4.20ProceduresðŸ”—â„¹

 procedure v : any/c
Returns #t if v is a procedure, #f otherwise.

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

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) 0.0+3.1622776601683795i > ((compose list split-path) (bytes->path #"/a" 'unix)) '(# # #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 [realm]) → procedure? proc : procedure? name : symbol? realm : symbol? = 'racket
Returns a procedure that is like proc, except that its name as returned by object-name (and as printed for debugging) is name and its realm (potentially used for adjusting error messages) is realm.

The given name and realm are used for printing and adjusting 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.

Changed in version 8.4.0.2 of package base: Added the realm argument.

 procedure(procedure-realm proc) → symbol? proc : procedure?
Reports the realm of a procedure, which can depend on the module where the procedure was created, the current-compile-realm value when the procedure’s code was compiled, or a realm explicitly assigned through a function like procedure-rename.

Added in version 8.4.0.2 of package base.

 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?

4.20.1Keywords and ArityðŸ”—â„¹

procedure

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

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 both in kw-lst 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 proc : procedure?

 procedure 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 proc : procedure?
Returns the same information as procedure-arity, but encoded differently. The arity is encoded as an exact integer mask where (bitwise-bit-set? mask n) returns true if proc accepts n arguments.

The mask encoding of an arity is often easier to test and manipulate, and procedure-arity-mask is sometimes faster than procedure-arity while always being at least as fast.

Added in version 7.0.0.11 of package base.

 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 [ name realm]) → procedure?
proc : procedure?
arity : procedure-arity?
name : (or/c symbol? #f) = #f
realm : symbol? = 'racket
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 the normalized form of 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.

If name is not #f, then object-name of the result procedure produces name, and procedure-realm of the result produced produces realm. Otherwise, object-name and procedure-realm of the result procedure produce the same result as for proc.

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

Changed in version 7.0.0.11 of package base: Added the optional name argument.
Changed in version 8.4.0.2: Added the realm argument.

procedure

proc : procedure?
name : (or/c symbol? #f) = #f
realm : symbol? = 'racket
The same as procedure-reduce-arity, but using the representation of arity described with procedure-arity-mask.

The mask encoding of an arity is often easier to test and manipulate, and procedure-reduce-arity-mask is sometimes faster than procedure-reduce-arity while always being at least as fast.

Added in version 7.0.0.11 of package base.
Changed in version 8.4.0.2: Added the realm argument.

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(procedure-result-arity proc) → (or/c #f procedure-arity?) proc : procedure?
Returns the arity of the result of the procedure proc or #f if the number of results are not known, perhaps due to shortcomings in the implementation of procedure-result-arity or because proc’s behavior is not sufficiently simple.

Examples:
> (procedure-result-arity car)

1

> (procedure-result-arity values)

#f

 > (procedure-result-arity (λ (x) (apply values (let loop () (cond [(zero? (random 10)) '()] [else (cons 1 (loop))])))))

#f

Added in version 6.4.0.3 of package base.

procedure

(make-keyword-procedure proc [plain-proc])  procedure?

proc : ((listof keyword?) list? any/c ... . -> . 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 [ name realm]) → procedure?
proc : procedure?
arity : procedure-arity?
required-kws : (listof keyword?)
allowed-kws :
 (or/c (listof keyword?) #f)
name : (or/c symbol? #f) = #f
realm : symbol? = 'racket
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)

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: #<procedure>

given keyword: #:more

arguments...:

1

2

3

#:extra 4

#:init 0

#:more 7

Changed in version 8.4.0.2 of package base: Added the realm argument.

procedure

proc : procedure?
required-kws : (listof keyword?)
allowed-kws :
 (or/c (listof keyword?) #f)
name : (or/c symbol? #f) = #f
realm : symbol? = 'racket
The same as procedure-reduce-keyword-arity, but using the representation of arity described with procedure-arity-mask.

Added in version 7.0.0.11 of package base.
Changed in version 8.4.0.2: Added the realm argument.

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
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)

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 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
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)

arity mismatch;

the expected number of arguments does not match the given

number

expected: an even number of arguments

given: 1

arguments...:

5

 value
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.

 procedure proc : procedure?
Returns proc or its equivalent, but provides a hint to the run-time system that it should spend extra time and memory to specialize the implementation of proc.

The hint is currently used when proc is the value of a lambda or case-lambda form that references variables bound outside of the lambda or case-lambda, and when proc has not been previously applied.

Added in version 6.3.0.10 of package base.

4.20.2Reflecting on PrimitivesðŸ”—â„¹

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

 procedure v : any/c
Returns #t if v is a primitive procedure, #f otherwise.

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

 procedure 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.

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

Examples:
 > ((const 'foo)) 'foo > ((const 'foo) 1 2 3) 'foo > ((const 'foo) 'a 'b #:c 'c) 'foo

 procedure(const* v ...) → procedure? v : any/c
Similar to const, except it returns vs.

Examples:
> ((const*))
> ((const*) 1 2 3)
> ((const*) 'a 'b #:c 'c)
> ((const* 'foo))

'foo

> ((const* 'foo) 1 2 3)

'foo

> ((const* 'foo) 'a 'b #:c 'c)

'foo

> ((const* 'foo 'foo))
 'foo 'foo
> ((const* 'foo 'foo) 1 2 3)
 'foo 'foo
> ((const* 'foo 'foo) 'a 'b #:c 'c)
 'foo 'foo

Added in version 8.7.0.5 of package base.

 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 > (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((conjoin f ...) x ...) → any f : procedure? x : any/c
Combines calls to each function with and. Equivalent to (and (f x ...) ...)

Examples:
(define f (conjoin exact? integer?))

> (f 1)

#t

> (f 1.0)

#f

> (f 1/2)

#f

> (f 0.5)

#f

> ((conjoin (λ (x) (values 1 2))) 0)
 1 2

 procedure((disjoin f ...) x ...) → any f : procedure? x : any/c
Combines calls to each function with or. Equivalent to (or (f x ...) ...)

Examples:
(define f (disjoin exact? integer?))

> (f 1)

#t

> (f 1.0)

#t

> (f 1/2)

#t

> (f 0.5)

#f

> ((disjoin (λ (x) (values 1 2))) 0)
 1 2

 procedure(curry proc) → procedure? proc : procedure? (curry proc v ...+) → any proc : procedure? v : any/c
The result of (curry proc) is 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 according to (procedure-arity proc), the result is a procedure to accept additional arguments.

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

After the first application of the result of (curry proc), each further application accumulates arguments until an acceptable number of arguments according to (procedure-arity proc) 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 > (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)

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

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)

The curry function also supports functions with keyword arguments: keyword arguments will be accumulated in the same way as positional arguments until all required keyword arguments according to (procedure-keywords proc) have been supplied.

Examples:
 (define (f #:a a #:b b #:c c) (list a b c))

> ((((curry f) #:a 1) #:b 2) #:c 3)

'(1 2 3)

> ((((curry f) #:b 1) #:c 2) #:a 3)

'(3 1 2)

> ((curry f #:a 1 #:c 2) #:b 3)

'(1 3 2)

Changed in version 7.0.0.7 of package base: Added support for keyword arguments.

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

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=? (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 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? (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