8.10.0.5

### 1Type Reference

 type
Any Racket value. All other types are subtypes of Any.

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Any number of Racket values of any type.

 type
The empty type. No values inhabit this type, and any expression of this type will not evaluate to a value.

#### 1.1Base Types

##### 1.1.1Numeric Types

These types represent the hierarchy of numbers of Racket. The diagram below shows the relationships between the types in the hierarchy. The regions with a solid border are layers of the numeric hierarchy corresponding to sets of numbers such as integers or rationals. Layers contained within another are subtypes of the layer containing them. For example, Exact-Rational is a subtype of Exact-Number.

The Real layer is also divided into positive and negative types (shown with a dotted line). The Integer layer is subdivided into several fixed-width integers types, detailed later in this section.

 type
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Number and Complex are synonyms. This is the most general numeric type, including all Racket numbers, both exact and inexact, including complex numbers.

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Includes Racket’s exact integers and corresponds to the exact-integer? predicate. This is the most general type that is still valid for indexing and other operations that require integral values.

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Includes Racket’s double-precision (default) floating-point numbers and corresponds to the flonum? predicate. This type excludes single-precision floating-point numbers.

 type
Includes Racket’s single-precision floating-point numbers and corresponds to the single-flonum? predicate. This type excludes double-precision floating-point numbers.

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Includes all of Racket’s floating-point numbers, both single- and double-precision.

 type
Includes Racket’s exact rationals, which include fractions and exact integers.

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Includes all of Racket’s real numbers, which include both exact rationals and all floating-point numbers. This is the most general type for which comparisons (e.g. <) are defined.

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These types correspond to Racket’s complex numbers.

Changed in version 1.7 of package typed-racket-lib: Added Imaginary, Inexact-Complex, Exact-Complex, Exact-Imaginary, Inexact-Imaginary.

The above types can be subdivided into more precise types if you want to enforce tighter constraints. Typed Racket provides types for the positive, negative, non-negative and non-positive subsets of the above types (where applicable).

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Natural and Exact-Nonnegative-Integer are synonyms. So are the integer and exact-integer types, and the float and flonum types. Zero includes only the integer 0. Real-Zero includes exact 0 and all the floating-point zeroes.

These types are useful when enforcing that values have a specific sign. However, programs using them may require additional dynamic checks when the type-checker cannot guarantee that the sign constraints will be respected.

In addition to being divided by sign, integers are further subdivided into range-bounded types. The relationships between most of the range-bounded types are shown in this diagram: Like the previous diagram, types nested inside of another in the diagram are subtypes of its containing types.

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One includes only the integer 1. Byte includes numbers from 0 to 255. Index is bounded by 0 and by the length of the longest possible Racket vector. Fixnum includes all numbers represented by Racket as machine integers. For the latter two families, the sets of values included in the types are architecture-dependent, but typechecking is architecture-independent.

These types are useful to enforce bounds on numeric values, but given the limited amount of closure properties these types offer, dynamic checks may be needed to check the desired bounds at runtime.

Examples:
 > 7 - : Integer [more precisely: Positive-Byte] 7 > 8.3 - : Flonum [more precisely: Positive-Float-No-NaN] 8.3 > (/ 8 3) - : Exact-Rational [more precisely: Positive-Exact-Rational] 8/3 > 0 - : Integer [more precisely: Zero] 0 > -12 - : Integer [more precisely: Negative-Fixnum] -12 > 3+4i - : Exact-Number 3+4i

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80-bit extflonum types, for the values operated on by racket/extflonum exports. These are not part of the numeric tower.

##### 1.1.2Other Base Types

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These types represent primitive Racket data.

Examples:
 > #t - : True #t > #f - : False #f > "hello" - : String "hello" > (current-input-port) - : Input-Port # > (current-output-port) - : Output-Port # > (string->path "/") - : Path # > #rx"a*b*" - : Regexp #rx"a*b*" > #px"a*b*" - : PRegexp #px"a*b*" > '#"bytes" - : Bytes #"bytes" > (current-namespace) - : Namespace # > #\b - : Char #\b > (thread (lambda () (add1 7))) - : Thread #

 type
The union of the Path and String types. Note that this does not match exactly what the predicate path-string? recognizes. For example, strings that contain the character #\nul have the type Path-String but path-string? returns #f for those strings. For a complete specification of which strings path-string? accepts, see its documentation.

#### 1.2Singleton Types

Some kinds of data are given singleton types by default. In particular, booleans, symbols, and keywords have types which consist only of the particular boolean, symbol, or keyword. These types are subtypes of Boolean, Symbol and Keyword, respectively.

Examples:
 > #t - : True #t > '#:foo - : '#:foo '#:foo > 'bar - : 'bar 'bar

#### 1.3Base Type Constructors and Supertypes

 type constructor(Pairof s t)
Returns a pair type containing s as the car and t as the cdr

Examples:
 > (cons 1 2) - : (Pairof One Positive-Byte) '(1 . 2) > (cons 1 "one") - : (Pairof One String) '(1 . "one")

 type constructor(Listof t)
Returns the type of a homogeneous list of t
 type constructor(List t ...)
Returns a list type with one element, in order, for each type provided to the List type constructor.
 type constructor(List t ... trest ... bound)
Returns the type of a list with one element for each of the ts, plus a sequence of elements corresponding to trest, where bound must be an identifier denoting a type variable bound with ....
 type constructor(List* t t1 ... s)
Is equivalent to (Pairof t (List* t1 ... s)). (List* s) is equivalent to s itself.

Examples:
> (list 'a 'b 'c)

- : (List 'a 'b 'c)

'(a b c)

 > (plambda: (a ...) ([sym : Symbol] boxes : (Boxof a) ... a) (ann (cons sym boxes) (List Symbol (Boxof a) ... a)))
 - : (All (a ...) (-> Symbol (Boxof a) ... a (Pairof Symbol (List (Boxof a) ... a))))

#<procedure>

> (map symbol->string (list 'a 'b 'c))

- : (Pairof String (Listof String))

'("a" "b" "c")

 type constructor(MListof t)
Returns the type of a homogeneous mutable list of t.
 type constructor(MPairof t u)
Returns the type of a Mutable pair of t and u.

 type
Is the type of a mutable pair with unknown element types and is the supertype of all mutable pair types. This type typically appears in programs via the combination of occurrence typing and mpair?.

Example:
 > (lambda: ([x : Any]) (if (mpair? x) x (error "not an mpair!"))) - : (-> Any MPairTop) #

 type constructor(Boxof t)
Returns the type of a box of t

Example:
 > (box "hello world") - : (Boxof String) '#&"hello world"

 type
Is the type of a box with an unknown element type and is the supertype of all box types. Only read-only box operations (e.g. unbox) are allowed on values of this type. This type typically appears in programs via the combination of occurrence typing and box?.

Example:
 > (lambda: ([x : Any]) (if (box? x) x (error "not a box!"))) - : (-> Any BoxTop) #

 type constructor(Vectorof t)
Returns the type of a homogeneous vector list of t (mutable or immutable).
 type constructor
Returns the type of a homogeneous immutable vector of t.

Added in version 1.9 of package typed-racket-lib.

 type constructor
Returns the type of a homogeneous mutable vector of t.

Added in version 1.9 of package typed-racket-lib.

 type constructor(Vector t ...)
Returns the type of a mutable or immutable vector with one element, in order, for each type provided to the Vector type constructor.

Example:
 > (ann (vector 1 'A) (Vector Fixnum 'A)) - : (U (Immutable-Vector Fixnum 'A) (Mutable-Vector Fixnum 'A)) '#(1 A)

 type constructor(Immutable-Vector t ...)
Similar to (Vector t ...), but for immutable vectors.

Example:
 > (vector-immutable 1 2 3) - : (Immutable-Vector One Positive-Byte Positive-Byte) '#(1 2 3)

Added in version 1.9 of package typed-racket-lib.

 type constructor(Mutable-Vector t ...)
Similar to (Vector t ...), but for mutable vectors.

Example:
 > (vector 1 2 3) - : (Mutable-Vector Integer Integer Integer) '#(1 2 3)

Added in version 1.9 of package typed-racket-lib.

 type

Example:
 > (flvector 1.0 2.0 3.0) - : FlVector (flvector 1.0 2.0 3.0)

 type

Example:
 > (extflvector 1.0t0 2.0t0 3.0t0) - : ExtFlVector #

 type

Example:
 > (fxvector 1 2 3) - : FxVector (fxvector 1 2 3)

 type
Is the type of a vector with unknown length and element types and is the supertype of all vector types. Only read-only vector operations (e.g. vector-ref) are allowed on values of this type. This type typically appears in programs via the combination of occurrence typing and vector?.

Example:
 > (lambda: ([x : Any]) (if (vector? x) x (error "not a vector!"))) - : (-> Any VectorTop) #

 type
Is the type of a mutable vector with unknown length and element types.

 type constructor(HashTable k v)
Returns the type of a mutable or immutable hash table with key type k and value type v.

Example:
 > (ann (make-hash '((a . 1) (b . 2))) (HashTable Symbol Integer)) - : (HashTable Symbol Integer) '#hash((a . 1) (b . 2))

 type constructor
Returns the type of an immutable hash table with key type k and value type v.

Example:
 > #hash((a . 1) (b . 2)) - : (Immutable-HashTable Symbol Integer) '#hash((a . 1) (b . 2))

Added in version 1.8 of package typed-racket-lib.

 type constructor
Returns the type of a mutable hash table that holds keys strongly (see Weak Boxes) with key type k and value type v.

Example:
 > (make-hash '((a . 1) (b . 2))) - : (Mutable-HashTable Symbol Integer) '#hash((a . 1) (b . 2))

Added in version 1.8 of package typed-racket-lib.

 type constructor(Weak-HashTable k v)
Returns the type of a mutable hash table that holds keys weakly with key type k and value type v.

Example:
 > (make-weak-hash '((a . 1) (b . 2))) - : (Weak-HashTable Symbol Integer) '#hash((a . 1) (b . 2))

Added in version 1.8 of package typed-racket-lib.

 type
Is the type of a hash table with unknown key and value types and is the supertype of all hash table types. Only read-only hash table operations (e.g. hash-ref) are allowed on values of this type. This type typically appears in programs via the combination of occurrence typing and hash?.

Example:
 > (lambda: ([x : Any]) (if (hash? x) x (error "not a hash table!"))) - : (-> Any HashTableTop) #

 type
Is the type of a mutable hash table that holds keys strongly with unknown key and value types.

 type
Is the type of a mutable hash table that holds keys weakly with unknown key and value types.

 type constructor(Setof t)
Returns the type of a hash set of t. This includes custom hash sets, but not mutable hash set or sets that are implemented using gen:set.

Example:
 > (set 0 1 2 3) - : (Setof Byte) (set 0 1 2 3)

Example:
 > (seteq 0 1 2 3) - : (Setof Byte) (seteq 0 1 2 3)

 type constructor(Channelof t)
Returns the type of a channel on which only ts can be sent.

Example:
 > (ann (make-channel) (Channelof Symbol)) - : (Channelof Symbol) #

 type
Is the type of a channel with unknown message type and is the supertype of all channel types. This type typically appears in programs via the combination of occurrence typing and channel?.

Example:
 > (lambda: ([x : Any]) (if (channel? x) x (error "not a channel!"))) - : (-> Any ChannelTop) #

 type constructor
Returns the type of an asynchronous channel on which only ts can be sent.

Examples:
 > (require typed/racket/async-channel) > (ann (make-async-channel) (Async-Channelof Symbol)) - : (Async-Channelof Symbol) #

Added in version 1.1 of package typed-racket-lib.

 type
Is the type of an asynchronous channel with unknown message type and is the supertype of all asynchronous channel types. This type typically appears in programs via the combination of occurrence typing and async-channel?.

Examples:
 > (require typed/racket/async-channel) > (lambda: ([x : Any]) (if (async-channel? x) x (error "not an async-channel!"))) - : (-> Any Async-ChannelTop) #

Added in version 1.1 of package typed-racket-lib.

 type constructor(Parameterof t) (Parameterof s t)
Returns the type of a parameter of t. If two type arguments are supplied, the first is the type the parameter accepts, and the second is the type returned.

Examples:
 > current-input-port - : (Parameterof Input-Port) # > current-directory - : (Parameterof Path-String Path) #

 type constructor(Promise t)
Returns the type of promise of t.

Example:
 > (delay 3) - : (Promise Positive-Byte) #

 type constructor(Futureof t)
Returns the type of future which produce a value of type t when touched.

 type constructor(Sequenceof t t ...)
Returns the type of sequence that produces (Values t t ...) on each iteration. E.g., (Sequenceof String) is a sequence of strings, (Sequenceof Number String) is a sequence which produces two values—a number and a string—on each iteration, etc.

 type
Is the type of a sequence with unknown element type and is the supertype of all sequences. This type typically appears in programs via the combination of ocurrence typing ang sequence?.

Example:
 > (lambda: ([x : Any]) (if (sequence? x) x (error "not a sequence!"))) - : (-> Any SequenceTop) #

Added in version 1.10 of package typed-racket-lib.

 type constructor
Returns the type of custodian box of t.

 type constructor
Returns the type of thread cell of t.

 type
Is the type of a thread cell with unknown element type and is the supertype of all thread cell types. This type typically appears in programs via the combination of occurrence typing and thread-cell?.

Example:
 > (lambda: ([x : Any]) (if (thread-cell? x) x (error "not a thread cell!"))) - : (-> Any Thread-CellTop) #

 type constructor(Weak-Boxof t)
Returns the type for a weak box whose value is of type t.

Examples:
 > (make-weak-box 5) - : (Weak-Boxof Integer) # > (weak-box-value (make-weak-box 5)) - : (U False Integer) 5

 type
Is the type of a weak box with an unknown element type and is the supertype of all weak box types. This type typically appears in programs via the combination of occurrence typing and weak-box?.

Example:
 > (lambda: ([x : Any]) (if (weak-box? x) x (error "not a box!"))) - : (-> Any Weak-BoxTop) #

 type constructor(Ephemeronof t)
Returns the type of an ephemeron whose value is of type t.

 type constructor(Evtof t)

Examples:
> always-evt

- : (Rec x (Evtof x))

#<always-evt>

> (system-idle-evt)

- : (Evtof Void)

#<system-idle-evt>

 - : (Evtof Thread) hello world

#### 1.4Syntax Objects

The following type constructors and types respectively create and represent syntax objects and their content.

 type constructor(Syntaxof t)
Returns the type of syntax object with content of type t. Applying syntax-e to a value of type (Syntaxof t) produces a value of type t.

 type
A syntax object containing a symbol. Equivalent to (Syntaxof Symbol).

 type
A syntax object containing only symbols, keywords, strings, byte strings, characters, booleans, numbers, boxes containing Syntax, vectors of Syntax, or (possibly improper) lists of Syntax. Equivalent to (Syntaxof Syntax-E).

 type
The content of syntax objects of type Syntax. Applying syntax-e to a value of type Syntax produces a value of type Syntax-E.

 type constructor(Sexpof t)
Returns the recursive union of t with symbols, keywords, strings, byte strings, characters, booleans, numbers, boxes, vectors, and (possibly improper) lists.

 type
Applying syntax->datum to a value of type Syntax produces a value of type Sexp. Equivalent to (Sexpof Nothing).

 type
Applying datum->syntax to a value of type Datum produces a value of type Syntax. Equivalent to (Sexpof Syntax).

#### 1.5Control

The following type constructors and type respectively create and represent prompt tags and keys for continuation marks for use with delimited continuation functions and continuation mark functions.

 type constructor(Prompt-Tagof s t)
Returns the type of a prompt tag to be used in a continuation prompt whose body produces the type s and whose handler has the type t. The type t must be a function type.

The domain of t determines the type of the values that can be aborted, using abort-current-continuation, to a prompt with this prompt tag.

Example:
 > (make-continuation-prompt-tag 'prompt-tag) - : (Prompt-Tagof Any Any) #

 type
is the type of a prompt tag with unknown body and handler types and is the supertype of all prompt tag types. This type typically appears in programs via the combination of occurrence typing and continuation-prompt-tag?.

Example:
 > (lambda: ([x : Any]) (if (continuation-prompt-tag? x) x (error "not a prompt tag!"))) - : (-> Any Prompt-TagTop) #

 type constructor
Returns the type of a continuation mark key that is used for continuation mark operations such as with-continuation-mark and continuation-mark-set->list. The type t represents the type of the data that is stored in the continuation mark with this key.

Example:
 > (make-continuation-mark-key 'mark-key) - : (Continuation-Mark-Keyof Any) #

 type
Is the type of a continuation mark key with unknown element type and is the supertype of all continuation mark key types. This type typically appears in programs via the combination of occurrence typing and continuation-mark-key?.

Example:
 > (lambda: ([x : Any]) (if (continuation-mark-key? x) x (error "not a mark key!"))) - : (-> Any Continuation-Mark-KeyTop) #

#### 1.6Other Type Constructors

type constructor

(-> dom ... rng opt-proposition)

(-> dom ... rest * rng)
(-> dom ... rest ooo bound rng)
(dom ... -> rng opt-proposition)
(dom ... rest * -> rng)
(dom ... rest ooo bound -> rng)

ooo = ...

dom = type
| mandatory-kw
| opt-kw

rng = type
| (Some (a ...) type : #:+ proposition)
| (Values type ...)

mandatory-kw = keyword type

opt-kw = [keyword type]

opt-proposition =
| : type
|
 : pos-proposition neg-proposition object

pos-proposition =
| #:+ proposition ...

neg-proposition =
| #:- proposition ...

object =
| #:object index

proposition = Top
| Bot
| type
| (! type)
| (type @ path-elem ... index)
| (! type @ path-elem ... index)
| (and proposition ...)
| (or proposition ...)
| (implies proposition ...)

path-elem = car
| cdr

index = positive-integer
| (positive-integer positive-integer)
| identifier
The type of functions from the (possibly-empty) sequence dom .... to the rng type.

Examples:
 > (λ ([x : Number]) x) - : (-> Number Number) # > (λ () 'hello) - : (-> 'hello) #

The second form specifies a uniform rest argument of type rest, and the third form specifies a non-uniform rest argument of type rest with bound bound. The bound refers to the type variable that is in scope within the rest argument type.

Examples:
> (λ ([x : Number]    y : String *)  (length y))

- : (-> Number String * Index)

#<procedure>

> ormap
 - : (All (a c b ...) (-> (-> a b ... b c) (Listof a) (Listof b) ... b (U False c)))

#<procedure:ormap>

In the third form, the ... introduced by ooo is literal, and bound must be an identifier denoting a type variable.

The doms can include both mandatory and optional keyword arguments. Mandatory keyword arguments are a pair of keyword and type, while optional arguments are surrounded by a pair of parentheses.

Examples:
> (:print-type file->string)

(-> Path-String [#:mode (U 'binary 'text)] String)

> (: is-zero? : (-> Number #:equality (-> Number Number Any) [#:zero Number] Any))
 > (define (is-zero? n #:equality equality #:zero [zero 0]) (equality n zero))
> (is-zero? 2 #:equality =)

- : Any

#f

> (is-zero? 2 #:equality eq? #:zero 2.0)

- : Any

#f

When opt-proposition is provided, it specifies the proposition for the function type (for an introduction to propositions in Typed Racket, see Propositions and Predicates). For almost all use cases, only the simplest form of propositions, with a single type after a :, are necessary:

Example:
 > string? - : (-> Any Boolean : String) #

The proposition specifies that when (string? x) evaluates to a true value for a conditional branch, the variable x in that branch can be assumed to have type String. Likewise, if the expression evaluates to #f in a branch, the variable does not have type String.

In some cases, asymmetric type information is useful in the propositions. For example, the filter function’s first argument is specified with only a positive proposition:

Example:
> filter
 - : (All (a b) (case-> (-> (-> a Any : #:+ b) (Listof a) (Listof b)) (-> (-> a Any) (Listof a) (Listof a))))

#<procedure:filter>

The use of #:+ indicates that when the function applied to a variable evaluates to a true value, the given type can be assumed for the variable. However, the type-checker gains no information in branches in which the result is #f.

Conversely, #:- specifies that a function provides information for the false branch of a conditional.

The other proposition cases are rarely needed, but the grammar documents them for completeness. They correspond to logical operations on the propositions.

The type of functions can also be specified with an infix -> which comes immediately before the rng type. The fourth through sixth forms match the first three cases, but with the infix style of arrow.

Examples:
 > (: add2 (Number -> Number)) > (define (add2 n) (+ n 2))

Currently, because explicit packing operations for existential types are not supported, existential type results are only used to annotate accessors for Struct-Property

(Some (a ...) type : #:+ proposition) for rng specifies an existential type result, where the type variables a ... may appear in type and opt-proposition. Unpacking the existential type result is done automatically while checking application of the function.

Changed in version 1.12 of package typed-racket-lib: Added existential type results

type constructor

(->* (mandatory-dom ...) optional-doms rest rng)

mandatory-dom = type
| keyword type

optional-doms =
| (optional-dom ...)

optional-dom = type
| keyword type

rest =
| #:rest type
| #:rest-star (type ...)
Constructs the type of functions with optional or rest arguments. The first list of mandatory-doms correspond to mandatory argument types. The list optional-doms, if provided, specifies the optional argument types.

Examples:
> (: append-bar (->* (String) (Positive-Integer) String))
 > (define (append-bar str [how-many 1]) (apply string-append str (make-list how-many "bar")))

If provided, the #:rest type specifies the type of elements in the rest argument list.

Examples:
> (: +all (->* (Integer) #:rest Integer (Listof Integer)))
 > (define (+all inc . rst) (map (λ ([x : Integer]) (+ x inc)) rst))
> (+all 20 1 2 3)

- : (Listof Integer)

'(21 22 23)

A #:rest-star (type ...) specifies the rest list is a sequence of types which occurs 0 or more times (i.e. the Kleene closure of the sequence).

Examples:
> (: print-name+ages (->* () #:rest-star (String Natural) Void))
 > (define (print-name+ages . names+ages) (let loop ([names+ages : (Rec x (U Null (List* String Natural x))) names+ages]) (when (pair? names+ages) (printf "~a is ~a years old!\n" (first names+ages) (second names+ages)) (loop (cddr names+ages)))) (printf "done printing ~a ages" (/ (length names+ages) 2)))
> (print-name+ages)

done printing 0 ages

> (print-name+ages "Charlotte" 8 "Harrison" 5 "Sydney" 3)
 Charlotte is 8 years old! Harrison is 5 years old! Sydney is 3 years old! done printing 3 ages

Both the mandatory and optional argument lists may contain keywords paired with types.

Examples:
 > (: kw-f (->* (#:x Integer) (#:y Integer) Integer)) > (define (kw-f #:x x #:y [y 0]) (+ x y))

The syntax for this type constructor matches the syntax of the ->* contract combinator, but with types instead of contracts.

 type
 type
These are propositions that can be used with ->. Top is the propositions with no information. Bot is the propositions which means the result cannot happen.

 type
is the supertype of all function types. The Procedure type corresponds to values that satisfy the procedure? predicate. Because this type encodes only the fact that the value is a procedure, and not its argument types or even arity, the type-checker cannot allow values of this type to be applied.

For the types of functions with known arity and argument types, see the -> type constructor.

Examples:
 > (: my-list Procedure) > (define my-list list) > (my-list "zwiebelkuchen" "socca") eval:91:0: Type Checker: cannot apply a function with unknown arity; function `my-list' has type Procedure which cannot be applied in: "socca"

 type constructor(U t ...)
is the union of the types t ....

Example:
 > (λ ([x : Real]) (if (> 0 x) "yes" 'no)) - : (-> Real (U 'no String)) #

 type constructor(∩ t ...)
is the intersection of the types t ....

Example:
 > ((λ #:forall (A) ([x : (∩ Symbol A)]) x) 'foo) - : 'foo 'foo

 type constructor(case-> fun-ty ...)
is a function that behaves like all of the fun-tys, considered in order from first to last. The fun-tys must all be non-dependent function types (i.e. no preconditions or dependencies between arguments are currently allowed).

Example:
 > (: add-map : (case-> [(Listof Integer) -> (Listof Integer)] [(Listof Integer) (Listof Integer) -> (Listof Integer)]))

For the definition of add-map look into case-lambda:.

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

 type(All (a ...) t) (All (a ... a ooo) t)
is a parameterization of type t, with type variables a .... If t is a function type constructed with infix ->, the outer pair of parentheses around the function type may be omitted.

Examples:
> (: list-length : (All (A) (Listof A) -> Natural))
 > (define (list-length lst) (if (null? lst) 0 (add1 (list-length (cdr lst)))))
> (list-length (list 1 2 3))

- : Integer [more precisely: Nonnegative-Integer]

3

 type(Some (a ...) t)

Added in version 1.10 of package typed-racket-lib.

 type constructor(Values t ...)
Returns the type of a sequence of multiple values, with types t .... This can only appear as the return type of a function.

Example:
> (values 1 2 3)

- : (values Integer Integer Integer) [more precisely: (Values One Positive-Byte Positive-Byte)]

 1 2 3

Note that a type variable cannot be instantiated with a (Values ....) type. For example, the type (All (A) (-> A)) describes a thunk that returns exactly one value.
 typev
where v is a number, boolean or string, is the singleton type containing only that value
 type(quote val)
where val is a Racket value, is the singleton type containing only that value
 typei
where i is an identifier can be a reference to a type name or a type variable

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

Examples:
 > (define-type IntList (Rec List (Pair Integer (U List Null)))) > (define-type (List A) (Rec List (Pair A (U List Null))))

 type(Struct st)
is a type which is a supertype of all instances of the potentially-polymorphic structure type st. Note that structure accessors for st will not accept (Struct st) as an argument.

 type(Struct-Type st)
is a type for the structure type descriptor value for the structure type st. Values of this type are used with reflective operations such as struct-type-info.

Examples:
> struct:arity-at-least

- : (StructType arity-at-least)

#<struct-type:arity-at-least>

> (struct-type-info struct:arity-at-least)
 - : (values Symbol Integer Integer (-> arity-at-least Nonnegative-Integer Any) (-> arity-at-least Nonnegative-Integer Nothing Void) (Listof Nonnegative-Integer) (U False Struct-TypeTop) Boolean) [more precisely: (values Symbol Nonnegative-Integer Nonnegative-Integer (-> arity-at-least Nonnegative-Integer Any) (-> arity-at-least Nonnegative-Integer Nothing Void) (Listof Nonnegative-Integer) (U False Struct-TypeTop) Boolean)]
 'arity-at-least 1 0 # # '(0) #f #f

 type
is the supertype of all types for structure type descriptor values. The corresponding structure type is unknown for values of this top type.

Example:
> (struct-info (arity-at-least 0))

- : (values (U False Struct-TypeTop) Boolean)

 # #f

 type constructor(Prefab key type ...)
Describes a prefab structure with the given (implicitly quoted) prefab key key and specified field types.

Prefabs are more-or-less tagged polymorphic tuples which can be directly serialized and whose fields can be accessed by anyone. Subtyping is covariant for immutable fields and invariant for mutable fields.

When a prefab struct is defined with struct the struct name is bound at the type-level to the Prefab type with the corresponding key and field types and the constructor expects types corresponding to those declared for each field. The defined predicate, however, only tests whether a value is a prefab structure with the same key and number of fields, but does not inspect the fields’ values.

Examples:
> (struct person ([name : String]) #:prefab)
> person

- : (-> String person)

#<procedure:person>

> person?

- : (-> Any Boolean : (Prefab person Any))

#<procedure:person?>

> person-name

- : (All (x) (case-> (-> (Prefab person x) x) (-> (Prefab person Any) Any)))

#<procedure:person-name>

> (person "Jim")

- : (Prefab person String)

'#s(person "Jim")

> (ann '#s(person "Dwight") person)

- : (Prefab person String)

'#s(person "Dwight")

> (ann '#s(person "Pam") (Prefab person String))

- : person

'#s(person "Pam")

> (ann '#s(person "Michael") (Prefab person Any))

- : (Prefab person Any)

'#s(person "Michael")

> (person 'Toby)

eval:112:0: Type Checker: type mismatch

expected: String

given: 'Toby

in: Toby

> (ann #s(person Toby) (Prefab person String))

eval:113:0: Type Checker: type mismatch

expected: person

given: (Prefab person 'Toby)

in: String

> (ann '#s(person Toby) (Prefab person Symbol))

- : (Prefab person Symbol)

'#s(person Toby)

> (person? '#s(person "Michael"))

- : True

#t

> (person? '#s(person Toby))

- : True

#t

> (struct employee person ([schrute-bucks : Natural]) #:prefab)
> (employee "Oscar" 10000)

- : (Prefab (employee person 1) String Nonnegative-Integer)

'#s((employee person 1) "Oscar" 10000)

> (ann '#s((employee person 1)             "Oscar" 10000) employee)

- : (Prefab (employee person 1) String Nonnegative-Integer)

'#s((employee person 1) "Oscar" 10000)

 > (ann '#s((employee person 1)             "Oscar" 10000) (Prefab (employee person 1) String Natural))

- : employee

'#s((employee person 1) "Oscar" 10000)

> (person? '#s((employee person 1)             "Oscar" 10000))

- : True

#t

> (employee? '#s((employee person 1)             "Oscar" 10000))

- : True

#t

> (employee 'Toby -1)

eval:123:0: Type Checker: type mismatch

expected: String

given: 'Toby

in: -1

 > (ann '#s((employee person 1)             Toby -1) (Prefab (employee person 1) Symbol Integer))

- : (Prefab (employee person 1) Symbol Integer)

'#s((employee person 1) Toby -1)

> (person? '#s((employee person 1)             Toby -1))

- : True

#t

> (employee? '#s((employee person 1)             Toby -1))

- : True

#t

 type(PrefabTop key field-count)
Describes all prefab types with the (implicitly quoted) prefab-key key and field-count many fields.

For immutable prefabs this is equivalent to (Prefab key Any ...) with field-count many occurrences of Any. For mutable prefabs, this describes a prefab that can be read from but not written to (since we do not know at what type other code may have the fields typed at).

Examples:
 > (struct point ([x : Number] [y : Number]) #:prefab #:mutable)
> point

- : (-> Number Number point)

#<procedure:point>

> point-x
 - : (All (a b) (case-> (-> (Prefab (point #(0 1)) a b) a) (-> (PrefabTop (point #(0 1)) 2) Any)))

#<procedure:point-x>

> point-y
 - : (All (a b) (case-> (-> (Prefab (point #(0 1)) a b) b) (-> (PrefabTop (point #(0 1)) 2) Any)))

#<procedure:point-y>

> point?

- : (-> Any Boolean : (PrefabTop (point #(0 1)) 2))

#<procedure:point?>

 > (define (maybe-read-x p) (if (point? p) (ann (point-x p) Any) 'not-a-point))
 > (define (read-some-x-num p) (if (point? p) (ann (point-x p) Number) -1))

eval:133:0: Type Checker: Polymorphic function `point-x'

could not be applied to arguments:

Types: (PrefabTop (point #(0 1)) 2)  -> Any

Arguments: (PrefabTop (point #(0 1)) 2)

Expected result: Number

in: -1

Added in version 1.7 of package typed-racket-lib.

 type constructor(Struct-Property ty)
Describes a property that can be attached to a structure type. The property value must match the type ty.

Example:
 > (:print-type prop:input-port) (Struct-Property (U Exact-Nonnegative-Integer Input-Port))

Added in version 1.10 of package typed-racket-lib.

 type
This type can only appear in a Struct-Property type. A struct property value is attached to an instance of a structure type; the Self type refers to this instance.

Example:
 > (:print-type prop:custom-write) (Struct-Property (-> Self Output-Port (U Boolean One Zero) AnyValues))

Added in version 1.10 of package typed-racket-lib.

 typeImp
This type can only appear in a Struct-Property type. An Imp value may be a structure subtype of the Self value, or another instance created by the same struct constructor.

Example:
> (:print-type prop:equal+hash)
 (Struct-Property (List (-> Self Imp (-> Any Any Boolean) Any) (-> Self (-> Any Integer) Integer) (-> Self (-> Any Integer) Integer)))

Added in version 1.10 of package typed-racket-lib.

 type(Has-Struct-Property prop)
This type describes an instance of a structure type associcated with a Struct-Property named prop.

 type constructor
An alias for U.
 type constructor
An alias for .
 type constructor
An alias for ->.
 type constructor
An alias for case->.
 type
An alias for All.

#### 1.7Other Types

 type constructor(Option t)
Either t or #f
 type constructor(Opaque t)
A type constructed using the #:opaque clause of require/typed.