discrimination.

Typed Scheme is also designed to integrate with the rest of your PLT Scheme system. It is possible to convert a single module to Typed Scheme, while leaving the rest of the program unchanged. The typed module is protected from the untyped code base via automatically-synthesized contracts.

Further information on Typed Scheme is available from the homepage.

1 Starting with Typed Scheme

If you already know PLT Scheme, or even some other Scheme, it should be easy to start using Typed Scheme.

1.1 A First Function

The following program defines the Fibonacci function in PLT Scheme:

  #lang scheme

  (define (fib n)

  (cond [(= 0 n) 1]

        [(= 1 n) 1]

        [else (+ (fib (- n 1)) (fib (- n 2)))]))

This program defines the same program using Typed Scheme.

  #lang typed-scheme

  (: fib (Number -> Number))

  (define (fib n)

  (cond [(= 0 n) 1]

        [(= 1 n) 1]

        [else (+ (fib (- n 1)) (fib (- n 2)))]))

There are two differences between these programs:

In general, these are most of the changes that have to be made to a PLT Scheme program to transform it into a Typed Scheme program.

Changes to uses of require may also be necessary - these are described later.

1.2 Adding more complexity

Other typed binding forms are also available. For example, we could have rewritten our fibonacci program as follows:

  #lang typed-scheme

  (: fib (Number -> Number))

  (define (fib n)

  (let ([base? (or (= 0 n) (= 1 n))])

  (if base?


      (+ (fib (- n 1)) (fib (- n 2))))))

This program uses the let binding form, but no new type annotations are required. Typed Scheme infers the type of base?.

We can also define mutually-recursive functions:

  #lang typed-scheme

  (: my-odd? (Number -> Boolean))

  (define (my-odd? n)

  (if (= 0 n) #f

      (my-even? (- n 1))))


  (: my-even? (Number -> Boolean))

  (define (my-even? n)

  (if (= 0 n) #t

      (my-odd? (- n 1))))


  (display (my-even? 12))

As expected, this program prints #t.

1.3 Defining New Datatypes

If our program requires anything more than atomic data, we must define new datatypes. In Typed Scheme, structures can be defined, similarly to PLT Scheme structures. The following program defines a date structure and a function that formats a date as a string, using PLT Scheme’s built-in format function.

  #lang typed-scheme

  (define-struct: Date ([day : Number] [month : String] [year : Number]))


  (: format-date (Date -> String))

  (define (format-date d)

  (format "Today is day ~a of ~a in the year ~a"

          (Date-day d) (Date-month d) (Date-year d)))


  (display (format-date (make-Date 28 "November" 2006)))

Here we see the new built-in type String as well as a definition of the new user-defined type my-date. To define my-date, we provide all the information usually found in a define-struct, but added type annotations to the fields using the define-struct: form. Then we can use the functions that this declaration creates, just as we would have with define-struct.

1.4 Recursive Datatypes and Unions

Many data structures involve multiple variants. In Typed Scheme, we represent these using union types, written (U t1 t2 ...).

  #lang typed-scheme

  (define-type-alias Tree (U leaf node))

  (define-struct: leaf ([val : Number]))

  (define-struct: node ([left : Tree] [right : Tree]))


  (: tree-height (Tree -> Number))

  (define (tree-height t)

  (cond [(leaf? t) 1]

        [else (max (tree-height (node-left t))

                   (tree-height (node-right t)))]))


  (: tree-sum (Tree -> Number))

  (define (tree-sum t)

  (cond [(leaf? t) (leaf-val t)]

        [else (+ (tree-sum (node-left t))

                 (tree-sum (node-right t)))]))

In this module, we have defined two new datatypes: leaf and node. We’ve also defined the type alias Tree to be (U node leaf), which represents a binary tree of numbers. In essence, we are saying that the tree-height function accepts a Tree, which is either a node or a leaf, and produces a number.

In order to calculate interesting facts about trees, we have to take them apart and get at their contents. But since accessors such as node-left require a node as input, not a Tree, we have to determine which kind of input we were passed.

For this purpose, we use the predicates that come with each defined structure. For example, the leaf? predicate distinguishes leafs from all other Typed Scheme values. Therefore, in the first branch of the cond clause in tree-sum, we know that t is a leaf, and therefore we can get its value with the leaf-val function.

In the else clauses of both functions, we know that t is not a leaf, and since the type of t was Tree by process of elimination we can determine that t must be a node. Therefore, we can use accessors such as node-left and node-right with t as input.

2 Polymorphism

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

  #lang typed-scheme

  (: 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-scheme

  (define-struct: Nothing ())

  (define-struct: (a) Just ([v : a]))


  (define-type-alias (Maybe a) (U Nothing (Just a)))


  (: find (Number (Listof Number) -> (Maybe Number)))

  (define (find v l)

  (cond [(null? l) (make-Nothing)]

        [(= v (car l)) (make-Just v)]

        [else (find v (cdr l))]))

The first define-struct: defines Nothing to be a structure with no contents.

The second definition

  (define-struct: (a) Just ([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 alias definiton

  (define-type-alias (Maybe a) (U Nothing (Just a)))

creates a parameterized alias – Maybe is a potential container for whatever type is supplied.

The find function takes a number v and list, and produces (make-Just v) when the number is found in the list, and (make-Nothing) otherwise. Therefore, it produces a (Maybe Number), just as the annotation specified.

3 Type Reference

Base Types

These types represent primitive Scheme data.


A number


An integer


Either #t or #f


A string


A literal keyword


A symbol




A port


A path


A character


Any value

The following base types are parameteric in their type arguments.

(Listof t)

Homogenous lists of t

(Boxof t)

A box of t

(Vectorof t)

Homogenous vectors of t

(Option t)

Either t of #f

(Pair s t)

is the pair containing s as the car and t as the cdr

Type Constructors

(dom ... -> rng)

is the type of functions from the (possibly-empty) sequence dom ... to the rng type.

(U t ...)

is the union of the types t ...

(case-lambda fun-ty ...)

is a function that behaves like all of the fun-tys. The fun-tys must all be function types constructed with ->.

(t t1 t2 ...)

is the instantiation of the parametric type t at types t1 t2 ...

(All (v ...) t)

is a parameterization of type t, with type variables v ...

(values t ...)

is the type of a sequence of multiple values, with types t .... This can only appear as the return type of a function.


where v is a number, boolean or string, is the singleton type containing only that value


where i is an identifier can be a reference to a type name or a type variable

(Rec n t)

is a recursive type where n is bound to the recursive type in the body t

Other types cannot be written by the programmer, but are used internally and may appear in error messages.

(struct:n (t ...))

is the type of structures named n with field types t. There may be multiple such types with the same printed representation.


is the printed representation of a reference to the type variable n

4 Special Form Reference

Typed Scheme provides a variety of special forms above and beyond those in PLT Scheme. They are used for annotating variables with types, creating new types, and annotating expressions.

4.1 Binding Forms

loop, f, a, and v are names, t is a type. e is an expression and body is a block.

(define: v : t e)

(define: (f [v : t] ...) : t . body)

(define: (a ...) (f [v : t] ...) : t . body)

These forms define variables, with annotated types. The first form defines v with type t and value e. The second and third forms defines a function f with appropriate types. In most cases, use of : is preferred to use of define:.

(let: ([v : t e] ...) . body)

(let: loop : t0 ([v : t e] ...) . body)

where t0 is the type of the result of loop (and thus the result of the entire expression).

(letrec: ([v : t e] ...) . body)

(let*: ([v : t e] ...) . body)

(lambda: ([v : t] ...) . body)

(lambda: ([v : t] ...    v : t)  . body)

(plambda: (a ...) ([v : t] ...) . body)

(plambda: (a ...) ([v : t] ...    v : t)  . body)

(case-lambda: [formals body] ...)

where formals is like the second element of a lambda:

(pcase-lambda: (a ...) [formals body] ...)

where formals is like the second element of a lambda:.

4.2 Structure Definitions

(define-struct: name ([f : t] ...))

(define-struct: (name parent) ([f : t] ...))

(define-struct: (v ...) name ([f : t] ...))

(define-struct: (v ...) (name parent) ([f : t] ...))

4.3 Type Aliases

(define-type-alias name t)

(define-type-alias (name v ...) t)

The first form defines name as type, with the same meaning as t. The second form is equivalent to (define-type-alias name (All (v ...) t)). Type aliases may refer to other type aliases or types defined in the same module, but cycles among type aliases are prohibited.

4.4 Type Annotation and Instantiation

(: v t)

This declares that v has type t. The definition of v must appear after this declaration. This can be used anywhere a definition form may be used.

#{v : t} This declares that the variable v has type t. This is legal only for binding occurences of v.

(ann e t)

Ensure that e has type t, or some subtype. The entire expression has type t. This is legal only in expression contexts.

#{e :: t} This is identical to (ann e t).

(inst e t ...)

Instantiate the type of e with types t .... e must have a polymorphic type with the appropriate number of type variables. This is legal only in expression contexts.

#{e @ t ...} This is identical to (inst e t ...).

4.5 Require

Here, m is a module spec, pred is an identifier naming a predicate, and r is an optionally-renamed identifier.

(require/typed r t m)

(require/typed m [r t] ...)

The first form requires r from module m, giving it type t. The second form generalizes this to multiple identifiers.

(require/opaque-type t pred m)

This defines a new type t. pred, imported from module m, is a predicate for this type. The type is defined as precisely those values to which pred produces #t. pred must have type (Any -> Boolean).

(require-typed-struct name ([f : t] ...) m)