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<h3 class="sectionHead"><span class="titlemark">6.7 </span> <a
id="x39-710006.7"></a>Lists and recursion</h3>
<!--l. 710--><p class="noindent" ><a name="nip_label_sec:lists"></a>
<!--l. 712--><p class="indent" > Functional programming languages do not have variables,
assignment or iteration. You can achieve the same effects
using just lists and recursion.
<!--l. 715--><p class="indent" > There are two main sorts of recursion over lists. The first
is called <span
class="ptmri7t-">mapping</span>: a function is applied to each element of a
list, producing a new list in which each element has been
transformed.
<div class="verbatim" id="verbatim-55">
map fn [a,b,c] == [fn a, fn b, fn c]
</div>
<!--l. 721--><p class="nopar" >
<!--l. 723--><p class="indent" > The second main sort of recursion is called <span
class="ptmri7t-">folding</span>: a list
is turned into a single value by joining pairs of elements
together with a function and a start value.
<div class="verbatim" id="verbatim-56">
foldr fn start [a,b .. c] ==
 <br />  (fn a (fn b (.. (fn c start))))
</div>
<!--l. 730--><p class="nopar" >
<!--l. 732--><p class="noindent" >(The function is called <span
class="phvr7t-x-x-80">foldr </span>as it folds the list up
right-to-left. There is an analogous function called <span
class="phvr7t-x-x-80">foldl</span>
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.)
<!--l. 738--><p class="indent" > <span
class="phvr7t-x-x-80">map </span>is defined in the standard list library for you:
<div class="verbatim" id="verbatim-57">
/⋆ map fn l: map function fn over list l
 <br /> ⋆/
 <br />
 <br />map fn l
 <br />  = [], l == []
 <br />  = fn (hd l) : map fn (tl l)
</div>
<!--l. 747--><p class="nopar" >
<!--l. 749--><p class="noindent" >So, for example, you could use <span
class="phvr7t-x-x-80">map </span>like this:
<div class="verbatim" id="verbatim-58">
map (add 2) [1..5] == [3,4,5,6,7,8]
</div>
<!--l. 754--><p class="nopar" >
<!--l. 756--><p class="indent" > <span
class="phvr7t-x-x-80">foldr </span>is defined in the standard list library for you:
<div class="verbatim" id="verbatim-59">
/⋆ foldr fn st l: fold up list l,
 <br /> ⋆ right to left with function fn and
 <br /> ⋆ start value st
 <br /> ⋆/
 <br />
 <br />foldr fn st l
 <br />  = st, l == []
 <br />  = fn (hd l) (foldr fn st (tl l))
</div>
<!--l. 767--><p class="nopar" >
<!--l. 769--><p class="noindent" >So, for example, you could use <span
class="phvr7t-x-x-80">foldr </span>like this:
<div class="verbatim" id="verbatim-60">
foldr add 0 [1..5] == 15
</div>
<!--l. 774--><p class="nopar" >
<!--l. 776--><p class="noindent" >(Mathematically, <span
class="phvr7t-x-x-80">foldr </span>is the more basic operation. You can
write <span
class="phvr7t-x-x-80">map </span>in terms of <span
class="phvr7t-x-x-80">foldr</span>, but you can’t write <span
class="phvr7t-x-x-80">foldr </span>in terms
of <span
class="phvr7t-x-x-80">map</span>.)
<!--l. 781--><p class="indent" > Unconstrained recursion over lists can be very hard to
understand, rather like <span
class="phvr7t-x-x-80">goto </span>in an imperative language. It’s
much better to use a combination of <span
class="phvr7t-x-x-80">map </span>and <span
class="phvr7t-x-x-80">foldr </span>if you
possibly can.
<!--l. 785--><p class="indent" > The toolkit <span
class="phvr7t-x-x-80">_list </span>contains definitions of most of the
standard list-processing functions. These are listed in
Table <a
href="#x39-710013">6.3<!--tex4ht:ref: tb:list --></a>. Check the source for detailed comments.
<!--l. 790--><p class="indent" > <a
id="x39-710013"></a><hr class="float"><div class="float"
>
<div class="center"
>
<!--l. 790--><p class="noindent" >
<div class="tabular"> <table id="TBL-5" class="tabular"
cellspacing="0" cellpadding="0" rules="groups"
><colgroup id="TBL-5-1g"><col
id="TBL-5-1"></colgroup><colgroup id="TBL-5-2g"><col
id="TBL-5-2"></colgroup><tr
class="hline"><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-5-1-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-1-1"
class="td11">Name </td><td style="white-space:nowrap; text-align:left;" id="TBL-5-1-2"
class="td11">Description </td>
</tr><tr
class="hline"><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-5-2-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-2-1"
class="td11"><span
class="phvr7t-x-x-80">all l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-2-2"
class="td11">and all the elements of list <span
class="phvr7t-x-x-80">l </span>together </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-3-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-3-1"
class="td11"><span
class="phvr7t-x-x-80">any l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-3-2"
class="td11">or all the elements of list <span
class="phvr7t-x-x-80">l </span>together </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-4-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-4-1"
class="td11"><span
class="phvr7t-x-x-80">concat l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-4-2"
class="td11">join a list of lists together </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-5-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-5-1"
class="td11"><span
class="phvr7t-x-x-80">drop n l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-5-2"
class="td11">drop the first <span
class="phvr7t-x-x-80">n </span>elements from list <span
class="phvr7t-x-x-80">l </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-6-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-6-1"
class="td11"><span
class="phvr7t-x-x-80">dropwhile fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-6-2"
class="td11">drop while <span
class="phvr7t-x-x-80">fn </span>is true </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-7-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-7-1"
class="td11"><span
class="phvr7t-x-x-80">extract n l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-7-2"
class="td11">extract element <span
class="phvr7t-x-x-80">n </span>from list <span
class="phvr7t-x-x-80">l </span></td></tr><tr
style="vertical-align:baseline;" id="TBL-5-8-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-8-1"
class="td11"><span
class="phvr7t-x-x-80">filter fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-8-2"
class="td11">all elements of <span
class="phvr7t-x-x-80">l </span>for which <span
class="phvr7t-x-x-80">fn </span>holds</td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-9-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-9-1"
class="td11"><span
class="phvr7t-x-x-80">foldl fn st l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-9-2"
class="td11">fold list <span
class="phvr7t-x-x-80">l </span>left-to-right with <span
class="phvr7t-x-x-80">fn </span>and <span
class="phvr7t-x-x-80">st </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-10-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-10-1"
class="td11"><span
class="phvr7t-x-x-80">foldl1 fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-10-2"
class="td11">like <span
class="phvr7t-x-x-80">foldl</span>, but use the first element of the list as the start value</td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-11-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-11-1"
class="td11"><span
class="phvr7t-x-x-80">foldr fn st l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-11-2"
class="td11">fold list <span
class="phvr7t-x-x-80">l </span>right-to-left with <span
class="phvr7t-x-x-80">fn </span>and <span
class="phvr7t-x-x-80">st </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-12-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-12-1"
class="td11"><span
class="phvr7t-x-x-80">foldr1 fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-12-2"
class="td11">like <span
class="phvr7t-x-x-80">foldr</span>, but use the first element of the list as the start value</td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-13-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-13-1"
class="td11"><span
class="phvr7t-x-x-80">index fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-13-2"
class="td11">search list <span
class="phvr7t-x-x-80">l </span>for index of first element matching predicate <span
class="phvr7t-x-x-80">fn </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-14-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-14-1"
class="td11"><span
class="phvr7t-x-x-80">init l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-14-2"
class="td11">remove last element of list <span
class="phvr7t-x-x-80">l </span></td></tr><tr
style="vertical-align:baseline;" id="TBL-5-15-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-15-1"
class="td11"><span
class="phvr7t-x-x-80">iterate f x </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-15-2"
class="td11">repeatedly apply <span
class="phvr7t-x-x-80">f </span>to <span
class="phvr7t-x-x-80">x</span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-16-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-16-1"
class="td11"><span
class="phvr7t-x-x-80">last l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-16-2"
class="td11">return the last element of list <span
class="phvr7t-x-x-80">l </span></td></tr><tr
style="vertical-align:baseline;" id="TBL-5-17-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-17-1"
class="td11"><span
class="phvr7t-x-x-80">len l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-17-2"
class="td11">find length of list <span
class="phvr7t-x-x-80">l</span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-18-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-18-1"
class="td11"><span
class="phvr7t-x-x-80">limit l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-18-2"
class="td11">find the first element of list <span
class="phvr7t-x-x-80">l </span>equal to its predecessor </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-19-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-19-1"
class="td11"><span
class="phvr7t-x-x-80">map fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-19-2"
class="td11">map function <span
class="phvr7t-x-x-80">fn </span>over list <span
class="phvr7t-x-x-80">l </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-20-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-20-1"
class="td11"><span
class="phvr7t-x-x-80">map2 fn l1 l2 </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-20-2"
class="td11">map 2-ary function <span
class="phvr7t-x-x-80">fn </span>over lists <span
class="phvr7t-x-x-80">l1 </span>and <span
class="phvr7t-x-x-80">l2 </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-21-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-21-1"
class="td11"><span
class="phvr7t-x-x-80">map3 fn l1 l2 l3</span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-21-2"
class="td11">map 3-ary function <span
class="phvr7t-x-x-80">fn </span>over lists <span
class="phvr7t-x-x-80">l1</span>, <span
class="phvr7t-x-x-80">l2 </span>and <span
class="phvr7t-x-x-80">l3 </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-22-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-22-1"
class="td11"><span
class="phvr7t-x-x-80">member l x </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-22-2"
class="td11">true if <span
class="phvr7t-x-x-80">x </span>is a member of list <span
class="phvr7t-x-x-80">l </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-23-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-23-1"
class="td11"><span
class="phvr7t-x-x-80">mkset l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-23-2"
class="td11">remove duplicates from list <span
class="phvr7t-x-x-80">l </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-24-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-24-1"
class="td11"><span
class="phvr7t-x-x-80">postfix l r </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-24-2"
class="td11">add element <span
class="phvr7t-x-x-80">r </span>to the end of list <span
class="phvr7t-x-x-80">l </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-25-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-25-1"
class="td11"><span
class="phvr7t-x-x-80">product l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-25-2"
class="td11">product of list l </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-26-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-26-1"
class="td11"><span
class="phvr7t-x-x-80">repeat x </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-26-2"
class="td11">make an infinite list of <span
class="phvr7t-x-x-80">x</span>es </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-27-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-27-1"
class="td11"><span
class="phvr7t-x-x-80">replicate n x </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-27-2"
class="td11">make <span
class="phvr7t-x-x-80">n </span>copies of <span
class="phvr7t-x-x-80">x </span>in a list </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-28-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-28-1"
class="td11"><span
class="phvr7t-x-x-80">reverse l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-28-2"
class="td11">reverse list <span
class="phvr7t-x-x-80">l </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-29-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-29-1"
class="td11"><span
class="phvr7t-x-x-80">scan fn st l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-29-2"
class="td11">apply <span
class="phvr7t-x-x-80">(foldr fn r) </span>to every initial segment of list <span
class="phvr7t-x-x-80">l </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-30-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-30-1"
class="td11"><span
class="phvr7t-x-x-80">sort l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-30-2"
class="td11">sort list <span
class="phvr7t-x-x-80">l </span>into ascending order </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-31-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-31-1"
class="td11"><span
class="phvr7t-x-x-80">sortc fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-31-2"
class="td11">sort list <span
class="phvr7t-x-x-80">l </span>into order by using a comparison function </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-32-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-32-1"
class="td11"><span
class="phvr7t-x-x-80">sortpl pl l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-32-2"
class="td11">sort list <span
class="phvr7t-x-x-80">l </span>by predicate list <span
class="phvr7t-x-x-80">pl </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-33-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-33-1"
class="td11"><span
class="phvr7t-x-x-80">sortr l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-33-2"
class="td11">sort list <span
class="phvr7t-x-x-80">l </span>into descending order </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-34-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-34-1"
class="td11"><span
class="phvr7t-x-x-80">split fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-34-2"
class="td11">break list <span
class="phvr7t-x-x-80">l </span>into sections separated by predicate <span
class="phvr7t-x-x-80">fn </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-35-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-35-1"
class="td11"><span
class="phvr7t-x-x-80">splits fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-35-2"
class="td11">break list <span
class="phvr7t-x-x-80">l </span>into single sections separated by predicate <span
class="phvr7t-x-x-80">fn </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-36-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-36-1"
class="td11"><span
class="phvr7t-x-x-80">splitpl pl l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-36-2"
class="td11">break list <span
class="phvr7t-x-x-80">l </span>up by predicate list <span
class="phvr7t-x-x-80">pl </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-37-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-37-1"
class="td11"><span
class="phvr7t-x-x-80">split</span><span
class="phvr7t-x-x-80">_lines n l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-37-2"
class="td11">break list <span
class="phvr7t-x-x-80">l </span>into lines of length <span
class="phvr7t-x-x-80">n </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-38-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-38-1"
class="td11"><span
class="phvr7t-x-x-80">sum l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-38-2"
class="td11">sum list l </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-39-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-39-1"
class="td11"><span
class="phvr7t-x-x-80">take n l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-39-2"
class="td11">take the first <span
class="phvr7t-x-x-80">n </span>elements from list <span
class="phvr7t-x-x-80">l </span></td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-40-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-40-1"
class="td11"><span
class="phvr7t-x-x-80">takewhile fn l </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-40-2"
class="td11">take from the front of <span
class="phvr7t-x-x-80">l </span>while <span
class="phvr7t-x-x-80">fn </span>holds </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-41-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-41-1"
class="td11"><span
class="phvr7t-x-x-80">zip2 l1 l2 </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-41-2"
class="td11">zip two lists together </td>
</tr><tr
style="vertical-align:baseline;" id="TBL-5-42-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-42-1"
class="td11"><span
class="phvr7t-x-x-80">zip3 l1 l2 l3 </span></td><td style="white-space:nowrap; text-align:left;" id="TBL-5-42-2"
class="td11">zip three lists together </td>
</tr><tr
class="hline"><td><hr></td><td><hr></td></tr><tr
style="vertical-align:baseline;" id="TBL-5-43-"><td style="white-space:nowrap; text-align:left;" id="TBL-5-43-1"
class="td11"> </td></tr></table></div></div>
<br /><div class="caption"
><span class="id">Table 6.3: </span><span
class="content">Functions in the standard list-processing toolkit</span></div><!--tex4ht:label?: x39-710013 -->
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