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### 6.7 Lists and recursion

Functional programming languages do not have variables, assignment or iteration. You can achieve the same effects using just lists and recursion.

There are two main sorts of recursion over lists. The first is called mapping: a function is applied to each element of a list, producing a new list in which each element has been transformed.

map fn [a,b,c] == [fn a, fn b, fn c]

The second main sort of recursion is called folding: a list is turned into a single value by joining pairs of elements together with a function and a start value.

foldr fn start [a,b .. c] ==
(fn a (fn b (.. (fn c start))))

(The function is called foldr as it folds the list up right-to-left. There is an analogous function called foldl which folds a list up left-to-right, but because of the way lists work, it is much slower and should be avoided if possible.)

map is defined in the standard list library for you:

/⋆ map fn l: map function fn over list l
⋆/
map fn l
= [], l == []
= fn (hd l) : map fn (tl l)

So, for example, you could use map like this:

map (add 2) [1..5] == [3,4,5,6,7,8]

foldr is defined in the standard list library for you:

/⋆ foldr fn st l: fold up list l,
⋆ right to left with function fn and
⋆ start value st
⋆/
foldr fn st l
= st, l == []
= fn (hd l) (foldr fn st (tl l))

So, for example, you could use foldr like this:

foldr add 0 [1..5] == 15

(Mathematically, foldr is the more basic operation. You can write map in terms of foldr, but you can’t write foldr in terms of map.)

Unconstrained recursion over lists can be very hard to understand, rather like goto in an imperative language. It’s much better to use a combination of map and foldr if you possibly can.

The toolkit _list contains definitions of most of the standard list-processing functions. These are listed in Table 6.3. Check the source for detailed comments.

Name | Description

all l | and all the elements of list l together
any l | or all the elements of list l together
concat l | join a list of lists together
drop n l | drop the first n elements from list l
dropwhile fn l | drop while fn is true
extract n l | extract element n from list l
filter fn l | all elements of l for which fn holds
foldl fn st l | fold list l left-to-right with fn and st
foldl1 fn l | like foldl, but use the first element of the list as the start value
foldr fn st l | fold list l right-to-left with fn and st
foldr1 fn l | like foldr, but use the first element of the list as the start value
index fn l | search list l for index of first element matching predicate fn
init l | remove last element of list l
iterate f x | repeatedly apply f to x
last l | return the last element of list l
len l | find length of list l
limit l | find the first element of list l equal to its predecessor
map fn l | map function fn over list l
map2 fn l1 l2 | map 2-ary function fn over lists l1 and l2
map3 fn l1 l2 l3 | map 3-ary function fn over lists l1, l2 and l3
member l x | true if x is a member of list l
mkset l | remove duplicates from list l
postfix l r | add element r to the end of list l
product l | product of list l
repeat x | make an infinite list of xes
replicate n x | make n copies of x in a list
reverse l | reverse list l
scan fn st l | apply (foldr fn r) to every initial segment of list l
sort l | sort list l into ascending order
sortc fn l | sort list l into order by using a comparison function
sortpl pl l | sort list l by predicate list pl
sortr l | sort list l into descending order
split fn l | break list l into sections separated by predicate fn
splits fn l | break list l into single sections separated by predicate fn
splitpl pl l | break list l up by predicate list pl
split_lines n l | break list l into lines of length n
sum l | sum list l
take n l | take the first n elements from list l
takewhile fn l | take from the front of l while fn holds
zip2 l1 l2 | zip two lists together
zip3 l1 l2 l3 | zip three lists together

Table 6.3: Functions in the standard list-processing toolkit

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