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<!-- %A  lists.msk                   GAP documentation            Martin Schönert -->
<!-- %A                                                           Alexander Hulpke -->
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<!-- %A  @(#)<M>Id: lists.msk,v 1.58 2006/06/19 10:13:43 stefan Exp </M> -->
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<!-- %Y  (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland -->
<!-- %Y  Copyright (C) 2002 The GAP Group -->
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<Chapter Label="Lists">
<Heading>Lists</Heading>

Lists are the most important way to treat objects together.
A <E>list</E> arranges objects in a definite order.
So each list implies a partial mapping from the integers to the elements
of the list.
I.e., there is a first element of a list, a second, a third, and so on.
Lists can occur in mutable or immutable form,
see&nbsp;<Ref Sect="Mutability and Copyability"/> for the concept of mutability,
and&nbsp;<Ref Sect="Duplication of Lists"/> for the case of lists.
<P/>
This chapter deals mainly with the aspect of lists in &GAP;
as <E>data structures</E>.
Chapter&nbsp;<Ref Chap="Collections"/> tells more about the <E>collection</E>
aspect of certain lists,
and more about lists as <E>arithmetic objects</E> can be found in the chapters
<Ref Chap="Row Vectors"/> and <Ref Chap="Matrices"/>.
<P/>
Lists are used to implement ranges (see&nbsp;<Ref Sect="Ranges"/>),
sets (see&nbsp;<Ref Sect="Sorted Lists and Sets"/>),<Index>Sets</Index>
strings (see&nbsp;<Ref Chap="Strings and Characters"/>),
row vectors (see&nbsp;<Ref Chap="Row Vectors"/>),
and matrices (see&nbsp;<Ref Chap="Matrices"/>);
Boolean lists (see&nbsp;<Ref Chap="Boolean Lists"/>) are a further
special kind of lists.
<P/>
Several operations for lists,
such as <Ref Func="Intersection" Label="for a list"/> and
<Ref Func="Random" Label="for a list or collection"/>,
will be described in Chapter&nbsp;<Ref Chap="Collections"/>,
in particular see&nbsp;<Ref Sect="Lists and Collections"/>.


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="List Categories">
<Heading>List Categories</Heading>

A list can be written by writing down the elements in order between
square brackets <C>[</C>, <C>]</C>, and separating them with commas <C>,</C>.
An <E>empty list</E>, i.e., a list with no elements, is written as <C>[]</C>.
<P/>
<Example><![CDATA[
gap> [ 1, 2, 3 ];              # a list with three elements
[ 1, 2, 3 ]
gap> [ [], [ 1 ], [ 1, 2 ] ];  # a list may contain other lists
[ [  ], [ 1 ], [ 1, 2 ] ]
]]></Example>
<P/>
Each list constructed this way is mutable
(see&nbsp;<Ref Sect="Mutability and Copyability"/>).

<#Include Label="IsList">
<#Include Label="IsDenseList">
<#Include Label="IsHomogeneousList">
<#Include Label="IsTable">
<#Include Label="IsRectangularTable">
<#Include Label="IsConstantTimeAccessList">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Basic Operations for Lists">
<Heading>Basic Operations for Lists</Heading>

The basic operations for lists are element access
(see&nbsp;<Ref Sect="List Elements"/>),
assignment of elements to a list (see&nbsp;<Ref Sect="List Assignment"/>),
fetching the length of a list (see&nbsp;<Ref Func="Length"/>),
the test for a hole at a given position, and unbinding an element at a given
position (see&nbsp;<Ref Sect="IsBound and Unbind for Lists"/>).
<P/>
The term basic operation means that each other list operation can be
formulated in terms of the basic operations.
(But note that often a more efficient method than this one is implemented.)
<P/>
Any &GAP; object <A>list</A> in the category <Ref Func="IsList"/> is regarded
as a list, and if methods for the basic list operations are installed for
<A>list</A> then <A>list</A> can be used also for the other list operations.
<P/>
For internally represented lists, kernel methods are provided for the basic
list operations with positive integer indices.
For other lists or other indices, it is possible to install appropriate methods for these
operations.
This permits the implementation of lists that do not need to store all list
elements (see also&nbsp;<Ref Sect="Enumerators"/>);
for example, the elements might be described by an algorithm, such as the
elements list of a group.
For this reduction of space requirements, however, a price in access time
may have to be paid (see&nbsp;<Ref Func="ConstantTimeAccessList"/>).
<P/>
<Index Subkey="operation">list element</Index>
<Index Subkey="operation">list boundedness test</Index>
<Index Subkey="operation">list assignment</Index>
<Index Subkey="operation">list unbind</Index>
<ManSection>
<Oper Name="\[\]" Arg='list, ix'/>
<Oper Name="IsBound\[\]" Arg='list, ix'/>
<Oper Name="\[\]\:\=" Arg='list, pos, ix'/>
<Oper Name="Unbind\[\]" Arg='list, ix'/>

<Description>
These operations implement element access, test for element boundedness,
list element assignment, and removal of the element with index  <A>ix</A>.
<P/>
Note that the special characters <C>[</C>, <C>]</C>, <C>:</C>,
and <C>=</C> must be escaped with a backslash <C>\</C>
(see&nbsp;<Ref Sect="Symbols"/>);
so <Ref Oper="\[\]"/> denotes the operation for element access in a list,
whereas <C>[]</C> denotes an empty list.
(Maybe the variable names involving special characters look strange,
but nevertheless they are quite suggestive.)
<P/>
<C>\[\]( <A>list</A>, <A>ix</A> )</C> is equivalent to
<C><A>list</A>[ <A>ix</A> ]</C>,
which clearly will usually be preferred;
the former is useful mainly if one wants to access the operation itself,
for example if one wants to install a method for element access in a
special kind of lists.
<P/>
The syntax <C><A>list</A>[ <A>ix1</A>, <A>ix2</A>,.... <A>ixn</A>
]</C>, with two or more indices is treated as a shorthand for
<C><A>list</A>[[ <A>ix1</A>, <A>ix2</A>,.... <A>ixn</A>]]</C>

This is intended to provide a nicer syntax for accessing elements of
matrices and tensors.

<P/>
Similarly,
<Ref Oper="IsBound\[\]"/> is used explicitly mainly in method installations.
In other situations, one can simply call
<Ref Func="IsBound" Label="for a list index"/>,
which then delegates to <Ref Oper="IsBound\[\]"/>
if the first argument is a list,
and to <Ref Oper="IsBound\."/> if the first argument is a record.
<P/>
Analogous statements hold for <Ref Oper="\[\]\:\="/>
and <Ref Oper="Unbind\[\]"/>.
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="List Elements">
<Heading>List Elements</Heading>

<Index Subkey="list elements">accessing</Index>
<Index Subkey="access">list element</Index>
<C><A>list</A>[ <A>ix</A> ]</C>
<P/>
The above construct evaluates to the element of the list
<A>list</A> with index <A>ix</A>.
For built-in list types and collections, indexing is done with origin 1,
i.e., the first element of the list is the element with index 1.
<Example><![CDATA[
gap> l := [ 2, 3, 5, 7, 11, 13 ];;  l[1];  l[2];  l[6];
2
3
13
]]></Example>
If <A>list</A> is not a built-in list, or <A>ix</A> does not evaluate to a
positive integer, method selection is invoked to try and find a way of
indexing <A>list</A> with index <A>ix</A>. If this fails, or the
selected method finds that
<C><A>list</A>[<A>ix</A>]</C> is unbound, an error is signalled.
<P/>
<Index>multiple indices</Index>
<C><A>list</A>[ <A>ix1</A>,<A>ix2</A>,...]</C> <P/> is a short-hand for <C><A>list</A>[[<A>ix1</A>,<A>ix2</A>,...]]</C>
<P/>

<Index>sublist</Index>
<Index Subkey="access">sublist</Index>
<C><A>list</A>{ <A>poss</A> }</C>
<P/>
The above construct evaluates to a new list <A>new</A> whose first element is
<C><A>list</A>[<A>poss</A>[1]]</C>,
whose second element is <C><A>list</A>[<A>poss</A>[2]]</C>, and so on.
However, it does not need to be sorted and may contain duplicate elements.
If for any <M>i</M>,
<C><A>list</A>[ <A>poss</A>[</C><M>i</M><C>] ]</C> is unbound,
an error is signalled.
<P/>
<Example><![CDATA[
gap> l := [ 2, 3, 5, 7, 11, 13, 17, 19 ];;
gap> l{[4..6]};  l{[1,7,1,8]};
[ 7, 11, 13 ]
[ 2, 17, 2, 19 ]
]]></Example>
<P/>
The result is a <E>new</E> list, that is not identical to any other list. The
elements of that list, however, are identical to the corresponding elements
of the left operand (see&nbsp;<Ref Sect="Identical Lists"/>).
<P/>
It is possible to nest such <E>sublist extractions</E>, as can be seen in the
example below.
<P/>
<Example><![CDATA[
gap> m := [ [1,2,3], [4,5,6], [7,8,9], [10,11,12] ];;  m{[1,2,3]}{[3,2]};
[ [ 3, 2 ], [ 6, 5 ], [ 9, 8 ] ]
gap> l := m{[1,2,3]};; l{[3,2]};
[ [ 7, 8, 9 ], [ 4, 5, 6 ] ]
]]></Example>
<P/>
Note the difference between the two examples.  The latter extracts
elements 1, 2, and 3 from <A>m</A> and then extracts the elements 3 and 2 from
<E>this list</E>.
The former extracts elements 1, 2, and 3 from <A>m</A> and then
extracts the elements 3 and 2 from <E>each of those element lists</E>.
<P/>
To be precise:
With each selector <C>[<A>pos</A>]</C> or <C>{<A>poss</A>}</C> we associate
a <E>level</E> that is defined as the number of selectors of the form
<C>{<A>poss</A>}</C> to its left in the same expression. For example
<P/>
<Listing><![CDATA[
    l[pos1]{poss2}{poss3}[pos4]{poss5}[pos6]
level   0      0      1     2      2     3
]]></Listing>
<P/>
Then  a selector <C><A>list</A>[<A>pos</A>]</C> of level <A>level</A> is
computed as <C>ListElement(<A>list</A>,<A>pos</A>,<A>level</A>)</C>,
where <C>ListElement</C> is defined as follows.
(Note that <C>ListElement</C> is <E>not</E> a &GAP; function.)
<P/>
<Log><![CDATA[
ListElement := function ( list, pos, level )
 if level = 0 then
  return list[pos];
 else
  return List( list, elm -> ListElement(elm,pos,level-1) );
 fi;
end;
]]></Log>
<P/>
and a selector <C><A>list</A>{<A>poss</A>}</C> of level <A>level</A> is
computed as <C>ListElements(<A>list</A>,<A>poss</A>,<A>level</A>)</C>,
where <C>ListElements</C> is defined as follows.
(Note that <C>ListElements</C> is <E>not</E> a &GAP; function.)
<P/>
<Log><![CDATA[
ListElements := function ( list, poss, level )
 if level = 0 then
  return list{poss};
  else
   return List( list, elm -> ListElements(elm,poss,level-1) );
  fi;
end;
]]></Log>
<P/>
<Index Subkey="operation">sublist</Index>
<ManSection>
<Oper Name="\{\}" Arg='list, poss'/>

<Description>
This operation implements <E>sublist access</E>.
For any list, the default method is to loop over the entries in the list
<A>poss</A>, and to delegate to the element access operation.
(For the somewhat strange variable name,
cf.&nbsp;<Ref Sect="Basic Operations for Lists"/>.)
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="List Assignment">
<Heading>List Assignment</Heading>

<Index Subkey="to a list">assignment</Index>
<Index Subkey="assignment">list element</Index>
<C><A>list</A>[ <A>ix</A> ] := <A>object</A>;</C>
<P/>
The list element assignment assigns the object <A>object</A>,
which can be of any type, to the list with index <A>ix</A>,
in the mutable (see&nbsp;<Ref Sect="Mutability and Copyability"/>) list
<A>list</A>.
That means that accessing the <A>ix</A>-th element of the list <A>list</A>
will return <A>object</A> after this assignment.
<P/>
<Example><![CDATA[
gap> l := [ 1, 2, 3 ];;
gap> l[1] := 3;; l;             # assign a new object
[ 3, 2, 3 ]
gap> l[2] := [ 4, 5, 6 ];; l;   # <object> may be of any type
[ 3, [ 4, 5, 6 ], 3 ]
gap> l[ l[1] ] := 10;; l;       # <index> may be an expression
[ 3, [ 4, 5, 6 ], 10 ]
]]></Example>
<P/>
If the index <A>ix</A> is an integer larger than the length of the list <A>list</A>
(see <Ref Func="Length"/>),
the list is automatically enlarged to make room for the new element.
Note that it is possible to generate lists with holes that way.
<P/>
<Example><![CDATA[
gap> l[4] := "another entry";; l;  # <list> is enlarged
[ 3, [ 4, 5, 6 ], 10, "another entry" ]
gap> l[ 10 ] := 1;; l;             # now <list> has a hole
[ 3, [ 4, 5, 6 ], 10, "another entry",,,,,, 1 ]
]]></Example>
<P/>
The  function <Ref Func="Add"/> should be used if you want to add an
element to the end of the list.
<P/>
Note that assigning to a list changes the list,
thus this list must be mutable
(see&nbsp;<Ref Sect="Mutability and Copyability"/>).
See&nbsp;<Ref Sect="Identical Lists"/> for subtleties of changing lists.
<P/>
If <A>list</A> does not evaluate to a list, <A>pos</A> does not evaluate to a
positive integer or <A>object</A> is a call to a function which does not
return a value (for example <C>Print</C>) an error is signalled.
<P/>
<Index Subkey="assignment">multiple indices</Index>
<C><A>list</A>[ <A>ix1</A>,<A>ix2</A>,...] := <A>obj</A></C>
<P/> is a short-hand for <C><A>list</A>[[<A>ix1</A>,<A>ix2</A>,...]]
:= <A>obj</A></C>.<P/>
<Index Subkey="assignment">sublist</Index>
<C><A>list</A>{ <A>poss</A> } := <A>objects</A>;</C>
<P/>
The sublist assignment assigns the object <C><A>objects</A>[1]</C>,
which can be of any type, to the list <A>list</A> at the position
<C><A>poss</A>[1]</C>, the object <C><A>objects</A>[2]</C> to
<C><A>list</A>[<A>poss</A>[2]]</C>, and so on. <A>poss</A> must be a dense
list of positive integers, it need, however, not be sorted and may
contain duplicate elements. <A>objects</A> must be a dense list and must have
the same length as <A>poss</A>.
<P/>
<Example><![CDATA[
gap> l := [ 2, 3, 5, 7, 11, 13, 17, 19 ];;
gap> l{[1..4]} := [10..13];; l;
[ 10, 11, 12, 13, 11, 13, 17, 19 ]
gap> l{[1,7,1,10]} := [ 1, 2, 3, 4 ];; l;
[ 3, 11, 12, 13, 11, 13, 2, 19,, 4 ]
]]></Example>
<P/>
The next example shows that it is possible to nest such sublist assignments.
<P/>
<Example><![CDATA[
gap> m := [ [1,2,3], [4,5,6], [7,8,9], [10,11,12] ];;
gap> m{[1,2,3]}{[3,2]} := [ [11,12], [13,14], [15,16] ];; m;
[ [ 1, 12, 11 ], [ 4, 14, 13 ], [ 7, 16, 15 ], [ 10, 11, 12 ] ]
]]></Example>
<P/>
The exact behaviour is defined in the same way as for list extractions
(see <Ref Sect="List Elements"/>).
Namely, with each selector <C>[<A>pos</A>]</C> or <C>{<A>poss</A>}</C>
we associate a <E>level</E> that is defined as the number of selectors
of the form <C>{<A>poss</A>}</C> to its left in the same expression.
For example
<P/>
<Log><![CDATA[
    l[pos1]{poss2}{poss3}[pos4]{poss5}[pos6]
level   0      0      1     1      1     2
]]></Log>
<P/>
Then a list assignment <C><A>list</A>[<A>pos</A>] := <A>vals</A>;</C> of
level <A>level</A> is computed as
<C>ListAssignment( <A>list</A>, <A>pos</A>, <A>vals</A>, <A>level</A> )</C>,
where <C>ListAssignment</C> is defined as follows.
(Note that <C>ListAssignment</C> is <E>not</E> a &GAP; function.)
<P/>
<Log><![CDATA[
ListAssignment := function ( list, pos, vals, level )
 local i;
 if level = 0 then
  list[pos] := vals;
 else
  for i in [1..Length(list)] do
   ListAssignment( list[i], pos, vals[i], level-1 );
  od;
 fi;
end;
]]></Log>
<P/>
and a list assignment <C><A>list</A>{<A>poss</A>} := <A>vals</A></C> of
level <A>level</A> is computed as
<C>ListAssignments( <A>list</A>, <A>poss</A>, <A>vals</A>, <A>level</A> )</C>,
where <C>ListAssignments</C> is defined as follows.
(Note that <C>ListAssignments</C> is <E>not</E> a &GAP; function.)
<P/>
<Log><![CDATA[
ListAssignments := function ( list, poss, vals, level )
 local i;
 if level = 0 then
  list{poss} := vals;
 else
  for i in [1..Length(list)] do
   ListAssignments( list[i], poss, vals[i], level-1 );
  od;
 fi;
end;
]]></Log>
<P/>
<Index Subkey="operation">sublist assignment</Index>
<ManSection>
<Oper Name="\{\}\:\=" Arg='list, poss, val'/>

<Description>
This operation implements sublist assignment.
For any list, the default method is to loop over the entries in the list
<A>poss</A>, and to delegate to the element assignment operation.
(For the somewhat strange variable name,
cf.&nbsp;<Ref Sect="Basic Operations for Lists"/>.)
</Description>
</ManSection>

<#Include Label="Add">
<#Include Label="Remove">

<ManSection>
<Func Name="CopyListEntries"
 Arg='fromlst, fromind, fromstep, tolst, toind, tostep, n'/>

<Description>
This function copies <A>n</A> elements from <A>fromlst</A>,
starting at position <A>fromind</A> and incrementing the position by
<A>fromstep</A> each time,
into <A>tolst</A> starting at position <A>toind</A>
and incrementing the position by <A>tostep</A> each time.
<A>fromlst</A> and <A>tolst</A> must be plain lists.
<A>fromstep</A> and/or <A>tostep</A> can be negative.
Unbound positions of <A>fromlst</A> are simply copied to <A>tolst</A>.
<P/>
<Ref Func="CopyListEntries"/> is used in methods for
the operations <Ref Func="Add"/> and <Ref Func="Remove"/>.
</Description>
</ManSection>

<#Include Label="Append">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="IsBound and Unbind for Lists">
<Heading>IsBound and Unbind for Lists</Heading>

<#Include Label="IsBound_list">
<#Include Label="Unbind_list">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Identical Lists">
<Heading>Identical Lists</Heading>

With the list assignment (see&nbsp;<Ref Sect="List Assignment"/>) it is
possible to change a mutable list.
This section describes the semantic consequences of this fact.
(See also&nbsp;<Ref Sect="Identical Objects"/>.)
<P/>
First we define what it means when we say that <Q>an object is changed</Q>.
You may think that in the following example the second assignment changes
the integer.
<P/>
<Log><![CDATA[
i := 3;
i := i + 1;
]]></Log>
<P/>
But in this example it is not the <E>integer</E> <C>3</C> which is changed,
by adding one to it.
Instead the <E>variable</E> <C>i</C> is changed by assigning the value of
<C>i+1</C>,
which happens to be <C>4</C>, to <C>i</C>.
The same thing happens in the example below.
<P/>
<Log><![CDATA[
l := [ 1, 2 ];
l := [ 1, 2, 3 ];
]]></Log>
<P/>
The second assignment does not change the first list, instead it assigns
a new list to the variable <C>l</C>.  On the other hand, in the following
example the list <E>is</E> changed by the second assignment.
<P/>
<Log><![CDATA[
l := [ 1, 2 ];
l[3] := 3;
]]></Log>
<P/>
To understand the difference, think of a variable as a name for an
object. The important point is that a list can have several names at the
same time.  An assignment <C><A>var</A>:= <A>list</A>;</C> means in this
interpretation that <A>var</A> is a name for the object <A>list</A>.
At the end of the following example <C>l2</C> still has the value
<C>[ 1, 2 ]</C> as this list has not been changed and nothing else has been
assigned to it.
<P/>
<Log><![CDATA[
l1 := [ 1, 2 ];
l2 := l1;
l1 := [ 1, 2, 3 ];
]]></Log>
<P/>
But after the following example the list for which <C>l2</C> is a name has
been changed and thus the value of <C>l2</C> is now <C>[ 1, 2, 3 ]</C>.
<P/>
<Log><![CDATA[
l1 := [ 1, 2 ];
l2 := l1;
l1[3] := 3;
]]></Log>
<P/>
We say that two lists are <E>identical</E> if changing one of them by a
list assignment also changes the other one.  This is slightly incorrect,
because if <E>two</E> lists are identical, there are actually only two names
for <E>one</E> list. However, the correct usage would be very awkward and
would only add to the confusion.  Note that two identical lists must be
equal, because there is only one list with two different names. Thus
identity is an equivalence relation that is a refinement of equality.
Identity of objects can be detected using <Ref Func="IsIdenticalObj"/>.
<P/>
Let us now consider under which circumstances two lists are identical.
<P/>
If you enter a list literal then the list denoted by this literal is a
new list that is not identical to any other list.
Thus in the following example <C>l1</C> and <C>l2</C> are not identical,
though they are equal of course.
<P/>
<Log><![CDATA[
l1 := [ 1, 2 ];
l2 := [ 1, 2 ];
]]></Log>
<P/>
Also in the following example, no lists in the list <C>l</C> are identical.
<P/>
<Log><![CDATA[
l := [];
for i in [1..10] do l[i] := [ 1, 2 ]; od;
]]></Log>
<P/>
If you assign a list to a variable no new list is created. Thus the list
value of the variable on the left hand side and the list on the right
hand side of the assignment are identical. So in the following example
<C>l1</C> and <C>l2</C> are identical lists.
<P/>
<Log><![CDATA[
l1 := [ 1, 2 ];
l2 := l1;
]]></Log>
<P/>
If you pass a list as an argument, the old list and the argument of the
function are identical. Also if you return a list from a function, the
old list and the value of the function call are identical. So in the
following example <C>l1</C> and <C>l2</C> are identical lists:
<P/>
<Log><![CDATA[
l1 := [ 1, 2 ];
f := function ( l ) return l; end;
l2 := f( l1 );
]]></Log>
<P/>
If you change a list it keeps its identity.  Thus if two lists are
identical and you change one of them, you also change the other, and they
are still identical afterwards. On the other hand, two lists that are
not identical will never become identical if you change one of them. So
in the following example both <C>l1</C> and <C>l2</C> are changed,
and are still identical.
<P/>
<Log><![CDATA[
l1 := [ 1, 2 ];
l2 := l1;
l1[1] := 2;
]]></Log>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Duplication of Lists">
<Heading>Duplication of Lists</Heading>

Here we describe the meaning of <Ref Func="ShallowCopy"/> and
<Ref Func="StructuralCopy"/> for lists.
For the general definition of these functions,
see&nbsp;<Ref Sect="Duplication of Objects"/>.
<P/>
<Index Key="ShallowCopy" Subkey="for lists"><C>ShallowCopy</C></Index>
The subobjects (see&nbsp;<Ref Func="ShallowCopy"/>) of a list are exactly
its elements.
<P/>
This means that for any list <A>list</A>,
<Ref Func="ShallowCopy"/> returns a mutable <E>new</E> list <A>new</A>
that is <E>not identical</E> to any other list
(see&nbsp;<Ref Sect="Identical Lists"/>),
and whose elements are identical to the elements of <A>list</A>.
<P/>
<Index Key="StructuralCopy" Subkey="for lists"><C>StructuralCopy</C></Index>
Analogously, for a <E>mutable</E> list <A>list</A>,
<Ref Func="StructuralCopy"/> returns a mutable <E>new</E> list <A>scp</A>
that is <E>not identical</E> to any other list,
and whose elements are structural copies (defined recursively)
of the elements of <A>list</A>;
an element of <A>scp</A> is mutable (and then a <E>new</E> list)
if and only if the corresponding element of <A>list</A> is mutable.
<P/>
In both cases, modifying the copy <A>new</A> resp.&nbsp;<A>scp</A> by
assignments (see&nbsp;<Ref Sect="List Assignment"/>) does not modify
the original object <A>list</A>.
<P/>
<Ref Func="ShallowCopy"/> basically executes the following code for lists.
<Log><![CDATA[
new := [];
for i in [ 1 .. Length( list ) ] do
  if IsBound( list[i] ) then
    new[i] := list[i];
  fi;
od;
]]></Log>
<P/>
<Example><![CDATA[
gap> list1 := [ [ 1, 2 ], [ 3, 4 ] ];;  list2 := ShallowCopy( list1 );;
gap> IsIdenticalObj( list1, list2 );
false
gap> IsIdenticalObj( list1[1], list2[1] );
true
gap> list2[1] := 0;;  list1;  list2;
[ [ 1, 2 ], [ 3, 4 ] ]
[ 0, [ 3, 4 ] ]
]]></Example>
<P/>
<Ref Func="StructuralCopy"/> basically executes the following code for lists.
<Log><![CDATA[
new := [];
for i in [ 1 .. Length( list ) ] do
  if IsBound( list[i] ) then
    new[i] := StructuralCopy( list[i] );
  fi;
od;
]]></Log>
<P/>
<Example><![CDATA[
gap> list1 := [ [ 1, 2 ], [ 3, 4 ] ];;  list2 := StructuralCopy( list1 );;
gap> IsIdenticalObj( list1, list2 );
false
gap> IsIdenticalObj( list1[1], list2[1] );
false
gap> list2[1][1] := 0;;  list1;  list2;
[ [ 1, 2 ], [ 3, 4 ] ]
[ [ 0, 2 ], [ 3, 4 ] ]
]]></Example>
<P/>
The above code is not entirely correct. If the object <A>list</A> contains a
mutable object twice this object is not copied twice,
as would happen with the above definition, but only once.
This means that the copy <A>new</A> and the object <A>list</A> have exactly
the same structure when viewed as a general graph.
<P/>
<Example><![CDATA[
gap> sub := [ 1, 2 ];; list1 := [ sub, sub ];;
gap> list2 := StructuralCopy( list1 );
[ [ 1, 2 ], [ 1, 2 ] ]
gap> list2[1][1] := 0;; list2;
[ [ 0, 2 ], [ 0, 2 ] ]
gap> list1;
[ [ 1, 2 ], [ 1, 2 ] ]
]]></Example>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Membership Test for Lists">
<Heading>Membership Test for Lists</Heading>

<ManSection>
<Oper Name="\in" Arg="obj, list" Label="element test for lists"/>

<Description>
<Index Key="in" Subkey="for lists"><K>in</K></Index>
<Index Subkey="for lists">element test</Index>
This function call or the infix variant
<A>obj</A>&nbsp;<K>in</K>&nbsp;<A>list</A>
tests whether there is a positive integer <M>i</M> such that
<A>list</A><M>[i] =</M> <A>obj</A> holds.
<P/>
If the list <A>list</A> knows that it is strictly sorted
(see&nbsp;<Ref Func="IsSSortedList"/>),
the membership test is much quicker, because a binary search can be used
instead of the linear search used for arbitrary lists,
see <Ref Meth="\in" Label="for strictly sorted lists"/>.
<P/>
<Example><![CDATA[
gap> 1 in [ 2, 2, 1, 3 ];  1 in [ 4, -1, 0, 3 ];
true
false
gap> s := SSortedList( [2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32] );;
gap> 17 in s;  # uses binary search and only 4 comparisons
false
]]></Example>
<P/>
For finding the position of an element in a list,
see&nbsp;<Ref Sect="Finding Positions in Lists"/>.
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Enlarging Internally Represented Lists">
<Heading>Enlarging Internally Represented Lists</Heading>

Section&nbsp;<Ref Sect="List Assignment"/> told you (among other things)
that it is possible to assign beyond the logical end of a mutable list,
automatically enlarging the list.
This section tells you how this is done for internally represented lists.
<P/>
It would be extremely wasteful to make all lists large enough so that
there is room for all assignments, because some lists may have more than
100000 elements, while most lists have less than 10 elements.
<P/>
On the other hand suppose every assignment beyond the end of a list would
be done by allocating new space for the list and copying all entries to
the new space. Then creating a list of 1000 elements by assigning them
in order, would take half a million copy operations and also create a lot
of garbage that the garbage collector would have to reclaim.
<P/>
So the following strategy is used.  If a list is created it is created
with exactly the correct size. If a  list is enlarged, because of an
assignment beyond  the end of the list,  it is enlarged  by at least
<C><A>length</A>/8 + 4</C> entries.
Therefore the next assignments beyond the end of the list do not need
to enlarge the list.
For example creating a list of 1000 elements by assigning them in order,
would now take only 32 enlargements.
<P/>
The result of this is of course that the <E>physical length</E> of a list
may be larger than the <E>logical length</E>,
which is usually called simply the length of the list.
Aside from the implications for the performance you need not be aware
of the physical length.
In fact all you can ever observe, for example by calling
<Ref Func="Length"/>, is the logical length.
<P/>
Suppose that <Ref Func="Length"/> would have to take the physical length
and then test how many entries at the end of a list are unassigned,
to compute the logical length of the list. That would take too much time.
In order to make <Ref Func="Length"/>, and other functions that need to know
the logical length, more efficient,
the length of a list is stored along with the list.
<P/>
For fine tuning code dealing with plain lists we provide the following 
two functions.
<P/>
<ManSection>
<Func Arg="len" Name="EmptyPlist"/>
<Returns>a plain list</Returns>
<Func Arg="l" Name="ShrinkAllocationPlist"/>
<Returns>nothing</Returns>
<Description>
The function <Ref Func="EmptyPlist"/> returns an empty plain list which
has enough memory allocated for <A>len</A> entries. This can be useful
for creating and filling a plain list with a known number of entries. 
<P/>
The function <Ref Func="ShrinkAllocationPlist"/> gives back to &GAP;'s
memory manager the physical memory which is allocated for the plain list
<A>l</A> but not needed by the current number of entries.
<P/>
Note that there are similar functions <Ref Func="EmptyString"/> and
<Ref Func="ShrinkAllocationString"/> for strings instead of plain lists.
<Example>
gap> l:=[]; for i in [1..160] do Add(l, i^2); od; 
[  ]
gap> m:=EmptyPlist(160); for i in [1..160] do Add(m, i^2); od;
[  ]
gap> # now l uses about 25% more memory than the equal list m
gap> ShrinkAllocationPlist(l);
gap> # now l and m use the same amount of memory
</Example>
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Comparisons of Lists">
<Heading>Comparisons of Lists</Heading>

<Index Subkey="of lists">comparisons</Index>
<Index Subkey="comparison">list equal</Index>
<C><A>list1</A> = <A>list2</A></C>
<P/>
<C><A>list1</A> &lt;> <A>list2</A></C>
<P/>
Two lists <A>list1</A> and <A>list2</A> are equal if and only if for
every index <M>i</M>,
either both entries <A>list1</A><M>[i]</M> and <A>list2</A><M>[i]</M>
are unbound, or both are bound and are equal, i.e.,
<A>list1</A><M>[i] =</M> <A>list2</A><M>[i]</M> is <K>true</K>.
<P/>
<Example><![CDATA[
gap> [ 1, 2, 3 ] = [ 1, 2, 3 ];
true
gap> [ , 2, 3 ] = [ 1, 2, ];
false
gap> [ 1, 2, 3 ] = [ 3, 2, 1 ];
false
]]></Example>
<P/>
This definition will cause problems with lists which are their own entries.
Comparing two such lists for equality may lead to an infinite recursion in
the kernel if the list comparison has to compare the list entries which are
in fact the lists themselves,
and then &GAP; crashes.
<P/>
<Index Subkey="comparison">list smaller</Index>
<C><A>list1</A> &lt; <A>list2</A></C>
<P/>
<C><A>list1</A> &lt;= <A>list2</A></C>
<P/>
Lists are ordered <E>lexicographically</E>.
Unbound entries are smaller than any bound entry.
That implies the following behaviour.
Let <M>i</M> be the smallest positive integer <M>i</M> such that <A>list1</A>
and <A>list2</A> at position <M>i</M> differ,
i.e., either exactly one of <A>list1</A><M>[i]</M>, <A>list2</A><M>[i]</M>
is bound or both entries are bound and differ.
Then <A>list1</A> is less than <A>list2</A> if either
<A>list1</A><M>[i]</M> is unbound (and <A>list2</A><M>[i]</M> is not)
or both are bound and
<A>list1</A><M>[i]</M> &lt; <A>list2</A><M>[i]</M> is <K>true</K>.
<P/>
<Example><![CDATA[
gap> [ 1, 2, 3, 4 ] < [ 1, 2, 4, 8 ]; # <list1>[3] < <list2>[3]
true
gap> [ 1, 2, 3 ] < [ 1, 2, 3, 5 ];  # <list1>[4] is unbound and thus < 5
true
gap> [ 1, , 3, 4 ] < [ 1, -1, 3 ];  # <list1>[2] is unbound and thus < -1
true
]]></Example>
<P/>
Note that for comparing two lists with <C>&lt;</C> or <C>&lt;=</C>,
the (relevant) list elements must be comparable with <C>&lt;</C>,
which is usually <E>not</E> the case for objects in different families,
see&nbsp;<Ref Sect="Families"/>.
Also for the possibility to compare lists with other objects,
see&nbsp;<Ref Sect="Families"/>.

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Arithmetic for Lists">
<Heading>Arithmetic for Lists</Heading>

<Index Subkey="for lists">operators</Index>

It is convenient to have arithmetic operations for lists,
in particular because in &GAP;
row vectors and matrices are special kinds of lists.
However, it is the wide variety of list objects because of which we
prescribe arithmetic operations <E>not for all</E> of them.
(Keep in mind that <Q>list</Q> means just an object in the category
<Ref Func="IsList"/>.)
<P/>
(Due to the intended generality and flexibility,
the definitions given in the following sections are quite technical.
But for not too complicated cases
such as matrices (see&nbsp;<Ref Sect="Operators for Matrices"/>)
and row vectors (see&nbsp;<Ref Sect="Operators for Row Vectors"/>)
whose entries aren't lists,
the resulting behaviour should be intuitive.)
<P/>
For example, we want to deal with matrices which can be added and
multiplied in the usual way, via the infix operators <C>+</C> and <C>*</C>;
and we want also Lie matrices, with the same additive behaviour but with
the multiplication defined by the Lie bracket.
Both kinds of matrices shall be lists, with the usual access to their rows,
with <Ref Func="Length"/> returning the number of rows etc.
<P/>
For the categories and attributes that control the arithmetic behaviour
of lists,
see&nbsp;<Ref Sect="Filters Controlling the Arithmetic Behaviour of Lists"/>.
<P/>
For the definition of return values of additive and multiplicative operations
whose arguments are lists in these filters,
see&nbsp;<Ref Sect="Additive Arithmetic for Lists"/> and
<Ref Sect="Multiplicative Arithmetic for Lists"/>, respectively.
It should be emphasized that these sections describe only what the return
values are, and not how they are computed.
<P/>
For the mutability status of the return values,
see&nbsp;<Ref Sect="Mutability Status and List Arithmetic"/>.
(Note that this is not dealt with in the sections about the result values.)
<P/>
Further details about the special cases of row vectors and matrices
can be found in&nbsp;<Ref Sect="Operators for Row Vectors"/> and
in&nbsp;<Ref Sect="Operators for Matrices"/>,
the compression status is dealt with
in&nbsp;<Ref Sect="Row Vectors over Finite Fields"/>
and&nbsp;<Ref Sect="Matrices over Finite Fields"/>.

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Filters Controlling the Arithmetic Behaviour of Lists">
<Heading>Filters Controlling the Arithmetic Behaviour of Lists</Heading>

<#Include Label="[1]{arith}">
<#Include Label="IsGeneralizedRowVector">
<#Include Label="IsMultiplicativeGeneralizedRowVector">
<#Include Label="IsListDefault">
<#Include Label="NestingDepthA">
<#Include Label="NestingDepthM">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Additive Arithmetic for Lists">
<Heading>Additive Arithmetic for Lists</Heading>

In this general context, we define the results of additive operations
only in the following situations.
For unary operations (zero and additive inverse),
the unique argument must be in <Ref Func="IsGeneralizedRowVector"/>;
for binary operations (addition and subtraction),
at least one argument must be in <Ref Func="IsGeneralizedRowVector"/>,
and the other either is not a list or also in
<Ref Func="IsGeneralizedRowVector"/>.
<P/>
(For non-list &GAP; objects, defining the results of unary operations is not
an issue here,
and if at least one argument is a list not in
<Ref Func="IsGeneralizedRowVector"/>,
it shall be left to this argument whether the result in question is defined
and what it is.)

<Subsection Label="Zero for lists">
<Heading>Zero for lists</Heading>

The zero (see&nbsp;<Ref Func="Zero"/>) of a list <M>x</M> in
<Ref Func="IsGeneralizedRowVector"/>
is defined as the list whose entry at position <M>i</M> is the zero of
<M>x[i]</M> if this entry is bound, and is unbound otherwise.
<P/>
<Example><![CDATA[
gap> Zero( [ 1, 2, 3 ] );  Zero( [ [ 1, 2 ], 3 ] );  Zero( liemat );
[ 0, 0, 0 ]
[ [ 0, 0 ], 0 ]
LieObject( [ [ 0, 0 ], [ 0, 0 ] ] )
]]></Example>

</Subsection>

<Subsection Label="AdditiveInverse for lists">
<Heading>AdditiveInverse for lists</Heading>

The additive inverse (see&nbsp;<Ref Func="AdditiveInverse"/>) of a list
<M>x</M> in <Ref Func="IsGeneralizedRowVector"/> is defined as the list
whose entry at position <M>i</M> is the additive inverse of <M>x[i]</M>
if this entry is bound, and is unbound otherwise.
<P/>
<Example><![CDATA[
gap> AdditiveInverse( [ 1, 2, 3 ] );  AdditiveInverse( [ [ 1, 2 ], 3 ] );
[ -1, -2, -3 ]
[ [ -1, -2 ], -3 ]
]]></Example>

</Subsection>

<Subsection Label="Addition of lists">
<Heading>Addition of lists</Heading>

<Index Subkey="list and non-list">addition</Index>
If <M>x</M> and <M>y</M> are in <Ref Func="IsGeneralizedRowVector"/>
and have the same additive nesting depth
(see&nbsp;<Ref Func="NestingDepthA"/>),
<!-- % By definition, this depth is nonzero. -->
the sum <M>x + y</M> is defined <E>pointwise</E>,
in the sense that the result is a list whose entry at position <M>i</M> is
<M>x[i] + y[i]</M> if these entries are bound,
is a shallow copy (see&nbsp;<Ref Func="ShallowCopy"/>) of <M>x[i]</M> or
<M>y[i]</M> if the other argument is not bound at position <M>i</M>,
and is unbound if both <M>x</M> and <M>y</M> are unbound at position <M>i</M>.
<P/>
If <M>x</M> is in <Ref Func="IsGeneralizedRowVector"/> and <M>y</M> is
in <Ref Func="IsGeneralizedRowVector"/> and has lower additive nesting depth,
or is neither a list nor a domain,
the sum <M>x + y</M> is defined as a list whose entry at position <M>i</M> is
<M>x[i] + y</M> if <M>x</M> is bound at position <M>i</M>,
and is unbound if not.
The equivalent holds in the reversed case,
where the order of the summands is kept,
as addition is not always commutative.
<P/>
<Example><![CDATA[
gap> 1 + [ 1, 2, 3 ];  [ 1, 2, 3 ] + [ 0, 2, 4 ];  [ 1, 2 ] + [ Z(2) ];
[ 2, 3, 4 ]
[ 1, 4, 7 ]
[ 0*Z(2), 2 ]
gap> l1:= [ 1, , 3, 4 ];;             l2:= [ , 2, 3, 4, 5 ];;
gap> l3:= [ [ 1, 2 ], , [ 5, 6 ] ];;  l4:= [ , [ 3, 4 ], [ 5, 6 ] ];;
gap> NestingDepthA( l1 );  NestingDepthA( l2 );
1
1
gap> NestingDepthA( l3 );  NestingDepthA( l4 );
2
2
gap> l1 + l2;
[ 1, 2, 6, 8, 5 ]
gap> l1 + l3;
[ [ 2, 2, 3, 4 ],, [ 6, 6, 3, 4 ] ]
gap> l2 + l4;
[ , [ 3, 6, 3, 4, 5 ], [ 5, 8, 3, 4, 5 ] ]
gap> l3 + l4;
[ [ 1, 2 ], [ 3, 4 ], [ 10, 12 ] ]
gap> l1 + [];
[ 1,, 3, 4 ]
]]></Example>

</Subsection>

<Subsection Label="Subtraction of lists">
<Heading>Subtraction of lists</Heading>

<Index Subkey="difference">list and non-list</Index>
For two &GAP; objects <M>x</M> and <M>y</M> of which one is in
<Ref Func="IsGeneralizedRowVector"/> and the other is also in
<Ref Func="IsGeneralizedRowVector"/>
or is neither a list nor a domain, <M>x - y</M> is defined as <M>x + (-y)</M>.
<P/>
<Example><![CDATA[
gap> l1 - l2;
[ 1, -2, 0, 0, -5 ]
gap> l1 - l3;
[ [ 0, -2, 3, 4 ],, [ -4, -6, 3, 4 ] ]
gap> l2 - l4;
[ , [ -3, -2, 3, 4, 5 ], [ -5, -4, 3, 4, 5 ] ]
gap> l3 - l4;
[ [ 1, 2 ], [ -3, -4 ], [ 0, 0 ] ]
gap> l1 - [];
[ 1,, 3, 4 ]
]]></Example>

</Subsection>
</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Multiplicative Arithmetic for Lists">
<Heading>Multiplicative Arithmetic for Lists</Heading>

In this general context, we define the results of multiplicative operations
only in the following situations.
For unary operations (one and inverse), the unique argument must be in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/>;
for binary operations (multiplication and division),
at least one argument must be in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/>,
and the other either not a list or also in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/>.
<P/>
(For non-list &GAP; objects, defining the results of unary operations is not
an issue here, and if at least one argument is a list not in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/>,
it shall be left to this argument whether the result in question is defined
and what it is.)


<Subsection Label="One for lists">
<Heading>One for lists</Heading>

The one (see&nbsp;<Ref Func="One"/>) of a dense list <A>x</A> in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/> such that <A>x</A> has
even multiplicative nesting depth and has the same length
as each of its rows is defined as the usual identity matrix
on the outer two levels,
that is, an identity matrix of the same dimensions, with diagonal entries
<C>One( <A>x</A>[1][1] )</C> and off-diagonal entries
<C>Zero( <A>x</A>[1][1] )</C>.
<P/>
<Example><![CDATA[
gap> One( [ [ 1, 2 ], [ 3, 4 ] ] );
[ [ 1, 0 ], [ 0, 1 ] ]
gap> One( [ [ [ [ 1 ] ], [ [ 2 ] ] ], [ [ [ 3 ] ], [ [ 4 ] ] ] ] );
[ [ [ [ 1 ] ], [ [ 0 ] ] ], [ [ [ 0 ] ], [ [ 1 ] ] ] ]
]]></Example>

</Subsection>

<Subsection Label="Inverse for lists">
<Heading>Inverse for lists</Heading>

The inverse (see&nbsp;<Ref Func="Inverse"/>) of an invertible square table
<A>x</A> in <Ref Func="IsMultiplicativeGeneralizedRowVector"/> whose entries
lie in a common field is defined as the usual inverse <M>y</M>, i.e.,
a square matrix over the same field such that <M><A>x</A> y</M> and
<M>y <A>x</A></M> is equal to <C>One( <A>x</A> )</C>.
<P/>
<Example><![CDATA[
gap> Inverse( [ [ 1, 2 ], [ 3, 4 ] ] );
[ [ -2, 1 ], [ 3/2, -1/2 ] ]
]]></Example>

</Subsection>

<Subsection Label="Multiplication of lists">
<Heading>Multiplication of lists</Heading>

<Index Subkey="product">list and non-list</Index>
There are three possible computations that might be triggered by a
multiplication involving a list in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/>.
Namely, <M>x * y</M> might be
<List>
<Mark>(I)</Mark>
<Item>
    the inner product
    <M>x[1] * y[1] + x[2] * y[2] + \cdots + x[n] * y[n]</M>,
    where summands are omitted for which the entry in <M>x</M> or <M>y</M>
    is unbound
    (if this leaves no summand then the multiplication is an error),
    or
</Item>
<Mark>(L)</Mark>
<Item>
    the left scalar multiple, i.e., a list whose entry at position <M>i</M>
    is <M>x * y[i]</M> if <M>y</M> is bound at position <M>i</M>,
    and is unbound if not, or
</Item>
<Mark>(R)</Mark>
<Item>
    the right scalar multiple, i.e., a list whose entry at position <M>i</M>
    is <M>x[i] * y</M> if <M>x</M> is bound at position <M>i</M>,
    and is unbound if not.
</Item>
</List>
<P/>
Our aim is to generalize the basic arithmetic of simple row vectors and
matrices, so we first summarize the situations that shall be covered.
<P/>
<Table Align="c|ccc">
<Row>
  <Item> </Item>
  <Item>scl</Item>
  <Item>vec</Item>
  <Item>mat</Item>
</Row>
<HorLine/>
<Row>
  <Item>scl</Item>
  <Item> </Item>
  <Item>(L)</Item>
  <Item>(L)</Item>
</Row>
<Row>
  <Item>vec</Item>
  <Item>(R)</Item>
  <Item>(I)</Item>
  <Item>(I)</Item>
</Row>
<Row>
  <Item>mat</Item>
  <Item>(R)</Item>
  <Item>(R)</Item>
  <Item>(R)</Item>
</Row>
</Table>
<P/>
This means for example that the product of a scalar (scl) with a vector (vec)
or a matrix (mat) is computed according to (L).
Note that this is asymmetric.
<P/>
Now we can state the general multiplication rules.
<P/>
If exactly one argument is in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/>
then we regard the other argument (which is then neither a list nor a domain)
as a scalar, and specify result (L) or (R), depending on ordering.
<P/>
In the remaining cases, both <M>x</M> and <M>y</M> are in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/>,
and we distinguish the possibilities by their multiplicative nesting depths.
An argument with <E>odd</E> multiplicative nesting depth is regarded
as a vector,
and an argument with <E>even</E> multiplicative nesting depth is regarded
as a scalar or a matrix.
<P/>
So if both arguments have odd multiplicative nesting depth,
we specify result (I).
<P/>
If exactly one argument has odd nesting depth,
the other is treated as a scalar if it has lower multiplicative nesting
depth, and as a matrix otherwise.
In the former case, we specify result (L) or (R), depending on ordering;
in the latter case, we specify result (L) or (I), depending on ordering.
<P/>
We are left with the case that each argument has even multiplicative
nesting depth.
<!-- % By definition, this depth is nonzero. -->
If the two depths are equal, we treat the computation as a matrix product,
and specify result (R).
Otherwise, we treat the less deeply nested argument as a scalar and the other
as a matrix, and specify result (L) or (R), depending on ordering.
<P/>
<Example><![CDATA[
gap> [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ] * (1,4);
[ (1,4), (1,4)(2,3), (1,2,4), (1,2,3,4), (1,3,2,4), (1,3,4) ]
gap> [ 1, 2, , 4 ] * 2;
[ 2, 4,, 8 ]
gap> [ 1, 2, 3 ] * [ 1, 3, 5, 7 ];
22
gap> m:= [ [ 1, 2 ], 3 ];;  m * m;
[ [ 7, 8 ], [ [ 3, 6 ], 9 ] ]
gap> m * m = [ m[1] * m, m[2] * m ];
true
gap> n:= [ 1, [ 2, 3 ] ];;  n * n;
14
gap> n * n = n[1] * n[1] + n[2] * n[2];
true
]]></Example>

</Subsection>

<Subsection Label="Division of lists">
<Heading>Division of lists</Heading>

<Index Subkey="quotient">list and non-list</Index>
For two &GAP; objects <M>x</M> and <M>y</M> of which one is in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/> and the other is also in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/> or is neither a list
nor a domain, <M>x / y</M> is defined as <M>x * y^{{-1}}</M>.
<P/>
<Example><![CDATA[
gap> [ 1, 2, 3 ] / 2;  [ 1, 2 ] / [ [ 1, 2 ], [ 3, 4 ] ];
[ 1/2, 1, 3/2 ]
[ 1, 0 ]
]]></Example>

</Subsection>

<Subsection Label="mod for lists">
<Heading>mod for lists</Heading>

<Index Subkey="mod">list and non-list</Index>
<Index Subkey="lists">mod</Index>
If <A>x</A> and <A>y</A> are in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/> and have the
same multiplicative nesting depth (see&nbsp;<Ref Func="NestingDepthM"/>),
<!-- % By definition, this depth is nonzero. -->
<C><A>x</A> mod <A>y</A></C> is defined <E>pointwise</E>,
in the sense that the result is a list whose entry at position <M>i</M> is
<C><A>x</A>[i] mod <A>y</A>[i]</C> if these entries are bound,
is a shallow copy (see&nbsp;<Ref Func="ShallowCopy"/>) of <M>x[i]</M> or
<M>y[i]</M> if the other argument is not bound at position <M>i</M>,
and is unbound if both <M>x</M> and <M>y</M> are unbound at position <M>i</M>.
<P/>
If <M>x</M> is in <Ref Func="IsMultiplicativeGeneralizedRowVector"/> and
<M>y</M> is in <Ref Func="IsMultiplicativeGeneralizedRowVector"/> and has
lower multiplicative nesting depth or is neither a list nor a domain,
<C><A>x</A> mod <A>y</A></C> is defined as a list whose entry at position
<M>i</M> is <C><A>x</A>[i] mod <A>y</A></C> if <A>x</A> is bound at position
<M>i</M>, and is unbound if not.
The equivalent holds in the reversed case,
where the order of the arguments is kept.
<P/>
<Example><![CDATA[
gap> 4711 mod [ 2, 3,, 5, 7 ];
[ 1, 1,, 1, 0 ]
gap> [ 2, 3, 4, 5, 6 ] mod 3;
[ 2, 0, 1, 2, 0 ]
gap> [ 10, 12, 14, 16 ] mod [ 3, 5, 7 ];
[ 1, 2, 0, 16 ]
]]></Example>

</Subsection>

<Subsection Label="Left quotients of lists">
<Heading>Left quotients of lists</Heading>

<Index Subkey="left quotient">list and non-list</Index>
For two &GAP; objects <M>x</M> and <M>y</M> of which one is in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/> and the other is also in
<Ref Func="IsMultiplicativeGeneralizedRowVector"/> or is neither a list nor
a domain,
<C>LeftQuotient( <A>x</A>, <A>y</A> )</C> is defined as <M>x^{{-1}} * y</M>.
<P/>
<Example><![CDATA[
gap> LeftQuotient( [ [ 1, 2 ], [ 3, 4 ] ], [ 1, 2 ] );
[ 0, 1/2 ]
]]></Example>

</Subsection>
</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Mutability Status and List Arithmetic">
<Heading>Mutability Status and List Arithmetic</Heading>

Many results of arithmetic operations, when applied to lists,
are again lists, and it is of interest whether their entries are mutable
or not (if applicable).
Note that the mutability status of the result itself is already defined
by the general rule for any result of an arithmetic operation, not
only for lists (see&nbsp;<Ref Sect="Mutability and Copyability"/>).
<P/>
However, we do <E>not</E> define exactly the mutability status for each
element on each level of a nested list returned by an arithmetic operation.
(Of course it would be possible to define this recursively,
but since the methods used are in general not recursive,
in particular for efficient multiplication of compressed matrices,
such a general definition would be a burden in these cases.)
Instead we consider,
for a list <M>x</M> in <Ref Func="IsGeneralizedRowVector"/>,
the sequence <M>x = x_1, x_2, \ldots x_n</M> where <M>x_{{i+1}}</M> is
the first bound entry in <M>x_i</M> if exists
(that is, if <M>x_i</M> is a nonempty list),
and <M>n</M> is the largest <M>i</M> such that <M>x_i</M> lies in
<Ref Func="IsGeneralizedRowVector"/>.
The <E>immutability level</E> of <M>x</M> is defined as infinity
if <M>x</M> is immutable,
and otherwise the number of <M>x_i</M> which are immutable.
(So the immutability level of a mutable empty list is <M>0</M>.)
<P/>
Thus a fully mutable matrix has immutability level <M>0</M>,
and a mutable matrix with immutable first row has immutability level <M>1</M>
(independent of the mutability of other rows).
<P/>
The immutability level of the result of any of the binary operations
discussed here is the minimum of the immutability levels of the arguments,
provided that objects of the required mutability status exist in &GAP;.
<!-- Note that this means to call <C>ShallowCopy</C> more often than
     necessary!
     Would it be possible to promise the mutability status for all entries
     only if the arguments are homogeneously mutable,
     as a refinement of the general mutability rule? -->
<P/>
Moreover, the results have a <Q>homogeneous</Q> mutability status,
that is, if the first bound entry at nesting depth <M>i</M> is immutable
(mutable)
then all entries at nesting depth <M>i</M> are immutable
(mutable, provided that a mutable version of this entry exists in &GAP;).
<P/>
Thus the sum of two mutable matrices whose first rows are mutable
is a matrix all of whose rows are mutable,
and the product of two matrices whose first rows are immutable
is a matrix all of whose rows are immutable,
independent of the mutability status of the other rows of the arguments.
<!-- Note that there are situations where this rule (and in fact already the
     general rule mentioned above) leads to counter-intuitive results. -->
<P/>
For example, the sum of a matrix (mutable or immutable, i.e.,
of immutability level one of <M>0</M>, <M>1</M>, or <M>2</M>) and a mutable
row vector (i.e., immutability level <M>0</M>) is a fully mutable matrix.
The product of two mutable row vectors of integers is an integer,
and since &GAP; does not support mutable integers, the result is immutable.
<P/>
For unary arithmetic operations, there are three operations available,
an attribute that returns an immutable result
(<Ref Func="Zero"/>, <Ref Func="AdditiveInverse"/>, <Ref Func="One"/>,
<Ref Func="Inverse"/>),
an operation that returns a result that is mutable
<!-- at least on the outer level, or shall more be guaranteed? -->
(<Ref Oper="ZeroOp"/>, <Ref Oper="AdditiveInverseOp"/>,
<Ref Oper="OneOp"/>, <Ref Oper="InverseOp"/>),
and an operation whose result has the same immutability level as the argument
(<Ref Func="ZeroSM"/>, <Ref Func="AdditiveInverseSM"/>,
<Ref Func="OneSM"/>, <Ref Func="InverseSM"/>).
The last kind of operations is equivalent to the corresponding infix
operations <C>0 * <A>list</A></C>, <C>- <A>list</A></C>,
<C><A>list</A>^0</C>, and <C><A>list</A>^-1</C>.
(This holds not only for lists,
see&nbsp;<Ref Sect="Mutability and Copyability"/>.)
<P/>
<Example><![CDATA[
gap> IsMutable( l1 );  IsMutable( 2 * Immutable( [ 1, 2, 3 ] ) );
true
false
gap> IsMutable( l2 );  IsMutable( l3 );
true
true
]]></Example>
<P/>
An example motivating the mutability rule is the use of syntactic constructs
such as <C><A>obj</A> * <A>list</A></C> and <C>- <A>list</A></C> as
an elegant and efficient way to
create mutable lists needed for further manipulations from mutable lists.
In particular one can construct a mutable zero vector of length <M>n</M>
by <C>0 * [ 1 .. </C><M>n</M><C> ]</C>.
The latter can be done also using <Ref Func="ListWithIdenticalEntries"/>.

<#Include Label="ListWithIdenticalEntries">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Finding Positions in Lists">
<Heading>Finding Positions in Lists</Heading>

<#Include Label="Position">
<#Include Label="Positions">
<#Include Label="PositionCanonical">
<#Include Label="PositionNthOccurrence">
<#Include Label="PositionSorted">
<#Include Label="PositionSet">
<#Include Label="PositionProperty">
<#Include Label="PositionsProperty">
<#Include Label="PositionBound">
<#Include Label="PositionNot">
<#Include Label="PositionNonZero">
<#Include Label="PositionSublist">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Properties and Attributes for Lists">
<Heading>Properties and Attributes for Lists</Heading>

A list that contains mutable objects (like lists or records)
<E>cannot</E> store attribute values that depend on the values of its entries,
such as whether it is homogeneous, sorted, or strictly sorted,
as changes in any of its entries could change such property values,
like the following example shows.
<P/>
<Example><![CDATA[
gap> l:=[[1],[2]];
[ [ 1 ], [ 2 ] ]
gap> IsSSortedList(l);
true
gap> l[1][1]:=3;
3
gap> IsSSortedList(l);
false
]]></Example>
<P/>
For such lists these property values must be computed anew
each time the property is asked for.
For example, if <A>list</A> is a list of mutable row vectors then the call of
<Ref Func="Position"/> with <A>list</A> as first argument
cannot take advantage of the fact that <A>list</A> is in fact sorted.
One solution is to call explicitly <Ref Func="PositionSorted"/>
in such a situation, another solution is to replace <A>list</A> by an immutable
copy using <Ref Func="Immutable"/>.

<#Include Label="IsMatchingSublist">
<#Include Label="IsDuplicateFree">
<#Include Label="IsSortedList">
<#Include Label="IsSSortedList">
<#Include Label="Length">
<#Include Label="ConstantTimeAccessList">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Sorting Lists">
<Heading>Sorting Lists</Heading>


<#Include Label="Sort">
<#Include Label="SortParallel">
<#Include Label="Sortex">
<#Include Label="SortingPerm">

The default methods for all of these sorting operations currently
use Shell sort as it has a comparable performance to Quicksort for lists of
length at most a few thousands, and has better worst-case behaviour.

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Sorted Lists and Sets">
<Heading>Sorted Lists and Sets</Heading>

<Index>sets</Index>
<Index>multisets</Index>

Searching objects in a list works much quicker if the list is known to be
sorted.
Currently &GAP; exploits the sortedness of a list automatically only if
the list is <E>strictly sorted</E>, which is indicated by the property
<Ref Func="IsSSortedList"/>.
<P/>
Remember that a list of <E>mutable</E> objects cannot store that it is
strictly sorted but has to test it anew whenever it is asked
whether it is sorted,
see the remark in&nbsp;<Ref Sect="Properties and Attributes for Lists"/>.
Therefore &GAP; cannot take advantage of the sortedness of a list if this
list has mutable entries.
Moreover, if a sorted list <A>list</A> with mutable elements is used
as an argument of a function that <E>expects</E> this argument to be sorted,
for example <Ref Func="UniteSet"/> or <Ref Func="RemoveSet"/>,
then it is checked whether <A>list</A> is in fact sorted;
this check can have the effect actually to slow down the computations,
compared to computations with sorted lists of immutable elements
or computations that do not involve functions that do automatically check
sortedness.
<P/>
Strictly sorted lists are used to represent <E>sets</E> in &GAP;.
More precisely, a strictly sorted list is called a <E>proper set</E>
in the following, in order to avoid confusion with domains
(see&nbsp;<Ref Sect="Domains"/>) which also represent sets.
<P/>
In short proper sets are represented by sorted lists without holes and
duplicates in &GAP;.
Note that we guarantee this representation, so  you may make use of
the fact that a set is represented by a sorted list in your functions.
<P/>
In some contexts (for example see&nbsp;<Ref Chap="Combinatorics"/>),
we also want to talk about multisets.
A <E>multiset</E> is like a set, except that an element may appear
several times in a multiset.
Such multisets are represented by sorted lists without holes
that may have duplicates.
<P/>
This section lists only those functions that are defined exclusively for
proper sets.
Set theoretic functions for general collections,
such as <Ref Func="Intersection" Label="for a list"/>
and <Ref Func="Union" Label="for a list"/>, 
are described in Chapter&nbsp;<Ref Chap="Collections"/>.
In particular, for the construction of proper sets,
see&nbsp;<Ref Func="SSortedList"/> and <Ref Func="AsSSortedList"/>.
For finding positions in sorted lists, see&nbsp;<Ref Func="PositionSorted"/>.
<P/>
There are nondestructive counterparts of the functions
<Ref Func="UniteSet"/>, <Ref Func="IntersectSet"/>, and
<Ref Func="SubtractSet"/> available for proper sets.
These are <C>UnionSet</C>, <C>IntersectionSet</C>, and
<Ref Oper="Difference"/>.
The former two are methods for the more general operations
<Ref Func="Union" Label="for a list"/>
and <Ref Func="Intersection" Label="for a list"/>,
the latter is itself an operation (see&nbsp;<Ref Oper="Difference"/>).
<P/>
The result of <C>IntersectionSet</C> and <C>UnionSet</C> is always a new list,
that is not identical to any other list.
The elements of that list however are identical to the corresponding
elements of the first argument <A>set</A>.
If <A>set</A> is not a proper set it is not specified to which of a number
of equal elements in <A>set</A> the element in the result is identical
(see&nbsp;<Ref Sect="Identical Lists"/>).

<#Include Label="[1]{set}">

<ManSection>
<Meth Name="\in" Arg="obj, list" Label="for strictly sorted lists"/>

<Description>
For a list <A>list</A> that stores that it is strictly sorted,
the test with <Ref Func="\in" Label="for strictly sorted lists"/>
whether the object <A>obj</A> is an entry of <A>list</A> uses binary search.
This test can be entered also with the infix notation
<A>obj</A> <K>in</K> <A>list</A>.
</Description>
</ManSection>

<#Include Label="IsEqualSet">
<#Include Label="IsSubsetSet">
<#Include Label="AddSet">
<#Include Label="RemoveSet">
<#Include Label="UniteSet">
<#Include Label="IntersectSet">
<#Include Label="SubtractSet">

<!-- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- % % (from the GAP 3.4 manual) -->
<!-- % </Section>
<Section Label="More about Sets">
<Heading>More about Sets</Heading> -->
<!-- %  -->
<!-- % In the previous section we defined a proper  set as a sorted list without -->
<!-- % holes or duplicates.  This representation is not  only nice to use, it is -->
<!-- % also a good internal representation supporting efficient algorithms.  For -->
<!-- % example the <K>in</K> operator can use binary instead of a linear search since -->
<!-- % a set is sorted.  For another example <Ref Func="Union"/> only has to merge the sets. -->
<!-- %  -->
<!-- % However, all  those set functions  also allow lists that are  not  proper -->
<!-- % sets,  silently making  a copy  of it and  converting this copy to a set. -->
<!-- % Suppose all the functions would have to test  their arguments every time, -->
<!-- % comparing  each element  with its  successor, to see if  they  are proper -->
<!-- % sets.  This would chew up most  of  the performance advantage again.  For -->
<!-- % example suppose <K>in</K> would have to run  over the whole list, to see if it -->
<!-- % is  a  proper set, so  it could  use the  binary search.   That  would be -->
<!-- % ridiculous. -->
<!-- %  -->
<!-- % To avoid this a  list that is  a proper set  may, but need  not, have  an -->
<!-- % internal flag set that tells  those functions that  this list is indeed a -->
<!-- % proper set.  Those functions do not have to check this argument then, and -->
<!-- % can use the more  efficient algorithms.  This  section tells  you  when a -->
<!-- % proper set obtains this flag,  so you can write your  functions in such a -->
<!-- % way that you make best use of the algorithms. -->
<!-- %  -->
<!-- % The results of <Ref Func="Set"/>, <Ref Func="Difference"/>, <Ref Func="Intersection"/>  and <Ref Func="Union"/> are known -->
<!-- % to be sets by construction, and thus have the flag set upon creation. -->
<!-- %  -->
<!-- % If an argument to <Ref Func="IsSet"/>, <Ref Func="IsEqualSet"/>, <Ref Func="IsSubset"/>, <Ref Func="Set"/>, <Ref Func="Difference"/>, -->
<!-- % <Ref Func="Intersection"/> or  <Ref Func="Union"/> is a proper  set, that does  not  yet have the -->
<!-- % flag set, those functions will notice that and set the flag for this set. -->
<!-- % Note that <K>in</K> will use linear search if the  right operand does not have -->
<!-- % the flag set, will therefore not detect  if it is  a proper set and will, -->
<!-- % unlike the functions above, never set the flag. -->
<!-- %  -->
<!-- % If you change a proper set, that does have this  flag set, by assignment, -->
<!-- % <C>Add</C>   or <C>Append</C> the  set  will generally lose  it  flag,  even if the -->
<!-- % change is such that the resulting list is still a proper set.  However if -->
<!-- % the set has more than 100 elements and the value assigned or added is not -->
<!-- % a list and not a record and the resulting list is still a proper set than -->
<!-- % it will keep  the flag.  Note that  changing a list  that is not a proper -->
<!-- % set will never set the flag, even if the resulting list  is a proper set. -->
<!-- % Such a set will obtain the flag only if it is passed to a set function. -->
<!-- %  -->
<!-- % Suppose you have built a proper set  in such a way that  it does not have -->
<!-- % the flag set, and that you now want  to perform lots of membership tests. -->
<!-- % Then you  should call <C>IsSet</C>  with that set   as an argument.   If it is -->
<!-- % indeed  a proper set  <C>IsSet</C> will set the flag,  and the subsequent <K>in</K> -->
<!-- % operations will use  the more efficient binary  search.  You can think of -->
<!-- % the call to <C>IsSet</C> as a hint to &GAP; that this list is a proper set. -->
<!-- %  -->
<!-- % There is no way you can set the flag for an ordinary  list  without going -->
<!-- % through the checking in <C>IsSet</C>.  The  internal  functions depend so much -->
<!-- % on the fact that a list with  this flag set  is indeed sorted and without -->
<!-- % holes and duplicates that the risk would be too high to allow setting the -->
<!-- % flag without such a check. -->

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Operations for Lists">
<Heading>Operations for Lists</Heading>

Several of the following functions expect the first argument to be either a
list or a collection (see&nbsp;<Ref Chap="Collections"/>),
with possibly slightly different meaning for lists and non-list collections.

<Index Subkey="of lists">concatenation</Index>
<#Include Label="Concatenation">
<#Include Label="Compacted">
<#Include Label="Collected">
<#Include Label="DuplicateFreeList">
<#Include Label="AsDuplicateFreeList">
<#Include Label="Flat">
<#Include Label="Reversed">
<#Include Label="Shuffle">
<#Include Label="IsLexicographicallyLess">
<#Include Label="Apply">
<#Include Label="Perform">
<#Include Label="PermListList">
<#Include Label="Maximum">
<#Include Label="Minimum">
<#Include Label="MaximumList">
<#Include Label="Cartesian">
<#Include Label="IteratorOfCartesianProduct">
<#Include Label="Permuted">
<#Include Label="List:list">
<#Include Label="Filtered">
<#Include Label="Number">
<#Include Label="First">
<#Include Label="ForAll">
<#Include Label="ForAny">
<#Include Label="Product">
<#Include Label="Sum">
<#Include Label="Iterated">
<#Include Label="ListN">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Advanced List Manipulations">
<Heading>Advanced List Manipulations</Heading>

The following functions are generalizations of
<Ref Func="List" Label="for a collection"/>,
<Ref Func="Set"/>, <Ref Func="Sum"/>, and <Ref Func="Product"/>.

<#Include Label="ListX">
<#Include Label="SetX">
<#Include Label="SumX">
<#Include Label="ProductX">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Ranges">
<Heading>Ranges</Heading>

<Index>range</Index>
A <E>range</E> is a dense list of integers in arithmetic progression (or
degression).
This is a list of integers such that the difference between
consecutive elements is a nonzero constant.  Ranges can be abbreviated
with the syntactic construct
<P/>
<C>[ <A>first</A>, <A>second</A> .. <A>last</A> ]</C>
<P/>
or, if the difference between consecutive elements is 1, as
<P/>
<C>[ <A>first</A> .. <A>last</A> ]</C>.
<P/>
If <C><A>first</A> > <A>last</A></C>,
<C>[ <A>first</A> .. <A>last</A> ]</C> is the empty list,
which by definition is also a range;
also, if <C><A>second</A> > <A>first</A> > <A>last</A></C> or
<C><A>second</A> &lt; <A>first</A> &lt; <A>last</A></C>,
then <C>[ <A>first</A>, <A>second</A> .. <A>last</A> ]</C> is the empty list.
If <C><A>first</A> = <A>last</A></C>,
<C>[ <A>first</A>, <A>second</A> .. <A>last</A> ]</C> is a singleton list,
which is a range, too.
Note that <C><A>last</A> - <A>first</A></C> must be divisible by the
increment <C><A>second</A> - <A>first</A></C>,
otherwise an error is signalled.
<P/>
Currently, the integers <A>first</A>, <A>second</A> and <A>last</A>
and the length of a range must be small integers,
that is at least <M>-2^d</M> and at most <M>2^d - 1</M> 
with <M>d = 28</M> on 32-bit architectures
and <M>d = 60</M> on 64-bit architectures.
<P/>
Note also that a range is just a special case of a list.
Thus you can access elements in a range (see <Ref Sect="List Elements"/>),
test for membership etc.
You can even assign to such a range if it is mutable
(see&nbsp;<Ref Sect="List Assignment"/>).
Of course, unless you assign
<C><A>last</A> + <A>second</A> - <A>first</A></C> to the entry
<C><A>range</A>[ Length( <A>range</A> ) + 1 ]</C>,
the resulting list will no longer be a range.
<P/>
<Example><![CDATA[
gap> r := [10..20];
[ 10 .. 20 ]
gap> Length( r );
11
gap> r[3];
12
gap> 17 in r;
true
gap> r[12] := 25;; r;  # r is no longer a range
[ 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 25 ]
gap> r := [1,3..17];
[ 1, 3 .. 17 ]
gap> Length( r );
9
gap> r[4];
7
gap> r := [0,-1..-9];
[ 0, -1 .. -9 ]
gap> r[5];
-4
gap> r := [ 1, 4 .. 32 ];
Error, Range: <last>-<first> (31) must be divisible by <inc> (3)
]]></Example>
<P/>
Most often ranges are used in connection with the <K>for</K>-loop
see&nbsp;<Ref Sect="For"/>).
Here the construct
<P/>
<C>for <A>var</A> in [ <A>first</A> .. <A>last</A> ] do <A>statements</A> od</C>
<P/>
replaces the
<P/>
<C>for <A>var</A> from <A>first</A> to <A>last</A> do <A>statements</A> od</C>
<P/>
which is more usual in other programming languages.
<P/>
<Example><![CDATA[
gap> s := [];; for i in [10..20] do Add( s, i^2 ); od; s;
[ 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400 ]
]]></Example>
<P/>
Note that a range with <C><A>last</A> >= <A>first</A></C>
is at the same time also a proper set
(see&nbsp;<Ref Sect="Sorted Lists and Sets"/>),
because it contains no holes or duplicates and is sorted,
and also a row vector (see&nbsp;<Ref Chap="Row Vectors"/>),
because it contains no holes and all elements are integers.
<P/>
<#Include Label="IsRange">
<#Include Label="ConvertToRangeRep">

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Enumerators">
<Heading>Enumerators</Heading>

An <E>enumerator</E> is an immutable list that need not store its elements
explicitly but knows, from a set of basic data,
how to determine the <M>i</M>-th element and the position of a given object.
A typical example of this is a vector space over a finite field with <M>q</M>
elements, say, for which it is very easy to enumerate all elements
using <M>q</M>-adic expansions of integers.
<P/>
Using this enumeration can be even quicker than a binary search in a sorted
list of vectors, see <Ref Func="IsQuickPositionList"/>.
<P/>
On the one hand, element access to an enumerator may take more time than
element access to an internally represented list containing the same
elements.
On the other hand, an enumerator may save a vast amount of memory.
Take for example a permutation group of size a few millions.
Even for moderate degree it is unlikely that a list of all its elements
will fit into memory whereas it is no problem to construct an enumerator
from a stabilizer chain (see&nbsp;<Ref Sect="Stabilizer Chains"/>).
<P/>
There are situations where one only wants to loop over the elements of a
domain, without using the special facilities of an enumerator,
namely the particular order of elements and the possibility to find the
position of elements.
For such cases, &GAP; provides iterators (see&nbsp;<Ref Sect="Iterators"/>).
<P/>
The functions <Ref Func="Enumerator"/> and
<Ref Func="EnumeratorSorted"/> return enumerators of domains.
Most of the special implementations of enumerators in the &GAP; library
are based on the general interface that is provided by
<Ref Func="EnumeratorByFunctions" Label="for a domain and a record"/>;
one generic example is <Ref Func="EnumeratorByBasis"/>,
which can be used to get an enumerator of a finite dimensional free module.
<P/>
Also enumerators for non-domains can be implemented via
<Ref Func="EnumeratorByFunctions" Label="for a domain and a record"/>;
for a discussion,
see&nbsp;<Ref Sect="Example -- Constructing Enumerators"/>.

<#Include Label="IsQuickPositionList">

</Section>
</Chapter>

<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<!-- %% -->
<!-- %E -->

