▼ |  | The Typed Racket Reference

1 | Type Reference
2 | Special Form Reference
3 | Libraries Provided With Typed Racket
4 | Typed Classes
5 | Typed Units
6 | Utilities
7 | Exploring Types
8 | Typed Racket Syntax Without Type Checking
9 | Typed Regions
10 | Optimization in Typed Racket
11 | Unsafe Typed Racket operations
12 | Legacy Forms
13 | Compatibility Languages
14 | Experimental Features

► | 1 | Type Reference

1.1 | Base Types
1.2 | Singleton Types
1.3 | Containers
1.4 | Syntax Objects
1.5 | Control
1.6 | Other Type Constructors
1.7 | Other Types

On this page:

Any
Any Values
Nothing
1.1`` Base Types
1.1.1`` Numeric Types
Number
Complex
Integer
Float
Flonum
Single-Flonum
Inexact-Real
Exact-Rational
Real
Exact-Number
Float-Complex
Single-Flonum-Complex
Inexact-Complex
Positive-Integer
Exact-Positive-Integer
Nonnegative-Integer
Exact-Nonnegative-Integer
Natural
Negative-Integer
Nonpositive-Integer
Zero
Positive-Float
Positive-Flonum
Nonnegative-Float
Nonnegative-Flonum
Negative-Float
Negative-Flonum
Nonpositive-Float
Nonpositive-Flonum
Float-Negative-Zero
Flonum-Negative-Zero
Float-Positive-Zero
Flonum-Positive-Zero
Float-Zero
Flonum-Zero
Float-Nan
Flonum-Nan
Positive-Single-Flonum
Nonnegative-Single-Flonum
Negative-Single-Flonum
Nonpositive-Single-Flonum
Single-Flonum-Negative-Zero
Single-Flonum-Positive-Zero
Single-Flonum-Zero
Single-Flonum-Nan
Positive-Inexact-Real
Nonnegative-Inexact-Real
Negative-Inexact-Real
Nonpositive-Inexact-Real
Inexact-Real-Negative-Zero
Inexact-Real-Positive-Zero
Inexact-Real-Zero
Inexact-Real-Nan
Positive-Exact-Rational
Nonnegative-Exact-Rational
Negative-Exact-Rational
Nonpositive-Exact-Rational
Positive-Real
Nonnegative-Real
Negative-Real
Nonpositive-Real
Real-Zero
One
Byte
Positive-Byte
Index
Positive-Index
Fixnum
Positive-Fixnum
Nonnegative-Fixnum
Negative-Fixnum
Nonpositive-Fixnum
Ext Flonum
Positive-Ext Flonum
Nonnegative-Ext Flonum
Negative-Ext Flonum
Nonpositive-Ext Flonum
Ext Flonum-Negative-Zero
Ext Flonum-Positive-Zero
Ext Flonum-Zero
Ext Flonum-Nan
1.1.2`` Other Base Types
Boolean
True
False
String
Keyword
Symbol
Char
Void
Input-Port
Output-Port
Port
Path
Path-For-Some-System
Regexp
PRegexp
Byte-Regexp
Byte-PRegexp
Bytes
Namespace
Namespace-Anchor
Variable-Reference
Null
EOF
Continuation-Mark-Set
Undefined
Module-Path
Module-Path-Index
Resolved-Module-Path
Compiled-Module-Expression
Compiled-Expression
Internal-Definition-Context
Pretty-Print-Style-Table
Special-Comment
Struct-Type-Property
Impersonator-Property
Read-Table
Bytes-Converter
Parameterization
Custodian
Inspector
Security-Guard
UDP-Socket
TCP-Listener
Logger
Log-Receiver
Log-Level
Thread
Thread-Group
Subprocess
Place
Place-Channel
Semaphore
FSemaphore
Will-Executor
Pseudo-Random-Generator
Environment-Variables
Path-String
1.2`` Singleton Types
1.3`` Containers
Pairof
Listof
List
List*
MListof
MPairof
MPair Top
Boxof
Box Top
Vectorof
Vector
Fl Vector
Ext Fl Vector
Fx Vector
Vector Top
Hash Table
Hash Table Top
Setof
Channelof
Channel Top
Async-Channelof
Async-Channel Top
Parameterof
Promise
Futureof
Sequenceof
Custodian-Boxof
Thread-Cellof
Thread-Cell Top
Weak-Boxof
Weak-Box Top
Ephemeronof
Evtof
1.4`` Syntax Objects
Syntaxof
Identifier
Syntax
Syntax-E
Sexpof
Sexp
Datum
1.5`` Control
Prompt-Tagof
Prompt-Tag Top
Continuation-Mark-Keyof
Continuation-Mark-Key Top
1.6`` Other Type Constructors
->
->*
Top
Bot
Procedure
U
∩
case->
All
Values
Rec
Struct
Struct-Type
Struct-Type Top
Prefab
Union
Intersection
→
case→
∀
1.7`` Other Types
Option
Opaque

6.8
top ← prev up next →

### 1`` Type Reference

> syntax Any

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

> syntax AnyValues

Any number of Racket values of any type.

> syntax Nothing

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

#### 1.1`` Base Types

##### 1.1.1`` Numeric 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.

> syntax Number
> syntax Complex

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

> syntax Integer

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.

> syntax Float
> syntax Flonum

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

> syntax Single-Flonum

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

> syntax Inexact-Real

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

> syntax Exact-Rational

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

> syntax Real

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.

> syntax Exact-Number
> syntax Float-Complex
> syntax Single-Flonum-Complex
> syntax Inexact-Complex

These types correspond to Racket’s complex numbers.

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).

> syntax Positive-Integer
> syntax Exact-Positive-Integer
> syntax Nonnegative-Integer
> syntax Exact-Nonnegative-Integer
> syntax Natural
> syntax Negative-Integer
> syntax Nonpositive-Integer
> syntax Zero
> syntax Positive-Float
> syntax Positive-Flonum
> syntax Nonnegative-Float
> syntax Nonnegative-Flonum
> syntax Negative-Float
> syntax Negative-Flonum
> syntax Nonpositive-Float
> syntax Nonpositive-Flonum
> syntax Float-Negative-Zero
> syntax Flonum-Negative-Zero
> syntax Float-Positive-Zero
> syntax Flonum-Positive-Zero
> syntax Float-Zero
> syntax Flonum-Zero
> syntax Float-Nan
> syntax Flonum-Nan
> syntax Positive-Single-Flonum
> syntax Nonnegative-Single-Flonum
> syntax Negative-Single-Flonum
> syntax Nonpositive-Single-Flonum
> syntax Single-Flonum-Negative-Zero
> syntax Single-Flonum-Positive-Zero
> syntax Single-Flonum-Zero
> syntax Single-Flonum-Nan
> syntax Positive-Inexact-Real
> syntax Nonnegative-Inexact-Real
> syntax Negative-Inexact-Real
> syntax Nonpositive-Inexact-Real
> syntax Inexact-Real-Negative-Zero
> syntax Inexact-Real-Positive-Zero
> syntax Inexact-Real-Zero
> syntax Inexact-Real-Nan
> syntax Positive-Exact-Rational
> syntax Nonnegative-Exact-Rational
> syntax Negative-Exact-Rational
> syntax Nonpositive-Exact-Rational
> syntax Positive-Real
> syntax Nonnegative-Real
> syntax Negative-Real
> syntax Nonpositive-Real
> syntax Real-Zero

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.

> syntax One
> syntax Byte
> syntax Positive-Byte
> syntax Index
> syntax Positive-Index
> syntax Fixnum
> syntax Positive-Fixnum
> syntax Nonnegative-Fixnum
> syntax Negative-Fixnum
> syntax Nonpositive-Fixnum

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-Flonum]
> 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

> syntax ExtFlonum
> syntax Positive-ExtFlonum
> syntax Nonnegative-ExtFlonum
> syntax Negative-ExtFlonum
> syntax Nonpositive-ExtFlonum
> syntax ExtFlonum-Negative-Zero
> syntax ExtFlonum-Positive-Zero
> syntax ExtFlonum-Zero
> syntax ExtFlonum-Nan

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

##### 1.1.2`` Other Base Types

> syntax Boolean
> syntax True
> syntax False
> syntax String
> syntax Keyword
> syntax Symbol
> syntax Char
> syntax Void
> syntax Input-Port
> syntax Output-Port
> syntax Port
> syntax Path
> syntax Path-For-Some-System
> syntax Regexp
> syntax PRegexp
> syntax Byte-Regexp
> syntax Byte-PRegexp
> syntax Bytes
> syntax Namespace
> syntax Namespace-Anchor
> syntax Variable-Reference
> syntax Null
> syntax EOF
> syntax Continuation-Mark-Set
> syntax Undefined
> syntax Module-Path
> syntax Module-Path-Index
> syntax Resolved-Module-Path
> syntax Compiled-Module-Expression
> syntax Compiled-Expression
> syntax Internal-Definition-Context
> syntax Pretty-Print-Style-Table
> syntax Special-Comment
> syntax Struct-Type-Property
> syntax Impersonator-Property
> syntax Read-Table
> syntax Bytes-Converter
> syntax Parameterization
> syntax Custodian
> syntax Inspector
> syntax Security-Guard
> syntax UDP-Socket
> syntax TCP-Listener
> syntax Logger
> syntax Log-Receiver
> syntax Log-Level
> syntax Thread
> syntax Thread-Group
> syntax Subprocess
> syntax Place
> syntax Place-Channel
> syntax Semaphore
> syntax FSemaphore
> syntax Will-Executor
> syntax Pseudo-Random-Generator
> syntax Environment-Variables

These types represent primitive Racket data.

Examples:

> > #t
> - : Boolean [more precisely: True]
> #t
> > #f
> - : False
> #f
> > "hello"
> - : String
> "hello"
> > (current-input-port)
> - : Input-Port
> #<input-port:string>
> > (current-output-port)
> - : Output-Port
> #<output-port:string>
> > (string->path "/")
> - : Path
> #<path:/>
> > #rx"a*b*"
> - : Regexp
> #rx"a*b*"
> > #px"a*b*"
> - : PRegexp
> #px"a*b*"
> > '#"bytes"
> - : Bytes
> #"bytes"
> > (current-namespace)
> - : Namespace
> #<namespace:0>
> > #\b
> - : Char
> #\b
> > (thread (lambda () (add1 7)))
> - : Thread
> #<thread>

> syntax Path-String

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.

#### 1.2`` Singleton 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
> - : Boolean [more precisely: True]
> #t
> > '#:foo
> - : '#:foo
> '#:foo
> > 'bar
> - : Symbol [more precisely: 'bar]
> 'bar

#### 1.3`` Containers

The following base types are parametric in their type arguments.

> syntax (Pairof s t)

is the pair 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")

> syntax (Listof t)

Homogenous lists of t

> syntax (List t ...)

is the type of the list with one element, in order, for each type provided to the List type constructor.

> syntax (List t ... trest ... bound)

is 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 ....

> syntax (List* t t1 ... s)

is equivalent to (Pairof t (List* t1 ... s)).

Examples:

> > (list 'a 'b 'c)
> - : (Listof (U 'a 'b 'c)) [more precisely: (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))
> - : (Listof String) [more precisely: (Pairof String (Listof String))]
> '("a" "b" "c")

> syntax (MListof t)

Homogenous mutable lists of t.

> syntax (MPairof t u)

Mutable pairs of t and u.

> syntax MPairTop

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)
> #<procedure>

> syntax (Boxof t)

A box of t

Example:

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

> syntax BoxTop

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)
> #<procedure>

> syntax (Vectorof t)

Homogenous vectors of t

> syntax (Vector t ...)

is the type of the vector with one element, in order, for each type provided to the Vector type constructor.

Examples:

> > (vector 1 2 3)
> - : (Vector Integer Integer Integer)
> '#(1 2 3)
> > #(a b c)
> - : (Vector Symbol Symbol Symbol)
> '#(a b c)

> syntax FlVector

An flvector.

Example:

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

> syntax ExtFlVector

An extflvector.

Example:

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

> syntax FxVector

An fxvector.

Example:

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

> syntax VectorTop

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)
> #<procedure>

> syntax (HashTable k v)

is the type of a hash table with key type k and value type v.

Example:

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

> syntax HashTableTop

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)
> #<procedure>

> syntax (Setof t)

is 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 1 3 2 0)

Example:

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

> syntax (Channelof t)

A channel on which only ts can be sent.

Example:

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

> syntax ChannelTop

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)
> #<procedure>

> syntax (Async-Channelof t)

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)
> #<async-channel>

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

> syntax Async-ChannelTop

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)
> #<procedure>

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

> syntax (Parameterof t)
> (Parameterof s t)

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)
> #<procedure:current-input-port>
> > current-directory
> - : (Parameterof Path-String Path)
> #<procedure:current-directory>

> syntax (Promise t)

A promise of t.

Example:

> > (delay 3)
> - : (Promise Positive-Byte)
> #<promise:eval:48:0>

> syntax (Futureof t)

A future which produce a value of type t when touched.

> syntax (Sequenceof t)

A sequence that produces values of type t on each iteration.

> syntax (Custodian-Boxof t)

A custodian box of t.

> syntax (Thread-Cellof t)

A thread cell of t.

> syntax Thread-CellTop

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)
> #<procedure>

> syntax (Weak-Boxof t)

The type for a weak box whose value is of type t.

Examples:

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

> syntax Weak-BoxTop

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)
> #<procedure>

> syntax (Ephemeronof t)

An ephemeron whose value is of type t.

> syntax (Evtof t)

A synchronizable event whose synchronization result is of type t.

Examples:

> > always-evt
> - : (Rec x (Evtof x))
> #<always-evt>
> > (system-idle-evt)
> - : (Evtof Void)
> #<evt>
> > (ann (thread (λ () (displayln "hello world"))) (Evtof Thread))
> - : (Evtof Thread)
> #<thread>

#### 1.4`` Syntax Objects

The following types represent syntax objects and their content.

> syntax (Syntaxof t)

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

> syntax Identifier

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

> syntax Syntax

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

> syntax Syntax-E

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

> syntax (Sexpof t)

The recursive union of t with symbols, keywords, strings, characters, booleans, numbers, boxes, vectors, and (possibly improper) lists.

> syntax Sexp

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

> syntax Datum

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

#### 1.5`` Control

The following types represent prompt tags and keys for continuation marks for use with delimited continuation functions and continuation mark functions.

> syntax (Prompt-Tagof s t)

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)
> #<continuation-prompt-tag:prompt-tag>

> syntax Prompt-TagTop

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)
> #<procedure>

> syntax (Continuation-Mark-Keyof t)

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)
> #<continuation-mark-key>

> syntax Continuation-Mark-KeyTop

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!")))
> hello world
> - : (-> Any Continuation-Mark-KeyTop)
> #<procedure>

#### 1.6`` Other Type Constructors

> syntax (-> 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
> 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)
> #<procedure>
> > (λ () 'hello)
> - : (-> 'hello)
> #<procedure>

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)
> #<procedure: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))

> syntax (->* (mandatory-dom ...) optional-doms rest rng)
> mandatory-dom |  | = |  | type
> |  | | |  | keyword type
> optional-doms |  | = |  |
> |  | | |  | (optional-dom ...)
> optional-dom |  | = |  | type
> |  | | |  | keyword type
> rest |  | = |  |
> |  | | |  | #:rest 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 expression 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)

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.

> syntax Top
> syntax Bot

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.

> syntax Procedure

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:82:0: Type Checker: cannot apply a function with
> unknown arity;
> function `my-list' has type Procedure which cannot be
> applied
> in: "socca"

> syntax (U t ...)

is the union of the types t ....

Example:

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

> syntax (∩ t ...)

is the intersection of the types t ....

Example:

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

> syntax (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 function types constructed with ->.

Example:

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

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

> syntax (t t1 t2 ...)

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

> syntax (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

> syntax (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.

Example:

> > (values 1 2 3)
> - : (values Integer Integer Integer) [more precisely: (Values One Positive-Byte Positive-Byte)]
> 1
> 2
> 3

> syntax v

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

> syntax (quote val)

where val is a Racket value, is the singleton type containing only that value

> syntax i

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

> syntax (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))))

> syntax (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.

> syntax (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
> #<procedure:arity-at-least-ref>
> #<procedure:arity-at-least-set!>
> '(0)
> #f

> syntax Struct-TypeTop

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

> syntax (Prefab key type ...)

Represents a prefab structure type with the given prefab structure key (such as one returned by prefab-struct-key or accepted by make-prefab-struct) and with the given types for each field.

In the case of prefab structure types with supertypes, the field types of the supertypes come before the field types of the child structure type. The order of types matches the order of arguments to a prefab struct constructor.

Examples:

> > #s(salad "potato" "mayo")
> - : (Prefab salad String String)
> '#s(salad "potato" "mayo")
> > (: q-salad (Prefab (salad food 1) String String Symbol))
> > (define q-salad
> #s((salad food 1) "quinoa" "EVOO" salad))

> syntax Union

An alias for U.

> syntax Intersection

An alias for ∩.

> syntax →

An alias for ->.

> syntax case→

An alias for case->.

> syntax ∀

An alias for All.

#### 1.7`` Other Types

> syntax (Option t)

Either t or #f

> syntax (Opaque t)

A type constructed using the #:opaque clause of require/typed.
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