► |  | The Racket Reference

1 | Language Model
2 | Notation for Documentation
3 | Syntactic Forms
4 | Datatypes
5 | Structures
6 | Classes and Objects
7 | Units
8 | Contracts
9 | Pattern Matching
10 | Control Flow
11 | Concurrency and Parallelism
12 | Macros
13 | Input and Output
14 | Reflection and Security
15 | Operating System
16 | Memory Management
17 | Unsafe Operations
18 | Running Racket
| Bibliography
| Index

▼ | 4 | Datatypes

4.1 | Booleans and Equality
4.2 | Numbers
4.3 | Strings
4.4 | Byte Strings
4.5 | Characters
4.6 | Symbols
4.7 | Regular Expressions
4.8 | Keywords
4.9 | Pairs and Lists
4.10 | Mutable Pairs and Lists
4.11 | Vectors
4.12 | Boxes
4.13 | Hash Tables
4.14 | Sequences and Streams
4.15 | Dictionaries
4.16 | Sets
4.17 | Procedures
4.18 | Void and Undefined

► | 4.17 | Procedures

4.17.1 | Keywords and Arity
4.17.2 | Reflecting on Primitives
4.17.3 | Additional Higher-Order Functions

On this page:

procedure?
apply
compose
compose1
procedure-rename
procedure->method
procedure-closure-contents-eq?
4.17.1`` Keywords and Arity
keyword-apply
procedure-arity
procedure-arity?
procedure-arity-includes?
procedure-reduce-arity
procedure-keywords
make-keyword-procedure
procedure-reduce-keyword-arity
arity-at-least
prop: procedure
procedure-struct-type?
procedure-extract-target
prop: arity-string
prop: checked-procedure
checked-procedure-check-and-extract
4.17.2`` Reflecting on Primitives
primitive?
primitive-closure?
primitive-result-arity
4.17.3`` Additional Higher-Order Functions
identity
const
thunk
thunk*
negate
curry
curryr
normalized-arity?
normalize-arity
arity=?
arity-includes?

top ← prev up next →

#### 4.17`` Procedures

> procedure (procedure? v) → boolean?
> v : any/c

Returns #t if v is a procedure, #f otherwise.

> procedure (apply proc v ... lst #:<kw> kw-arg ...) → any
> proc : procedure?
> v : any/c
> lst : list?
> kw-arg : any/c

> > > The apply Function in The Racket Guide introduces apply.

Applies proc using the content of (list* v ... lst) as the (by-position) arguments. The #:<kw> kw-arg sequence is also supplied as keyword arguments to proc, where #:<kw> stands for any keyword.

The given proc must accept as many arguments as the number of vs plus length of lst, it must accept the supplied keyword arguments, and it must not require any other keyword arguments; otherwise, the exn:fail:contract exception is raised. The given proc is called in tail position with respect to the apply call.

Examples:
> > (apply + '(1 2 3))
> 6
> > (apply + 1 2 '(3))
> 6
> > (apply + '())
> 0
> > (apply sort (list (list '(2) '(1)) <) #:key car)
> '((1) (2))

> procedure (compose proc ...) → procedure?
> proc : procedure?
> procedure (compose1 proc ...) → procedure?
> proc : procedure?

Returns a procedure that composes the given functions, applying the last proc first and the first proc last. The compose function allows the given functions to consume and produce any number of values, as long as each function produces as many values as the preceding function consumes, while compose1 restricts the internal value passing to a single value. In both cases, the input arity of the last function and the output arity of the first are unrestricted, and they become the corresponding arity of the resulting composition (including keyword arguments for the input side).

When no proc arguments are given, the result is values. When exactly one is given, it is returned.

Examples:
> > ((compose1 - sqrt) 10)
> -3.1622776601683795
> > ((compose1 sqrt -) 10)
> 0+3.1622776601683795i
> > ((compose list split-path) (bytes->path #"/a" 'unix))
> '(#<path:/> #<path:a> #f)

Note that in many cases, compose1 is preferred. For example, using compose with two library functions may lead to problems when one function is extended to return two values, and the preceding one has an optional input with different semantics. In addition, compose1 may create faster compositions.

> procedure (procedure-rename proc name) → procedure?
> proc : procedure?
> name : symbol?

Returns a procedure that is like proc, except that its name as returned by object-name (and as printed for debugging) is name.

The given name is used for printing an error message if the resulting procedure is applied to the wrong number of arguments. In addition, if proc is an accessor or mutator produced by struct, make-struct-field-accessor, or make-struct-field-mutator, the resulting procedure also uses name when its (first) argument has the wrong type. More typically, however, name is not used for reporting errors, since the procedure name is typically hard-wired into an internal check.

> procedure (procedure->method proc) → procedure?
> proc : procedure?

Returns a procedure that is like proc except that, when applied to the wrong number of arguments, the resulting error hides the first argument as if the procedure had been compiled with the 'method-arity-error syntax property.

> > procedure
> > (procedure-closure-contents-eq? |  | proc1 |  |  |  |
> > |  | proc2) |  | → |  | boolean?
> proc1 : procedure?
> proc2 : procedure?

Compares the contents of the closures of proc1 and proc2 for equality by comparing closure elements pointwise using eq?

##### 4.17.1`` Keywords and Arity

> > procedure
> > (keyword-apply |  | proc |  |  |  |
> > |  | kw-lst |  |  |  |
> > |  | kw-val-lst |  |  |  |
> > |  | v ... |  |  |  |
> > |  | lst |  |  |  |
> > |  | #:<kw> kw-arg ...) |  | → |  | any
> proc : procedure?
> kw-lst : (listof keyword?)
> kw-val-lst : list?
> v : any/c
> lst : list?
> kw-arg : any/c

> > > The apply Function in The Racket Guide introduces keyword-apply.

Like apply, but kw-lst and kw-val-lst supply by-keyword arguments in addition to the by-position arguments of the vs and lst, and in addition to the directly supplied keyword arguments in the #:<kw> kw-arg sequence, where #:<kw> stands for any keyword.

The given kw-lst must be sorted using keyword<?. No keyword can appear twice in kw-lst or in both kw-list and as a #:<kw>, otherwise, the exn:fail:contract exception is raised. The given kw-val-lst must have the same length as kw-lst, otherwise, the exn:fail:contract exception is raised. The given proc must accept all of the keywords in kw-lst plus the #:<kw>s, it must not require any other keywords, and it must accept as many by-position arguments as supplied via the vs and lst; otherwise, the exn:fail:contract exception is raised.

Examples:
> (define (f x #:y y #:z [z 10])
> (list x y z))
> > (keyword-apply f '(#:y) '(2) '(1))
> '(1 2 10)
> > (keyword-apply f '(#:y #:z) '(2 3) '(1))
> '(1 2 3)
> > (keyword-apply f #:z 7 '(#:y) '(2) '(1))
> '(1 2 7)

> procedure (procedure-arity proc) → normalized-arity?
> proc : procedure?

Returns information about the number of by-position arguments accepted by proc. See also procedure-arity? and normalized-arity?.

> procedure (procedure-arity? v) → boolean?
> v : any/c

A valid arity a is one of the following:

- An exact non-negative integer, which means that the procedure accepts a arguments, only.

- A arity-at-least instance, which means that the procedure accepts (arity-at-least-value a) or more arguments.

- A list containing integers and arity-at-least instances, which means that the procedure accepts any number of arguments that can match one of the elements of a.

The result of procedure-arity is always normalized in the sense of normalized-arity?.

Examples:
> > (procedure-arity cons)
> 2
> > (procedure-arity list)
> (arity-at-least 0)
> > (arity-at-least? (procedure-arity list))
> #t
> > (arity-at-least-value (procedure-arity list))
> 0
> > (arity-at-least-value (procedure-arity (lambda (x . y) x)))
> 1
> > (procedure-arity (case-lambda [(x) 0] [(x y) 1]))
> '(1 2)

> procedure (procedure-arity-includes? proc k [kws-ok?]) → boolean?
> proc : procedure?
> k : exact-nonnegative-integer?
> kws-ok? : any/c = #f

Returns #t if the procedure can accept k by-position arguments, #f otherwise. If kws-ok? is #f, the result is #t only if proc has no required keyword arguments.

Examples:
> > (procedure-arity-includes? cons 2)
> #t
> > (procedure-arity-includes? display 3)
> #f
> > (procedure-arity-includes? (lambda (x #:y y) x) 1)
> #f
> > (procedure-arity-includes? (lambda (x #:y y) x) 1 #t)
> #t

> procedure (procedure-reduce-arity proc arity) → procedure?
> proc : procedure?
> arity : procedure-arity?

Returns a procedure that is the same as proc (including the same name returned by object-name), but that accepts only arguments consistent with arity. In particular, when procedure-arity is applied to the generated procedure, it returns a value that is equal? to arity.

If the arity specification allows arguments that are not in (procedure-arity proc), the exn:fail:contract exception is raised. If proc accepts keyword argument, either the keyword arguments must be all optional (and they are not accepted in by the arity-reduced procedure) or arity must be the empty list (which makes a procedure that cannot be called); otherwise, the exn:fail:contract exception is raised.

Examples:
> > (define my+ (procedure-reduce-arity + 2))
> > (my+ 1 2)
> 3
> > (my+ 1 2 3)
> +: arity mismatch;
> the expected number of arguments does not match the given
> number
> expected: 2
> given: 3
> arguments...:
> 1
> 2
> 3

> > procedure
> > (procedure-keywords proc) |  | → |  |
> > (listof keyword?)
> > (or/c (listof keyword?) #f)
> proc : procedure?

Returns information about the keyword arguments required and accepted by a procedure. The first result is a list of distinct keywords (sorted by keyword<?) that are required when applying proc. The second result is a list of distinct accepted keywords (sorted by keyword<?), or #f to mean that any keyword is accepted. When the second result is a list, every element in the first list is also in the second list.

Examples:
> > (procedure-keywords +)
> '()
> '()
> > (procedure-keywords (lambda (#:tag t #:mode m) t))
> '(#:mode #:tag)
> '(#:mode #:tag)
> > (procedure-keywords (lambda (#:tag t #:mode [m #f]) t))
> '(#:tag)
> '(#:mode #:tag)

> procedure (make-keyword-procedure proc [plain-proc]) → procedure?
> proc : (((listof keyword?) list?) () #:rest list? . ->* . any)
> | plain-proc |  | : |  | procedure?
> |  |  | = |  | (lambda args (apply proc null null args))

Returns a procedure that accepts all keyword arguments (without requiring any keyword arguments).

When the procedure returned by make-keyword-procedure is called with keyword arguments, then proc is called; the first argument is a list of distinct keywords sorted by keyword<?, the second argument is a parallel list containing a value for each keyword, and the remaining arguments are the by-position arguments.

When the procedure returned by make-keyword-procedure is called without keyword arguments, then plain-proc is called—possibly more efficiently than dispatching through proc. Normally, plain-proc should have the same behavior as calling proc with empty lists as the first two arguments, but that correspondence is in no way enforced.

The result of procedure-arity and object-name on the new procedure is the same as for plain-proc. See also procedure-reduce-keyword-arity and procedure-rename.

Examples:
> (define show
> (make-keyword-procedure (lambda (kws kw-args . rest)
> (list kws kw-args rest))))
> > (show 1)
> '(() () (1))
> > (show #:init 0 1 2 3 #:extra 4)
> '((#:extra #:init) (4 0) (1 2 3))
> (define show2
> (make-keyword-procedure (lambda (kws kw-args . rest)
> (list kws kw-args rest))
> (lambda args
> (list->vector args))))
> > (show2 1)
> '#(1)
> > (show2 #:init 0 1 2 3 #:extra 4)
> '((#:extra #:init) (4 0) (1 2 3))

> > procedure
> > (procedure-reduce-keyword-arity |  | proc |  |  |  |
> > |  | arity |  |  |  |
> > |  | required-kws |  |  |  |
> > |  | allowed-kws) |  | → |  | procedure?
> proc : procedure?
> arity : procedure-arity?
> required-kws : (listof keyword?)
> | allowed-kws |  | : |  |
> (or/c (listof keyword?)
> #f)

Like procedure-reduce-arity, but constrains the keyword arguments according to required-kws and allowed-kws, which must be sorted using keyword<? and contain no duplicates. If allowed-kws is #f, then the resulting procedure still accepts any keyword, otherwise the keywords in required-kws must be a subset of those in allowed-kws. The original proc must require no more keywords than the ones listed in required-kws, and it must allow at least the keywords in allowed-kws (or it must allow all keywords if allowed-kws is #f).

Examples:
> (define orig-show
> (make-keyword-procedure (lambda (kws kw-args . rest)
> (list kws kw-args rest))))
> (define show (procedure-reduce-keyword-arity
> orig-show 3 '(#:init) '(#:extra #:init)))
> > (show #:init 0 1 2 3 #:extra 4)
> '((#:extra #:init) (4 0) (1 2 3))
> > (show 1)
> ...t/private/kw.rkt:196:14: arity mismatch;
> the expected number of arguments does not match the given
> number
> expected: 3 plus an argument with keyword #:init plus an
> optional argument with keyword #:extra
> given: 1
> arguments...:
> 1
> > (show #:init 0 1 2 3 #:extra 4 #:more 7)
> application: procedure does not expect an argument with
> given keyword
> procedure: ...t/private/kw.rkt:196:14
> given keyword: #:more
> arguments...:
> 1
> 2
> 3
> #:extra 4
> #:init 0
> #:more 7

> > struct
> > (struct |  | arity-at-least (value)
> > |  | #:extra-constructor-name make-arity-at-least)
> value : exact-nonnegative-integer?

A structure type used for the result of procedure-arity. See also procedure-arity?.

> value prop:procedure : struct-type-property?

A structure type property to identify structure types whose instances can be applied as procedures. In particular, when procedure? is applied to the instance, the result will be #t, and when an instance is used in the function position of an application expression, a procedure is extracted from the instance and used to complete the procedure call.

If the prop:procedure property value is an exact non-negative integer, it designates a field within the structure that should contain a procedure. The integer must be between 0 (inclusive) and the number of non-automatic fields in the structure type (exclusive, not counting supertype fields). The designated field must also be specified as immutable, so that after an instance of the structure is created, its procedure cannot be changed. (Otherwise, the arity and name of the instance could change, and such mutations are generally not allowed for procedures.) When the instance is used as the procedure in an application expression, the value of the designated field in the instance is used to complete the procedure call. (This procedure can be another structure that acts as a procedure; the immutability of procedure fields disallows cycles in the procedure graph, so that the procedure call will eventually continue with a non-structure procedure.) That procedure receives all of the arguments from the application expression. The procedure’s name (see object-name), arity (see procedure-arity), and keyword protocol (see procedure-keywords) are also used for the name, arity, and keyword protocol of the structure. If the value in the designated field is not a procedure, then the instance behaves like (case-lambda) (i.e., a procedure which does not accept any number of arguments). See also procedure-extract-target.

Providing an integer proc-spec argument to make-struct-type is the same as both supplying the value with the prop:procedure property and designating the field as immutable (so that a property binding or immutable designation is redundant and disallowed).

Examples:
> > (struct annotated-proc (base note)
> #:property prop:procedure
> (struct-field-index base))
> > (define plus1 (annotated-proc
> (lambda (x) (+ x 1))
> "adds 1 to its argument"))
> > (procedure? plus1)
> #t
> > (annotated-proc? plus1)
> #t
> > (plus1 10)
> 11
> > (annotated-proc-note plus1)
> "adds 1 to its argument"

When the prop:procedure value is a procedure, it should accept at least one non-keyword argument. When an instance of the structure is used in an application expression, the property-value procedure is called with the instance as the first argument. The remaining arguments to the property-value procedure are the arguments from the application expression (including keyword arguments). Thus, if the application expression provides five non-keyword arguments, the property-value procedure is called with six non-keyword arguments. The name of the instance (see object-name) and its keyword protocol (see procedure-keywords) are unaffected by the property-value procedure, but the instance’s arity is determined by subtracting one from every possible non-keyword argument count of the property-value procedure. If the property-value procedure cannot accept at least one argument, then the instance behaves like (case-lambda).

Providing a procedure proc-spec argument to make-struct-type is the same as supplying the value with the prop:procedure property (so that a specific property binding is disallowed).

Examples:
> > (struct fish (weight color)
> #:mutable
> #:property
> prop:procedure
> (lambda (f n)
> (let ([w (fish-weight f)])
> (set-fish-weight! f (+ n w)))))
> > (define wanda (fish 12 'red))
> > (fish? wanda)
> #t
> > (procedure? wanda)
> #t
> > (fish-weight wanda)
> 12
> > (for-each wanda '(1 2 3))
> > (fish-weight wanda)
> 18

If the value supplied for the prop:procedure property is not an exact non-negative integer or a procedure, the exn:fail:contract exception is raised.

> procedure (procedure-struct-type? type) → boolean?
> type : struct-type?

Returns #t if instances of the structure type represented by type are procedures (according to procedure?), #f otherwise.

> procedure (procedure-extract-target proc) → (or/c #f procedure?)
> proc : procedure?

If proc is an instance of a structure type with property prop:procedure, and if the property value indicates a field of the structure, and if the field value is a procedure, then procedure-extract-target returns the field value. Otherwise, the result is #f.

When a prop:procedure property value is a procedure, the procedure is not returned by procedure-extract-target. Such a procedure is different from one accessed through a structure field, because it consumes an extra argument, which is always the structure that was applied as a procedure. Keeping the procedure private ensures that is it always called with a suitable first argument.

> value prop:arity-string : struct-type-property?

A structure type property that is used for reporting arity-mismatch errors when a structure type with the prop:procedure property is applied to the wrong number of arguments. The value of the prop:arity-string property must be a procedure that takes a single argument, which is the misapplied structure, and returns a string. The result string is used after the word “expects,” and it is followed in the error message by the number of actual arguments.

Arity-mismatch reporting automatically uses procedure-extract-target when the prop:arity-string property is not associated with a procedure structure type.

Examples:
> > (struct evens (proc)
> #:property prop:procedure (struct-field-index proc)
> #:property prop:arity-string
> (lambda (p)
> "an even number of arguments"))
> > (define pairs
> (evens
> (case-lambda
> [() null]
> [(a b . more)
> (cons (cons a b)
> (apply pairs more))])))
> > (pairs 1 2 3 4)
> '((1 . 2) (3 . 4))
> > (pairs 5)
> #<procedure>: arity mismatch;
> the expected number of arguments does not match the given
> number
> expected: an even number of arguments
> given: 1
> arguments...:
> 5

> value prop:checked-procedure : struct-type-property?

A structure type property that is used with checked-procedure-check-and-extract, which is a hook to allow the compiler to improve the performance of keyword arguments. The property can only be attached to a structure type without a supertype and with at least two fields.

> > procedure
> > (checked-procedure-check-and-extract |  | type |  |  |  |
> > |  | v |  |  |  |
> > |  | proc |  |  |  |
> > |  | v1 |  |  |  |
> > |  | v2) |  | → |  | any/c
> type : struct-type?
> v : any/c
> proc : (any/c any/c any/c . -> . any/c)
> v1 : any/c
> v2 : any/c

Extracts a value from v if it is an instance of type, which must have the property prop:checked-procedure. If v is such an instance, then the first field of v is extracted and applied to v1 and v2; if the result is a true value, the result is the value of the second field of v.

If v is not an instance of type, or if the first field of v applied to v1 and v2 produces #f, then proc is applied to v, v1, and v2, and its result is returned by checked-procedure-check-and-extract.

##### 4.17.2`` Reflecting on Primitives

A primitive procedure is a built-in procedure that is implemented in low-level language. Not all procedures of racket/base are primitives, but many are. The distinction is mainly useful to other low-level code.

> procedure (primitive? v) → boolean?
> v : any/c

Returns #t if v is a primitive procedure, #f otherwise.

> procedure (primitive-closure? v) → boolean
> v : any/c

Returns #t if v is internally implemented as a primitive closure rather than a simple primitive procedure, #f otherwise.

> procedure (primitive-result-arity prim) → procedure-arity?
> prim : primitive?

Returns the arity of the result of the primitive procedure prim (as opposed to the procedure’s input arity as returned by procedure-arity). For most primitives, this procedure returns 1, since most primitives return a single value when applied.

##### 4.17.3`` Additional Higher-Order Functions

(require racket/function) | package: base

The bindings documented in this section are provided by the racket/function and racket libraries, but not racket/base.

> procedure (identity v) → any/c
> v : any/c

Returns v.

> procedure (const v) → procedure?
> v : any

Returns a procedure that accepts any arguments (including keyword arguments) and returns v.

Examples:
> > ((const 'foo) 1 2 3)
> 'foo
> > ((const 'foo))
> 'foo

> syntax (thunk body ...+)
> syntax (thunk* body ...+)

The thunk form creates a nullary function that evaluates the given body. The thunk* form is similar, except that the resulting function accepts any arguments (including keyword arguments).

Examples:
> (define th1 (thunk (define x 1) (printf "~a\n" x)))
> > (th1)
> 1
> > (th1 'x)
> th1: arity mismatch;
> the expected number of arguments does not match the given
> number
> expected: 0
> given: 1
> arguments...:
> 'x
> > (th1 #:y 'z)
> application: procedure does not accept keyword arguments
> procedure: th1
> arguments...:
> #:y 'z
> (define th2 (thunk* (define x 1) (printf "~a\n" x)))
> > (th2)
> 1
> > (th2 'x)
> 1
> > (th2 #:y 'z)
> 1

> procedure (negate proc) → procedure?
> proc : procedure?

Returns a procedure that is just like proc, except that it returns the not of proc’s result.

Examples:
> > (filter (negate symbol?) '(1 a 2 b 3 c))
> '(1 2 3)
> > (map (negate =) '(1 2 3) '(1 1 1))
> '(#f #t #t)

> procedure (curry proc) → procedure?
> proc : procedure?
> (curry proc v ...+) → any/c
> proc : procedure?
> v : any/c

Returns a procedure that is a curried version of proc. When the resulting procedure is first applied, unless it is given the maximum number of arguments that it can accept, the result is a procedure to accept additional arguments.

Examples:
> > ((curry list) 1 2)
> #<procedure:curried>
> > ((curry cons) 1)
> #<procedure:curried>
> > ((curry cons) 1 2)
> '(1 . 2)

After the first application of the result of curry, each further application accumulates arguments until an acceptable number of arguments have been accumulated, at which point the original proc is called.

Examples:
> > (((curry list) 1 2) 3)
> '(1 2 3)
> > (((curry list) 1) 3)
> '(1 3)
> > ((((curry foldl) +) 0) '(1 2 3))
> 6

A function call (curry proc v ...) is equivalent to ((curry proc) v ...). In other words, curry itself is curried.

The curry function provides limited support for keyworded functions: only the curry call itself can receive keyworded arguments to be propagated eventually to proc.

Examples:
> > (map ((curry +) 10) '(1 2 3))
> '(11 12 13)
> > (map (curry + 10) '(1 2 3))
> '(11 12 13)
> > (map (compose (curry * 2) (curry + 10)) '(1 2 3))
> '(22 24 26)
> > (define foo (curry (lambda (x y z) (list x y z))))
> > (foo 1 2 3)
> '(1 2 3)
> > (((((foo) 1) 2)) 3)
> '(1 2 3)

> procedure (curryr proc) → procedure?
> proc : procedure?
> (curryr proc v ...+) → any/c
> proc : procedure?
> v : any/c

Like curry, except that the arguments are collected in the opposite direction: the first step collects the rightmost group of arguments, and following steps add arguments to the left of these.

Example:
> > (map (curryr list 'foo) '(1 2 3))
> '((1 foo) (2 foo) (3 foo))

> procedure (normalized-arity? arity) → boolean?
> arity : any/c

A normalized arity has one of the following forms:

- the empty list;

- an exact non-negative integer;

- an arity-at-least instance;

- a list of two or more strictly increasing, exact non-negative integers; or

- a list of one or more strictly increasing, exact non-negative integers followed by a single arity-at-least instance whose value is greater than the preceding integer by at least 2.

Every normalized arity is a valid procedure arity and satisfies procedure-arity?. Any two normalized arity values that are arity=? must also be equal?.

Examples:
> > (normalized-arity? (arity-at-least 1))
> #t
> > (normalized-arity? (list (arity-at-least 1)))
> #f
> > (normalized-arity? (list 0 (arity-at-least 2)))
> #t
> > (normalized-arity? (list (arity-at-least 2) 0))
> #f
> > (normalized-arity? (list 0 2 (arity-at-least 3)))
> #f

> procedure (normalize-arity arity)
> → (and/c normalized-arity? (lambda (x) (arity=? x arity)))
> arity : procedure-arity?

Produces a normalized form of arity. See also normalized-arity? and arity=?.

Examples:
> > (normalize-arity 1)
> 1
> > (normalize-arity (list 1))
> 1
> > (normalize-arity (arity-at-least 2))
> (arity-at-least 2)
> > (normalize-arity (list (arity-at-least 2)))
> (arity-at-least 2)
> > (normalize-arity (list 1 (arity-at-least 2)))
> (arity-at-least 1)
> > (normalize-arity (list (arity-at-least 2) 1))
> (arity-at-least 1)
> > (normalize-arity (list (arity-at-least 2) 3))
> (arity-at-least 2)
> > (normalize-arity (list 3 (arity-at-least 2)))
> (arity-at-least 2)
> > (normalize-arity (list (arity-at-least 6) 0 2 (arity-at-least 4)))
> (list 0 2 (arity-at-least 4))

> procedure (arity=? a b) → boolean?
> a : procedure-arity?
> b : procedure-arity?

Returns #true if procedures with arity a and b accept the same numbers of arguments, and #false otherwise. Equivalent to both (and (arity-includes? a b) (arity-includes? b a)) and (equal? (normalize-arity a) (normalize-arity b)).

Examples:
> > (arity=? 1 1)
> #t
> > (arity=? (list 1) 1)
> #t
> > (arity=? 1 (list 1))
> #t
> > (arity=? 1 (arity-at-least 1))
> #f
> > (arity=? (arity-at-least 1) 1)
> #f
> > (arity=? 1 (arity-at-least 1))
> #f
> > (arity=? (arity-at-least 1) (list 1 (arity-at-least 2)))
> #t
> > (arity=? (list 1 (arity-at-least 2)) (arity-at-least 1))
> #t
> > (arity=? (arity-at-least 1) (list 1 (arity-at-least 3)))
> #f
> > (arity=? (list 1 (arity-at-least 3)) (arity-at-least 1))
> #f
> > (arity=? (list 0 1 2 (arity-at-least 3)) (list (arity-at-least 0)))
> #t
> > (arity=? (list (arity-at-least 0)) (list 0 1 2 (arity-at-least 3)))
> #t
> > (arity=? (list 0 2 (arity-at-least 3)) (list (arity-at-least 0)))
> #f
> > (arity=? (list (arity-at-least 0)) (list 0 2 (arity-at-least 3)))
> #f

> procedure (arity-includes? a b) → boolean?
> a : procedure-arity?
> b : procedure-arity?

Returns #true if procedures with arity a accept any number of arguments that procedures with arity b accept.

Examples:
> > (arity-includes? 1 1)
> #t
> > (arity-includes? (list 1) 1)
> #t
> > (arity-includes? 1 (list 1))
> #t
> > (arity-includes? 1 (arity-at-least 1))
> #f
> > (arity-includes? (arity-at-least 1) 1)
> #t
> > (arity-includes? 1 (arity-at-least 1))
> #f
> > (arity-includes? (arity-at-least 1) (list 1 (arity-at-least 2)))
> #t
> > (arity-includes? (list 1 (arity-at-least 2)) (arity-at-least 1))
> #t
> > (arity-includes? (arity-at-least 1) (list 1 (arity-at-least 3)))
> #t
> > (arity-includes? (list 1 (arity-at-least 3)) (arity-at-least 1))
> #f
> > (arity-includes? (list 0 1 2 (arity-at-least 3)) (list (arity-at-least 0)))
> #t
> > (arity-includes? (list (arity-at-least 0)) (list 0 1 2 (arity-at-least 3)))
> #t
> > (arity-includes? (list 0 2 (arity-at-least 3)) (list (arity-at-least 0)))
> #f
> > (arity-includes? (list (arity-at-least 0)) (list 0 2 (arity-at-least 3)))
> #t

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