4.9 Pairs and Lists

► |  | The Racket Reference

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2 | Notation for Documentation
3 | Syntactic Forms
4 | Datatypes
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7 | Units
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12 | Macros
13 | Input and Output
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15 | Operating System
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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
4.19 | Undefined

► | 4.9 | Pairs and Lists

4.9.1 | Pair Constructors and Selectors
4.9.2 | List Operations
4.9.3 | List Iteration
4.9.4 | List Filtering
4.9.5 | List Searching
4.9.6 | Pair Accessor Shorthands
4.9.7 | Additional List Functions and Synonyms
4.9.8 | Immutable Cyclic Data

On this page:

4.9.1`` Pair Constructors and Selectors
pair?
null?
cons
car
cdr
null
list?
list
list*
build-list
4.9.2`` List Operations
length
list-ref
list-tail
append
reverse
4.9.3`` List Iteration
map
andmap
ormap
for-each
foldl
foldr
4.9.4`` List Filtering
filter
remove
remq
remv
remove*
remq*
remv*
sort
4.9.5`` List Searching
member
memv
memq
memf
findf
assoc
assv
assq
assf
4.9.6`` Pair Accessor Shorthands
caar
cadr
cdar
cddr
caaar
caadr
cadar
caddr
cdaar
cdadr
cddar
cdddr
caaaar
caaadr
caadar
caaddr
cadaar
cadadr
caddar
cadddr
cdaaar
cdaadr
cdadar
cdaddr
cddaar
cddadr
cdddar
cddddr
4.9.7`` Additional List Functions and Synonyms
empty
cons?
empty?
first
rest
second
third
fourth
fifth
sixth
seventh
eighth
ninth
tenth
last
last-pair
make-list
list-update
list-set
take
drop
split-at
takef
dropf
splitf-at
take-right
drop-right
split-at-right
takef-right
dropf-right
splitf-at-right
list-prefix?
take-common-prefix
drop-common-prefix
split-common-prefix
add-between
append*
flatten
check-duplicates
remove-duplicates
filter-map
count
partition
range
append-map
filter-not
shuffle
combinations
in-combinations
permutations
in-permutations
argmin
argmax
group-by
cartesian-product
remf
remf*
4.9.8`` Immutable Cyclic Data
make-reader-graph
placeholder?
make-placeholder
placeholder-set!
placeholder-get
hash-placeholder?
make-hash-placeholder
make-hasheq-placeholder
make-hasheqv-placeholder

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#### 4.9`` Pairs and Lists

> > > Pairs and Lists in The Racket Guide introduces pairs and lists.

A pair combines exactly two values. The first value is accessed with the car procedure, and the second value is accessed with the cdr procedure. Pairs are not mutable (but see Mutable Pairs and Lists).

A list is recursively defined: it is either the constant null, or it is a pair whose second value is a list.

A list can be used as a single-valued sequence (see Sequences). The elements of the list serve as elements of the sequence. See also in-list.

Cyclic data structures can be created using only immutable pairs via read or make-reader-graph. If starting with a pair and using some number of cdrs returns to the starting pair, then the pair is not a list.

See Reading Pairs and Lists for information on reading pairs and lists and Printing Pairs and Lists for information on printing pairs and lists.

##### 4.9.1`` Pair Constructors and Selectors

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

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

Examples:

> > (pair? 1)
> #f
> > (pair? (cons 1 2))
> #t
> > (pair? (list 1 2))
> #t
> > (pair? '(1 2))
> #t
> > (pair? '())
> #f

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

Returns #t if v is the empty list, #f otherwise.

Examples:

> > (null? 1)
> #f
> > (null? '(1 2))
> #f
> > (null? '())
> #t
> > (null? (cdr (list 1)))
> #t

> procedure (cons a d) → pair?
> a : any/c
> d : any/c

Returns a newly allocated pair whose first element is a and second element is d.

Examples:

> > (cons 1 2)
> '(1 . 2)
> > (cons 1 '())
> '(1)

> procedure (car p) → any/c
> p : pair?

Returns the first element of the pair p.

Examples:

> > (car '(1 2))
> 1
> > (car (cons 2 3))
> 2

> procedure (cdr p) → any/c
> p : pair?

Returns the second element of the pair p.

Examples:

> > (cdr '(1 2))
> '(2)
> > (cdr '(1))
> '()

> value null : null?

The empty list.

Examples:

> > null
> '()
> > '()
> '()
> > (eq? '() null)
> #t

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

Returns #t if v is a list: either the empty list, or a pair whose second element is a list. This procedure effectively takes constant time due to internal caching (so that any necessary traversals of pairs can in principle count as an extra cost of allocating the pairs).

Examples:

> > (list? '(1 2))
> #t
> > (list? (cons 1 (cons 2 '())))
> #t
> > (list? (cons 1 2))
> #f

> procedure (list v ...) → list?
> v : any/c

Returns a newly allocated list containing the vs as its elements.

Examples:

> > (list 1 2 3 4)
> '(1 2 3 4)
> > (list (list 1 2) (list 3 4))
> '((1 2) (3 4))

> procedure (list* v ... tail) → any/c
> v : any/c
> tail : any/c

Like list, but the last argument is used as the tail of the result, instead of the final element. The result is a list only if the last argument is a list.

Examples:

> > (list* 1 2)
> '(1 . 2)
> > (list* 1 2 (list 3 4))
> '(1 2 3 4)

> procedure (build-list n proc) → list?
> n : exact-nonnegative-integer?
> proc : (exact-nonnegative-integer? . -> . any)

Creates a list of n elements by applying proc to the integers from 0 to (sub1 n) in order. If lst is the resulting list, then (list-ref lst i) is the value produced by (proc i).

Examples:

> > (build-list 10 values)
> '(0 1 2 3 4 5 6 7 8 9)
> > (build-list 5 (lambda (x) (* x x)))
> '(0 1 4 9 16)

##### 4.9.2`` List Operations

> procedure (length lst) → exact-nonnegative-integer?
> lst : list?

Returns the number of elements in lst.

Examples:

> > (length (list 1 2 3 4))
> 4
> > (length '())
> 0

> procedure (list-ref lst pos) → any/c
> lst : pair?
> pos : exact-nonnegative-integer?

Returns the element of lst at position pos, where the list’s first element is position 0. If the list has pos or fewer elements, then the exn:fail:contract exception is raised.

The lst argument need not actually be a list; lst must merely start with a chain of at least (add1 pos) pairs.

Examples:

> > (list-ref (list 'a 'b 'c) 0)
> 'a
> > (list-ref (list 'a 'b 'c) 1)
> 'b
> > (list-ref (list 'a 'b 'c) 2)
> 'c
> > (list-ref (cons 1 2) 0)
> 1
> > (list-ref (cons 1 2) 1)
> list-ref: index reaches a non-pair
> index: 1
> in: '(1 . 2)

> procedure (list-tail lst pos) → any/c
> lst : any/c
> pos : exact-nonnegative-integer?

Returns the list after the first pos elements of lst. If the list has fewer than pos elements, then the exn:fail:contract exception is raised.

The lst argument need not actually be a list; lst must merely start with a chain of at least pos pairs.

Examples:

> > (list-tail (list 1 2 3 4) 2)
> '(3 4)
> > (list-tail (cons 1 2) 1)
> 2
> > (list-tail (cons 1 2) 2)
> list-tail: index reaches a non-pair
> index: 2
> in: '(1 . 2)
> > (list-tail 'not-a-pair 0)
> 'not-a-pair

> procedure (append lst ...) → list?
> lst : list?
> (append lst ... v) → any/c
> lst : list?
> v : any/c

When given all list arguments, the result is a list that contains all of the elements of the given lists in order. The last argument is used directly in the tail of the result.

The last argument need not be a list, in which case the result is an “improper list.”

Examples:

> > (append (list 1 2) (list 3 4))
> '(1 2 3 4)
> > (append (list 1 2) (list 3 4) (list 5 6) (list 7 8))
> '(1 2 3 4 5 6 7 8)

> procedure (reverse lst) → list?
> lst : list?

Returns a list that has the same elements as lst, but in reverse order.

Example:

> > (reverse (list 1 2 3 4))
> '(4 3 2 1)

##### 4.9.3`` List Iteration

> procedure (map proc lst ...+) → list?
> proc : procedure?
> lst : list?

Applies proc to the elements of the lsts from the first elements to the last. The proc argument must accept the same number of arguments as the number of supplied lsts, and all lsts must have the same number of elements. The result is a list containing each result of proc in order.

Examples:

> > (map (lambda (number)
> (+ 1 number))
> '(1 2 3 4))
> '(2 3 4 5)
> > (map (lambda (number1 number2)
> (+ number1 number2))
> '(1 2 3 4)
> '(10 100 1000 10000))
> '(11 102 1003 10004)

> procedure (andmap proc lst ...+) → any
> proc : procedure?
> lst : list?

Similar to map in the sense that proc is applied to each element of lst, but

> > > The andmap function is actually closer to foldl than map, since andmap doesn’t produce a list. Still, (andmap f (list x y z)) is equivalent to (and (f x) (f y) (f z)) in the same way that (map f (list x y z)) is equivalent to (list (f x) (f y) (f z)).

- the result is #f if any application of proc produces #f, in which case proc is not applied to later elements of the lsts; and

- the result is that of proc applied to the last elements of the lsts; more specifically, the application of proc to the last elements in the lsts is in tail position with respect to the andmap call.

If the lsts are empty, then #t is returned.

Examples:

> > (andmap positive? '(1 2 3))
> #t
> > (andmap positive? '(1 2 a))
> positive?: contract violation
> expected: real?
> given: 'a
> > (andmap positive? '(1 -2 a))
> #f
> > (andmap + '(1 2 3) '(4 5 6))
> 9

> procedure (ormap proc lst ...+) → any
> proc : procedure?
> lst : list?

Similar to map in the sense that proc is applied to each element of lst, but

> > > To continue the andmap note above, (ormap f (list x y z)) is equivalent to (or (f x) (f y) (f z)).

- the result is #f if every application of proc produces #f; and

- the result is that of the first application of proc producing a value other than #f, in which case proc is not applied to later elements of the lsts; the application of proc to the last elements of the lsts is in tail position with respect to the ormap call.

If the lsts are empty, then #f is returned.

Examples:

> > (ormap eq? '(a b c) '(a b c))
> #t
> > (ormap positive? '(1 2 a))
> #t
> > (ormap + '(1 2 3) '(4 5 6))
> 5

> procedure (for-each proc lst ...+) → void?
> proc : procedure?
> lst : list?

Similar to map, but proc is called only for its effect, and its result (which can be any number of values) is ignored.

Example:

> > (for-each (lambda (arg)
> (printf "Got ~a\n" arg)
> 23)
> '(1 2 3 4))
> Got 1
> Got 2
> Got 3
> Got 4

> procedure (foldl proc init lst ...+) → any/c
> proc : procedure?
> init : any/c
> lst : list?

Like map, foldl applies a procedure to the elements of one or more lists. Whereas map combines the return values into a list, foldl combines the return values in an arbitrary way that is determined by proc.

If foldl is called with n lists, then proc must take n+1 arguments. The extra argument is the combined return values so far. The proc is initially invoked with the first item of each list, and the final argument is init. In subsequent invocations of proc, the last argument is the return value from the previous invocation of proc. The input lsts are traversed from left to right, and the result of the whole foldl application is the result of the last application of proc. If the lsts are empty, the result is init.

Unlike foldr, foldl processes the lsts in constant space (plus the space for each call to proc).

Examples:

> > (foldl cons '() '(1 2 3 4))
> '(4 3 2 1)
> > (foldl + 0 '(1 2 3 4))
> 10
> > (foldl (lambda (a b result)
> (* result (- a b)))
> 1
> '(1 2 3)
> '(4 5 6))
> -27

> procedure (foldr proc init lst ...+) → any/c
> proc : procedure?
> init : any/c
> lst : list?

Like foldl, but the lists are traversed from right to left. Unlike foldl, foldr processes the lsts in space proportional to the length of lsts (plus the space for each call to proc).

Examples:

> > (foldr cons '() '(1 2 3 4))
> '(1 2 3 4)
> > (foldr (lambda (v l) (cons (add1 v) l)) '() '(1 2 3 4))
> '(2 3 4 5)

##### 4.9.4`` List Filtering

> procedure (filter pred lst) → list?
> pred : procedure?
> lst : list?

Returns a list with the elements of lst for which pred produces a true value. The pred procedure is applied to each element from first to last.

Example:

> > (filter positive? '(1 -2 3 4 -5))
> '(1 3 4)

> procedure (remove v lst [proc]) → list?
> v : any/c
> lst : list?
> proc : procedure? = equal?

Returns a list that is like lst, omitting the first element of lst that is equal to v using the comparison procedure proc (which must accept two arguments).

Examples:

> > (remove 2 (list 1 2 3 2 4))
> '(1 3 2 4)
> > (remove 2 (list 1 2 3 2 4) =)
> '(1 3 2 4)
> > (remove '(2) (list '(1) '(2) '(3)))
> '((1) (3))
> > (remove "2" (list "1" "2" "3"))
> '("1" "3")
> > (remove #\c (list #\a #\b #\c))
> '(#\a #\b)

> procedure (remq v lst) → list?
> v : any/c
> lst : list?

Returns (remove v lst eq?).

Examples:

> > (remq 2 (list 1 2 3 4 5))
> '(1 3 4 5)
> > (remq '(2) (list '(1) '(2) '(3)))
> '((1) (2) (3))
> > (remq "2" (list "1" "2" "3"))
> '("1" "3")
> > (remq #\c (list #\a #\b #\c))
> '(#\a #\b)

> procedure (remv v lst) → list?
> v : any/c
> lst : list?

Returns (remove v lst eqv?).

Examples:

> > (remv 2 (list 1 2 3 4 5))
> '(1 3 4 5)
> > (remv '(2) (list '(1) '(2) '(3)))
> '((1) (2) (3))
> > (remv "2" (list "1" "2" "3"))
> '("1" "3")
> > (remv #\c (list #\a #\b #\c))
> '(#\a #\b)

> procedure (remove* v-lst lst [proc]) → list?
> v-lst : list?
> lst : list?
> proc : procedure? = equal?

Like remove, but removes from lst every instance of every element of v-lst.

Example:

> > (remove* (list 1 2) (list 1 2 3 2 4 5 2))
> '(3 4 5)

> procedure (remq* v-lst lst) → list?
> v-lst : list?
> lst : list?

Returns (remove* v-lst lst eq?).

Example:

> > (remq* (list 1 2) (list 1 2 3 2 4 5 2))
> '(3 4 5)

> procedure (remv* v-lst lst) → list?
> v-lst : list?
> lst : list?

Returns (remove* v-lst lst eqv?).

Example:

> > (remv* (list 1 2) (list 1 2 3 2 4 5 2))
> '(3 4 5)

> > procedure
> > (sort |  | lst |  |  |  |
> > |  | less-than? |  |  |  |
> > | [ | #:key extract-key |  |  |  |
> > |  | #:cache-keys? cache-keys?]) |  | → |  | list?
> lst : list?
> less-than? : (any/c any/c . -> . any/c)
> extract-key : (any/c . -> . any/c) = (lambda (x) x)
> cache-keys? : boolean? = #f

Returns a list sorted according to the less-than? procedure, which takes two elements of lst and returns a true value if the first is less (i.e., should be sorted earlier) than the second.

The sort is stable; if two elements of lst are “equal” (i.e., proc does not return a true value when given the pair in either order), then the elements preserve their relative order from lst in the output list. To preserve this guarantee, use sort with a strict comparison functions (e.g., < or string<?; not <= or string<=?).

> > > Because of the peculiar fact that the IEEE-754 number system specifies that +nan.0 is neither greater nor less than nor equal to any other number, sorting lists containing this value may produce a surprising result.

The #:key argument extract-key is used to extract a key value for comparison from each list element. That is, the full comparison procedure is essentially

> (lambda (x y)
> (less-than? (extract-key x) (extract-key y)))

By default, extract-key is applied to two list elements for every comparison, but if cache-keys? is true, then the extract-key function is used exactly once for each list item. Supply a true value for cache-keys? when extract-key is an expensive operation; for example, if file-or-directory-modify-seconds is used to extract a timestamp for every file in a list, then cache-keys? should be #t to minimize file-system calls, but if extract-key is car, then cache-keys? should be #f. As another example, providing extract-key as (lambda (x) (random)) and #t for cache-keys? effectively shuffles the list.

Examples:

> > (sort '(1 3 4 2) <)
> '(1 2 3 4)
> > (sort '("aardvark" "dingo" "cow" "bear") string<?)
> '("aardvark" "bear" "cow" "dingo")
> > (sort '(("aardvark") ("dingo") ("cow") ("bear"))
> #:key car string<?)
> '(("aardvark") ("bear") ("cow") ("dingo"))

##### 4.9.5`` List Searching

> procedure (member v lst [is-equal?]) → (or/c list? #f)
> v : any/c
> lst : list?
> is-equal? : (any/c any/c -> any/c) = equal?

Locates the first element of lst that is equal? to v. If such an element exists, the tail of lst starting with that element is returned. Otherwise, the result is #f.

Examples:

> > (member 2 (list 1 2 3 4))
> '(2 3 4)
> > (member 9 (list 1 2 3 4))
> #f
> > (member #'x (list #'x #'y) free-identifier=?)
> '(#<syntax:499:0 x> #<syntax:499:0 y>)
> > (member #'a (list #'x #'y) free-identifier=?)
> #f

> procedure (memv v lst) → (or/c list? #f)
> v : any/c
> lst : list?

Like member, but finds an element using eqv?.

Examples:

> > (memv 2 (list 1 2 3 4))
> '(2 3 4)
> > (memv 9 (list 1 2 3 4))
> #f

> procedure (memq v lst) → (or/c list? #f)
> v : any/c
> lst : list?

Like member, but finds an element using eq?.

Examples:

> > (memq 2 (list 1 2 3 4))
> '(2 3 4)
> > (memq 9 (list 1 2 3 4))
> #f

> procedure (memf proc lst) → (or/c list? #f)
> proc : procedure?
> lst : list?

Like member, but finds an element using the predicate proc; an element is found when proc applied to the element returns a true value.

Example:

> > (memf (lambda (arg)
> (> arg 9))
> '(7 8 9 10 11))
> '(10 11)

> procedure (findf proc lst) → any/c
> proc : procedure?
> lst : list?

Like memf, but returns the element or #f instead of a tail of lst or #f.

Example:

> > (findf (lambda (arg)
> (> arg 9))
> '(7 8 9 10 11))
> 10

> procedure (assoc v lst [is-equal?]) → (or/c pair? #f)
> v : any/c
> lst : (listof pair?)
> is-equal? : (any/c any/c -> any/c) = equal?

Locates the first element of lst whose car is equal to v according to is-equal?. If such an element exists, the pair (i.e., an element of lst) is returned. Otherwise, the result is #f.

Examples:

> > (assoc 3 (list (list 1 2) (list 3 4) (list 5 6)))
> '(3 4)
> > (assoc 9 (list (list 1 2) (list 3 4) (list 5 6)))
> #f
> > (assoc 3.5
> (list (list 1 2) (list 3 4) (list 5 6))
> (lambda (a b) (< (abs (- a b)) 1)))
> '(3 4)

> procedure (assv v lst) → (or/c pair? #f)
> v : any/c
> lst : (listof pair?)

Like assoc, but finds an element using eqv?.

Example:

> > (assv 3 (list (list 1 2) (list 3 4) (list 5 6)))
> '(3 4)

> procedure (assq v lst) → (or/c pair? #f)
> v : any/c
> lst : (listof pair?)

Like assoc, but finds an element using eq?.

Example:

> > (assq 3 (list (list 1 2) (list 3 4) (list 5 6)))
> '(3 4)

> procedure (assf proc lst) → (or/c list? #f)
> proc : procedure?
> lst : list?

Like assoc, but finds an element using the predicate proc; an element is found when proc applied to the car of an lst element returns a true value.

Example:

> > (assf (lambda (arg)
> (> arg 2))
> (list (list 1 2) (list 3 4) (list 5 6)))
> '(3 4)

##### 4.9.6`` Pair Accessor Shorthands

> procedure (caar v) → any/c
> v : (cons/c pair? any/c)

Returns (car (car v)).

Example:

> > (caar '((1 2) 3 4))
> 1

> procedure (cadr v) → any/c
> v : (cons/c any/c pair?)

Returns (car (cdr v)).

Example:

> > (cadr '((1 2) 3 4))
> 3

> procedure (cdar v) → any/c
> v : (cons/c pair? any/c)

Returns (cdr (car v)).

Example:

> > (cdar '((7 6 5 4 3 2 1) 8 9))
> '(6 5 4 3 2 1)

> procedure (cddr v) → any/c
> v : (cons/c any/c pair?)

Returns (cdr (cdr v)).

Example:

> > (cddr '(2 1))
> '()

> procedure (caaar v) → any/c
> v : (cons/c (cons/c pair? any/c) any/c)

Returns (car (car (car v))).

Example:

> > (caaar '(((6 5 4 3 2 1) 7) 8 9))
> 6

> procedure (caadr v) → any/c
> v : (cons/c any/c (cons/c pair? any/c))

Returns (car (car (cdr v))).

Example:

> > (caadr '(9 (7 6 5 4 3 2 1) 8))
> 7

> procedure (cadar v) → any/c
> v : (cons/c (cons/c any/c pair?) any/c)

Returns (car (cdr (car v))).

Example:

> > (cadar '((7 6 5 4 3 2 1) 8 9))
> 6

> procedure (caddr v) → any/c
> v : (cons/c any/c (cons/c any/c pair?))

Returns (car (cdr (cdr v))).

Example:

> > (caddr '(3 2 1))
> 1

> procedure (cdaar v) → any/c
> v : (cons/c (cons/c pair? any/c) any/c)

Returns (cdr (car (car v))).

Example:

> > (cdaar '(((6 5 4 3 2 1) 7) 8 9))
> '(5 4 3 2 1)

> procedure (cdadr v) → any/c
> v : (cons/c any/c (cons/c pair? any/c))

Returns (cdr (car (cdr v))).

Example:

> > (cdadr '(9 (7 6 5 4 3 2 1) 8))
> '(6 5 4 3 2 1)

> procedure (cddar v) → any/c
> v : (cons/c (cons/c any/c pair?) any/c)

Returns (cdr (cdr (car v))).

Example:

> > (cddar '((7 6 5 4 3 2 1) 8 9))
> '(5 4 3 2 1)

> procedure (cdddr v) → any/c
> v : (cons/c any/c (cons/c any/c pair?))

Returns (cdr (cdr (cdr v))).

Example:

> > (cdddr '(3 2 1))
> '()

> procedure (caaaar v) → any/c
> v : (cons/c (cons/c (cons/c pair? any/c) any/c) any/c)

Returns (car (car (car (car v)))).

Example:

> > (caaaar '((((5 4 3 2 1) 6) 7) 8 9))
> 5

> procedure (caaadr v) → any/c
> v : (cons/c any/c (cons/c (cons/c pair? any/c) any/c))

Returns (car (car (car (cdr v)))).

Example:

> > (caaadr '(9 ((6 5 4 3 2 1) 7) 8))
> 6

> procedure (caadar v) → any/c
> v : (cons/c (cons/c any/c (cons/c pair? any/c)) any/c)

Returns (car (car (cdr (car v)))).

Example:

> > (caadar '((7 (5 4 3 2 1) 6) 8 9))
> 5

> procedure (caaddr v) → any/c
> v : (cons/c any/c (cons/c any/c (cons/c pair? any/c)))

Returns (car (car (cdr (cdr v)))).

Example:

> > (caaddr '(9 8 (6 5 4 3 2 1) 7))
> 6

> procedure (cadaar v) → any/c
> v : (cons/c (cons/c (cons/c any/c pair?) any/c) any/c)

Returns (car (cdr (car (car v)))).

Example:

> > (cadaar '(((6 5 4 3 2 1) 7) 8 9))
> 5

> procedure (cadadr v) → any/c
> v : (cons/c any/c (cons/c (cons/c any/c pair?) any/c))

Returns (car (cdr (car (cdr v)))).

Example:

> > (cadadr '(9 (7 6 5 4 3 2 1) 8))
> 6

> procedure (caddar v) → any/c
> v : (cons/c (cons/c any/c (cons/c any/c pair?)) any/c)

Returns (car (cdr (cdr (car v)))).

Example:

> > (caddar '((7 6 5 4 3 2 1) 8 9))
> 5

> procedure (cadddr v) → any/c
> v : (cons/c any/c (cons/c any/c (cons/c any/c pair?)))

Returns (car (cdr (cdr (cdr v)))).

Example:

> > (cadddr '(4 3 2 1))
> 1

> procedure (cdaaar v) → any/c
> v : (cons/c (cons/c (cons/c pair? any/c) any/c) any/c)

Returns (cdr (car (car (car v)))).

Example:

> > (cdaaar '((((5 4 3 2 1) 6) 7) 8 9))
> '(4 3 2 1)

> procedure (cdaadr v) → any/c
> v : (cons/c any/c (cons/c (cons/c pair? any/c) any/c))

Returns (cdr (car (car (cdr v)))).

Example:

> > (cdaadr '(9 ((6 5 4 3 2 1) 7) 8))
> '(5 4 3 2 1)

> procedure (cdadar v) → any/c
> v : (cons/c (cons/c any/c (cons/c pair? any/c)) any/c)

Returns (cdr (car (cdr (car v)))).

Example:

> > (cdadar '((7 (5 4 3 2 1) 6) 8 9))
> '(4 3 2 1)

> procedure (cdaddr v) → any/c
> v : (cons/c any/c (cons/c any/c (cons/c pair? any/c)))

Returns (cdr (car (cdr (cdr v)))).

Example:

> > (cdaddr '(9 8 (6 5 4 3 2 1) 7))
> '(5 4 3 2 1)

> procedure (cddaar v) → any/c
> v : (cons/c (cons/c (cons/c any/c pair?) any/c) any/c)

Returns (cdr (cdr (car (car v)))).

Example:

> > (cddaar '(((6 5 4 3 2 1) 7) 8 9))
> '(4 3 2 1)

> procedure (cddadr v) → any/c
> v : (cons/c any/c (cons/c (cons/c any/c pair?) any/c))

Returns (cdr (cdr (car (cdr v)))).

Example:

> > (cddadr '(9 (7 6 5 4 3 2 1) 8))
> '(5 4 3 2 1)

> procedure (cdddar v) → any/c
> v : (cons/c (cons/c any/c (cons/c any/c pair?)) any/c)

Returns (cdr (cdr (cdr (car v)))).

Example:

> > (cdddar '((7 6 5 4 3 2 1) 8 9))
> '(4 3 2 1)

> procedure (cddddr v) → any/c
> v : (cons/c any/c (cons/c any/c (cons/c any/c pair?)))

Returns (cdr (cdr (cdr (cdr v)))).

Example:

> > (cddddr '(4 3 2 1))
> '()

##### 4.9.7`` Additional List Functions and Synonyms

(require racket/list) | package: base

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

> value empty : null?

The empty list.

Examples:

> > empty
> '()
> > (eq? empty null)
> #t

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

The same as (pair? v).

Example:

> > (cons? '(1 2))
> #t

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

The same as (null? v).

Examples:

> > (empty? '(1 2))
> #f
> > (empty? '())
> #t

> procedure (first lst) → any/c
> lst : list?

The same as (car lst), but only for lists (that are not empty).

Example:

> > (first '(1 2 3 4 5 6 7 8 9 10))
> 1

> procedure (rest lst) → list?
> lst : list?

The same as (cdr lst), but only for lists (that are not empty).

Example:

> > (rest '(1 2 3 4 5 6 7 8 9 10))
> '(2 3 4 5 6 7 8 9 10)

> procedure (second lst) → any
> lst : list?

Returns the second element of the list.

Example:

> > (second '(1 2 3 4 5 6 7 8 9 10))
> 2

> procedure (third lst) → any
> lst : list?

Returns the third element of the list.

Example:

> > (third '(1 2 3 4 5 6 7 8 9 10))
> 3

> procedure (fourth lst) → any
> lst : list?

Returns the fourth element of the list.

Example:

> > (fourth '(1 2 3 4 5 6 7 8 9 10))
> 4

> procedure (fifth lst) → any
> lst : list?

Returns the fifth element of the list.

Example:

> > (fifth '(1 2 3 4 5 6 7 8 9 10))
> 5

> procedure (sixth lst) → any
> lst : list?

Returns the sixth element of the list.

Example:

> > (sixth '(1 2 3 4 5 6 7 8 9 10))
> 6

> procedure (seventh lst) → any
> lst : list?

Returns the seventh element of the list.

Example:

> > (seventh '(1 2 3 4 5 6 7 8 9 10))
> 7

> procedure (eighth lst) → any
> lst : list?

Returns the eighth element of the list.

Example:

> > (eighth '(1 2 3 4 5 6 7 8 9 10))
> 8

> procedure (ninth lst) → any
> lst : list?

Returns the ninth element of the list.

Example:

> > (ninth '(1 2 3 4 5 6 7 8 9 10))
> 9

> procedure (tenth lst) → any
> lst : list?

Returns the tenth element of the list.

Example:

> > (tenth '(1 2 3 4 5 6 7 8 9 10))
> 10

> procedure (last lst) → any
> lst : list?

Returns the last element of the list.

Example:

> > (last '(1 2 3 4 5 6 7 8 9 10))
> 10

> procedure (last-pair p) → pair?
> p : pair?

Returns the last pair of a (possibly improper) list.

Example:

> > (last-pair '(1 2 3 4))
> '(4)

> procedure (make-list k v) → list?
> k : exact-nonnegative-integer?
> v : any/c

Returns a newly constructed list of length k, holding v in all positions.

Example:

> > (make-list 7 'foo)
> '(foo foo foo foo foo foo foo)

> procedure (list-update lst pos updater) → list?
> lst : list?
> pos : (and/c (>=/c 0) (</c (length lst)))
> updater : (-> any/c any/c)

Returns a list that is the same as lst except at the specified index. The element at the specified index is (updater (list-ref lst pos)).

Example:

> > (list-update '(zero one two) 1 symbol->string)
> '(zero "one" two)

Added in version 6.3 of package base.

> procedure (list-set lst pos value) → list?
> lst : list?
> pos : (and/c (>=/c 0) (</c (length lst)))
> value : any/c

Returns a list that is the same as lst except at the specified index. The element at the specified index is value.

Example:

> > (list-set '(zero one two) 2 "two")
> '(zero one "two")

Added in version 6.3 of package base.

> procedure (take lst pos) → list?
> lst : any/c
> pos : exact-nonnegative-integer?

Returns a fresh list whose elements are the first pos elements of lst. If lst has fewer than pos elements, the exn:fail:contract exception is raised.

The lst argument need not actually be a list; lst must merely start with a chain of at least pos pairs.

Examples:

> > (take '(1 2 3 4) 2)
> '(1 2)
> > (take 'non-list 0)
> '()

> procedure (drop lst pos) → any/c
> lst : any/c
> pos : exact-nonnegative-integer?

Just like list-tail.

> > procedure
> > (split-at lst pos) |  | → |  |
> > list? |  | any/c
> lst : any/c
> pos : exact-nonnegative-integer?

Returns the same result as

> (values (take lst pos) (drop lst pos))

except that it can be faster.

> procedure (takef lst pred) → list?
> lst : any/c
> pred : procedure?

Returns a fresh list whose elements are taken successively from lst as long as they satisfy pred. The returned list includes up to, but not including, the first element in lst for which pred returns #f.

The lst argument need not actually be a list; the chain of pairs in lst will be traversed until a non-pair is encountered.

Examples:

> > (takef '(2 4 5 8) even?)
> '(2 4)
> > (takef '(2 4 6 8) odd?)
> '()
> > (takef '(2 4 . 6) even?)
> '(2 4)

> procedure (dropf lst pred) → any/c
> lst : any/c
> pred : procedure?

Drops elements from the front of lst as long as they satisfy pred.

Examples:

> > (dropf '(2 4 5 8) even?)
> '(5 8)
> > (dropf '(2 4 6 8) odd?)
> '(2 4 6 8)

> > procedure
> > (splitf-at lst pred) |  | → |  |
> > list? |  | any/c
> lst : any/c
> pred : procedure?

Returns the same result as

> (values (takef lst pred) (dropf lst pred))

except that it can be faster.

> procedure (take-right lst pos) → any/c
> lst : any/c
> pos : exact-nonnegative-integer?

Returns the list’s pos-length tail. If lst has fewer than pos elements, then the exn:fail:contract exception is raised.

The lst argument need not actually be a list; lst must merely end with a chain of at least pos pairs.

Examples:

> > (take-right '(1 2 3 4) 2)
> '(3 4)
> > (take-right 'non-list 0)
> 'non-list

> procedure (drop-right lst pos) → list?
> lst : any/c
> pos : exact-nonnegative-integer?

Returns a fresh list whose elements are the prefix of lst, dropping its pos-length tail. If lst has fewer than pos elements, then the exn:fail:contract exception is raised.

The lst argument need not actually be a list; lst must merely end with a chain of at least pos pairs.

Examples:

> > (drop-right '(1 2 3 4) 2)
> '(1 2)
> > (drop-right 'non-list 0)
> '()

> > procedure
> > (split-at-right lst pos) |  | → |  |
> > list? |  | any/c
> lst : any/c
> pos : exact-nonnegative-integer?

Returns the same result as

> (values (drop-right lst pos) (take-right lst pos))

except that it can be faster.

Examples:

> > (split-at-right '(1 2 3 4 5 6) 3)
> '(1 2 3)
> '(4 5 6)
> > (split-at-right '(1 2 3 4 5 6) 4)
> '(1 2)
> '(3 4 5 6)

> procedure (takef-right lst pred) → list?
> lst : any/c
> pred : procedure?
> procedure (dropf-right lst pred) → any/c
> lst : any/c
> pred : procedure?
> > procedure
> > (splitf-at-right lst pred) |  | → |  |
> > list? |  | any/c
> lst : any/c
> pred : procedure?

Like takef, dropf, and splitf-at, but combined with the from-right functionality of take-right, drop-right, and split-at-right.

> procedure (list-prefix? l r [same?]) → boolean?
> l : list?
> r : list?
> same? : (any/c any/c . -> . any/c) = equal?

True if l is a prefix of r.

Example:

> > (list-prefix? '(1 2) '(1 2 3 4 5))
> #t

Added in version 6.3 of package base.

> procedure (take-common-prefix l r [same?]) → list?
> l : list?
> r : list?
> same? : (any/c any/c . -> . any/c) = equal?

Returns the longest common prefix of l and r.

Example:

> > (take-common-prefix '(a b c d) '(a b x y z))
> '(a b)

Added in version 6.3 of package base.

> > procedure
> > (drop-common-prefix l r [same?]) |  | → |  |
> > list? |  | list?
> l : list?
> r : list?
> same? : (any/c any/c . -> . any/c) = equal?

Returns the tails of l and r with the common prefix removed.

Example:

> > (drop-common-prefix '(a b c d) '(a b x y z))
> '(c d)
> '(x y z)

Added in version 6.3 of package base.

> > procedure
> > (split-common-prefix l r [same?]) |  | → |  |
> > list? |  | list? |  | list?
> l : list?
> r : list?
> same? : (any/c any/c . -> . any/c) = equal?

Returns the longest common prefix together with the tails of l and r with the common prefix removed.

Example:

> > (split-common-prefix '(a b c d) '(a b x y z))
> '(a b)
> '(c d)
> '(x y z)

Added in version 6.3 of package base.

> > procedure
> > (add-between |  | lst |  |  |  |
> > |  | v |  |  |  |
> > | [ | #:before-first before-first |  |  |  |
> > |  | #:before-last before-last |  |  |  |
> > |  | #:after-last after-last |  |  |  |
> > |  | #:splice? splice?]) |  | → |  | list?
> lst : list?
> v : any/c
> before-first : list? = '()
> before-last : any/c = v
> after-last : list? = '()
> splice? : any/c = #f

Returns a list with the same elements as lst, but with v between each pair of elements in lst; the last pair of elements will have before-last between them, instead of v (but before-last defaults to v).

If splice? is true, then v and before-last should be lists, and the list elements are spliced into the result. In addition, when splice? is true, before-first and after-last are inserted before the first element and after the last element respectively.

Examples:

> > (add-between '(x y z) 'and)
> '(x and y and z)
> > (add-between '(x) 'and)
> '(x)
> > (add-between '("a" "b" "c" "d") "," #:before-last "and")
> '("a" "," "b" "," "c" "and" "d")
> > (add-between '(x y z) '(-) #:before-last '(- -)
> #:before-first '(begin) #:after-last '(end LF)
> #:splice? #t)
> '(begin x - y - - z end LF)

> procedure (append* lst ... lsts) → list?
> lst : list?
> lsts : (listof list?)
> (append* lst ... lsts) → any/c
> lst : list?
> lsts : list?

Like append, but the last argument is used as a list of arguments for append, so (append* lst ... lsts) is the same as (apply append lst ... lsts). In other words, the relationship between append and append* is similar to the one between list and list*.

Examples:

> > (append* '(a) '(b) '((c) (d)))
> '(a b c d)
> > (cdr (append* (map (lambda (x) (list ", " x))
> '("Alpha" "Beta" "Gamma"))))
> '("Alpha" ", " "Beta" ", " "Gamma")

> procedure (flatten v) → list?
> v : any/c

Flattens an arbitrary S-expression structure of pairs into a list. More precisely, v is treated as a binary tree where pairs are interior nodes, and the resulting list contains all of the non-null leaves of the tree in the same order as an inorder traversal.

Examples:

> > (flatten '((a) b (c (d) . e) ()))
> '(a b c d e)
> > (flatten 'a)
> '(a)

> > procedure
> > (check-duplicates |  | lst |  |  |  |
> > | [ | same? |  |  |  |
> > |  | #:key extract-key]) |  | → |  | (or/c any/c #f)
> lst : list?
> same? : (any/c any/c . -> . any/c) = equal?
> extract-key : (-> any/c any/c) = (lambda (x) x)

Returns the first duplicate item in lst. More precisely, it returns the first x such that there was a previous y where (same? (extract-key x) (extract-key y)).

The same? argument should be an equivalence predicate such as equal? or eqv? or a dictionary. The procedures equal?, eqv?, and eq? automatically use a dictionary for speed.

Examples:

> > (check-duplicates '(1 2 3 4))
> #f
> > (check-duplicates '(1 2 3 2 1))
> 2
> > (check-duplicates '((a 1) (b 2) (a 3)) #:key car)
> '(a 3)
> > (check-duplicates '(1 2 3 4 5 6)
> (lambda (x y) (equal? (modulo x 3) (modulo y 3))))
> 4

Added in version 6.3 of package base.

> > procedure
> > (remove-duplicates |  | lst |  |  |  |
> > | [ | same? |  |  |  |
> > |  | #:key extract-key]) |  | → |  | list?
> lst : list?
> same? : (any/c any/c . -> . any/c) = equal?
> extract-key : (any/c . -> . any/c) = (lambda (x) x)

Returns a list that has all items in lst, but without duplicate items, where same? determines whether two elements of the list are equivalent. The resulting list is in the same order as lst, and for any item that occurs multiple times, the first one is kept.

The #:key argument extract-key is used to extract a key value from each list element, so two items are considered equal if (same? (extract-key x) (extract-key y)) is true.

Examples:

> > (remove-duplicates '(a b b a))
> '(a b)
> > (remove-duplicates '(1 2 1.0 0))
> '(1 2 1.0 0)
> > (remove-duplicates '(1 2 1.0 0) =)
> '(1 2 0)

> procedure (filter-map proc lst ...+) → list?
> proc : procedure?
> lst : list?

Returns (filter (lambda (x) x) (map proc lst ...)), but without building the intermediate list.

Example:

> > (filter-map (lambda (x) (and (positive? x) x)) '(1 2 3 -2 8))
> '(1 2 3 8)

> procedure (count proc lst ...+) → exact-nonnegative-integer?
> proc : procedure?
> lst : list?

Returns (length (filter-map proc lst ...)), but without building the intermediate list.

Example:

> > (count positive? '(1 -1 2 3 -2 5))
> 4

> > procedure
> > (partition pred lst) |  | → |  |
> > list? |  | list?
> pred : procedure?
> lst : list?

Similar to filter, except that two values are returned: the items for which pred returns a true value, and the items for which pred returns #f.

The result is the same as

> (values (filter pred lst) (filter (negate pred) lst))

but pred is applied to each item in lst only once.

Example:

> > (partition even? '(1 2 3 4 5 6))
> '(2 4 6)
> '(1 3 5)

> procedure (range end) → list?
> end : real?
> (range start end [step]) → list?
> start : real?
> end : real?
> step : real? = 1

Similar to in-range, but returns lists.

The resulting list holds numbers starting at start and whose successive elements are computed by adding step to their predecessor until end (excluded) is reached. If no starting point is provided, 0 is used. If no step argument is provided, 1 is used.

Examples:

> > (range 10)
> '(0 1 2 3 4 5 6 7 8 9)
> > (range 10 20)
> '(10 11 12 13 14 15 16 17 18 19)
> > (range 20 40 2)
> '(20 22 24 26 28 30 32 34 36 38)
> > (range 20 10 -1)
> '(20 19 18 17 16 15 14 13 12 11)
> > (range 10 15 1.5)
> '(10 11.5 13.0 14.5)

> procedure (append-map proc lst ...+) → list?
> proc : procedure?
> lst : list?

Returns (append* (map proc lst ...)).

Example:

> > (append-map vector->list '(#(1) #(2 3) #(4)))
> '(1 2 3 4)

> procedure (filter-not pred lst) → list?
> pred : (any/c . -> . any/c)
> lst : list?

Like filter, but the meaning of the pred predicate is reversed: the result is a list of all items for which pred returns #f.

Example:

> > (filter-not even? '(1 2 3 4 5 6))
> '(1 3 5)

> procedure (shuffle lst) → list?
> lst : list?

Returns a list with all elements from lst, randomly shuffled.

Examples:

> > (shuffle '(1 2 3 4 5 6))
> '(6 4 2 5 1 3)
> > (shuffle '(1 2 3 4 5 6))
> '(6 2 5 3 1 4)
> > (shuffle '(1 2 3 4 5 6))
> '(3 5 1 4 6 2)

> procedure (combinations lst) → list?
> lst : list?
> (combinations lst size) → list?
> lst : list?
> size : exact-nonnegative-integer?

> > > Wikipedia combinations

Return a list of all combinations of elements in the input list (aka the powerset of lst). If size is given, limit results to combinations of size elements.

Examples:

> > (combinations '(1 2 3))
> '(() (1) (2) (1 2) (3) (1 3) (2 3) (1 2 3))
> > (combinations '(1 2 3) 2)
> '((1 2) (1 3) (2 3))

> procedure (in-combinations lst) → sequence?
> lst : list?
> (in-combinations lst size) → sequence?
> lst : list?
> size : exact-nonnegative-integer?

Returns a sequence of all combinations of elements in the input list, or all combinations of length size if size is given. Builds combinations one-by-one instead of all at once.

Examples:

> > (time (begin (combinations (range 15)) (void)))
> cpu time: 25 real time: 22 gc time: 0
> > (time (begin (in-combinations (range 15)) (void)))
> cpu time: 0 real time: 0 gc time: 0

> procedure (permutations lst) → list?
> lst : list?

Returns a list of all permutations of the input list. Note that this function works without inspecting the elements, and therefore it ignores repeated elements (which will result in repeated permutations).

Examples:

> > (permutations '(1 2 3))
> '((1 2 3) (2 1 3) (1 3 2) (3 1 2) (2 3 1) (3 2 1))
> > (permutations '(x x))
> '((x x) (x x))

> procedure (in-permutations lst) → sequence?
> lst : list?

Returns a sequence of all permutations of the input list. It is equivalent to (in-list (permutations l)) but much faster since it builds the permutations one-by-one on each iteration

> procedure (argmin proc lst) → any/c
> proc : (-> any/c real?)
> lst : (and/c pair? list?)

Returns the first element in the list lst that minimizes the result of proc. Signals an error on an empty list.

Examples:

> > (argmin car '((3 pears) (1 banana) (2 apples)))
> '(1 banana)
> > (argmin car '((1 banana) (1 orange)))
> '(1 banana)

> procedure (argmax proc lst) → any/c
> proc : (-> any/c real?)
> lst : (and/c pair? list?)

Returns the first element in the list lst that maximizes the result of proc. Signals an error on an empty list.

Examples:

> > (argmax car '((3 pears) (1 banana) (2 apples)))
> '(3 pears)
> > (argmax car '((3 pears) (3 oranges)))
> '(3 pears)

> procedure (group-by key lst [same?]) → (listof list?)
> key : (-> any/c any/c)
> lst : list?
> same? : (any/c any/c . -> . any/c) = equal?

Groups the given list into equivalence classes, with equivalence being determined by same?. Within each equivalence class, group-by preserves the ordering of the original list. Equivalence classes themselves are in order of first appearance in the input.

Example:

> > (group-by (lambda (x) (modulo x 3)) '(1 2 1 2 54 2 5 43 7 2 643 1 2 0))
> '((1 1 43 7 643 1) (2 2 2 5 2 2) (54 0))

Added in version 6.3 of package base.

> procedure (cartesian-product lst ...) → (listof list?)
> lst : list?

Computes the n-ary cartesian product of the given lists.

Examples:

> > (cartesian-product '(1 2 3) '(a b c))
> '((1 a) (1 b) (1 c) (2 a) (2 b) (2 c) (3 a) (3 b) (3 c))
> > (cartesian-product '(4 5 6) '(d e f) '(#t #f))
> '((4 d #t)
> (4 d #f)
> (4 e #t)
> (4 e #f)
> (4 f #t)
> (4 f #f)
> (5 d #t)
> (5 d #f)
> (5 e #t)
> (5 e #f)
> (5 f #t)
> (5 f #f)
> (6 d #t)
> (6 d #f)
> (6 e #t)
> (6 e #f)
> (6 f #t)
> (6 f #f))

Added in version 6.3 of package base.

> procedure (remf pred lst) → list?
> pred : procedure?
> lst : list?

Returns a list that is like lst, omitting the first element of lst for which pred produces a true value.

Example:

> > (remf negative? '(1 -2 3 4 -5))
> '(1 3 4 -5)

Added in version 6.3 of package base.

> procedure (remf* pred lst) → list?
> pred : procedure?
> lst : list?

Like remf, but removes all the elements for which pred produces a true value.

Example:

> > (remf* negative? '(1 -2 3 4 -5))
> '(1 3 4)

Added in version 6.3 of package base.

##### 4.9.8`` Immutable Cyclic Data

> procedure (make-reader-graph v) → any/c
> v : any/c

Returns a value like v, with placeholders created by make-placeholder replaced with the values that they contain, and with placeholders created by make-hash-placeholder with an immutable hash table. No part of v is mutated; instead, parts of v are copied as necessary to construct the resulting graph, where at most one copy is created for any given value.

Since the copied values can be immutable, and since the copy is also immutable, make-reader-graph can create cycles involving only immutable pairs, vectors, boxes, and hash tables.

Only the following kinds of values are copied and traversed to detect placeholders:

- pairs

- vectors, both mutable and immutable

- boxes, both mutable and immutable

- hash tables, both mutable and immutable

- instances of a prefab structure type

- placeholders created by make-placeholder and make-hash-placeholder

Due to these restrictions, make-reader-graph creates exactly the same sort of cyclic values as read.

Example:

> > (let* ([ph (make-placeholder #f)]
> [x (cons 1 ph)])
> (placeholder-set! ph x)
> (make-reader-graph x))
> #0='(1 . #0#)

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

Returns #t if v is a placeholder created by make-placeholder, #f otherwise.

> procedure (make-placeholder v) → placeholder?
> v : any/c

Returns a placeholder for use with placeholder-set! and make-reader-graph. The v argument supplies the initial value for the placeholder.

> procedure (placeholder-set! ph datum) → void?
> ph : placeholder?
> datum : any/c

Changes the value of ph to v.

> procedure (placeholder-get ph) → any/c
> ph : placeholder?

Returns the value of ph.

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

Returns #t if v is a placeholder created by make-hash-placeholder, #f otherwise.

> procedure (make-hash-placeholder assocs) → hash-placeholder?
> assocs : (listof pair?)

Like make-immutable-hash, but produces a table placeholder for use with make-reader-graph.

> procedure (make-hasheq-placeholder assocs) → hash-placeholder?
> assocs : (listof pair?)

Like make-immutable-hasheq, but produces a table placeholder for use with make-reader-graph.

> procedure (make-hasheqv-placeholder assocs) → hash-placeholder?
> assocs : (listof pair?)

Like make-immutable-hasheqv, but produces a table placeholder for use with make-reader-graph.
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