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#############################################################################
##
#W coll.gd GAP library Martin Schoenert
#W & Thomas Breuer
##
#H @(#)$Id: coll.gd,v 4.108.2.2 2006/08/28 15:29:14 gap Exp $
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file declares the operations for collections.
##
Revision.coll_gd :=
"@(#)$Id: coll.gd,v 4.108.2.2 2006/08/28 15:29:14 gap Exp $";
#T change the installation of isomorphism and factor maintained methods
#T in the same way as that of subset maintained methods!
#1
## A *collection* in {\GAP} consists of elements in the same family
## (see~"Families").
## The most important kinds of collections are *homogeneous lists*
## (see~"Lists") and *domains* (see~"Domains").
## Note that a list is never a domain, and a domain is never a list.
## A list is a collection if and only if it is nonempty and homogeneous.
##
## Basic operations for collections are `Size' (see~"Size")
## and `Enumerator' (see~"Enumerator");
## for *finite* collections, `Enumerator' admits to delegate the other
## operations for collections
## (see~"Attributes and Properties for Collections"
## and~"Operations for Collections")
## to functions for lists (see~"Lists").
## Obviously, special methods depending on the arguments are needed for
## the computation of e.g.~the intersection of two *infinite* domains.
##
#############################################################################
##
#C IsListOrCollection( <obj> )
##
## Several functions are defined for both lists and collections,
## for example `Intersection' (see~"Intersection"), `Iterator'
## (see~"Iterator"), and `Random' (see~"Random").
## `IsListOrCollection' is a supercategory of `IsList' and `IsCollection'
## (that is, all lists and collections lie in this category),
## which is used to describe the arguments of functions such as the ones
## listed above.
##
DeclareCategory( "IsListOrCollection", IsObject );
#############################################################################
##
#C IsCollection( <obj> ) . . . . . . . . . test if an object is a collection
##
## tests whether an object is a collection.
##
DeclareCategory( "IsCollection", IsListOrCollection );
#############################################################################
##
#A CollectionsFamily( <Fam> ) . . . . . . . . . . make a collections family
##
## For a family <Fam>, `CollectionsFamily' returns the family of all
## collections that consist of elements in <Fam>.
##
## Note that families (see~"Families") are used to describe relations
## between objects.
## Important such relations are that between an element <elm> and each
## collection of elements that lie in the same family as <elm>,
## and that between two collections whose elements lie in the same family.
## Therefore, all collections of elements in the family <Fam> form the new
## family `CollectionsFamily( <Fam> )'.
##
DeclareAttribute( "CollectionsFamily", IsFamily );
#############################################################################
##
#C IsCollectionFamily( <Fam> ) test if an object is a family of collections
##
## is `true' if <Fam> is a family of collections, and `false' otherwise.
##
DeclareCategoryFamily( "IsCollection" );
#############################################################################
##
#A ElementsFamily( <Fam> ) . . . . . . . . . . . . fetch the elements family
##
## returns the family from which the collections family <Fam> was created
## by `CollectionsFamily'.
## The way a collections family is created, it always has its elements
## family stored.
## If <Fam> is not a collections family (see~"IsCollectionFamily")
## then an error is signalled.
##
DeclareAttribute( "ElementsFamily", IsFamily );
#############################################################################
##
#V CATEGORIES_COLLECTIONS . . . . . . global list of collections categories
##
BIND_GLOBAL( "CATEGORIES_COLLECTIONS", [] );
#############################################################################
##
#F CategoryCollections( <filter> ) . . . . . . . . . . collections category
##
## Let <filter> be a filter that is `true' for all elements of a family
## <Fam>, by construction of <Fam>.
## Then `CategoryCollections' returns a category that is `true' for all
## elements in `CollectionsFamily( <Fam> )'.
##
## For example, the construction of `PermutationsFamily' guarantees that
## each of its elements lies in the filter `IsPerm',
## and each collection of permutations lies in the category
## `CategoryCollections( IsPerm )'.
##
## Note that this works only if the collections category is created *before*
## the collections family.
## So it is necessary to construct interesting collections categories
## immediately after the underlying category has been created.
##
BIND_GLOBAL( "CategoryCollections", function ( elms_filter )
local pair, super, flags, name, coll_filter;
# Check whether the collections category is already defined.
for pair in CATEGORIES_COLLECTIONS do
if IsIdenticalObj( pair[1], elms_filter ) then
return pair[2];
fi;
od;
# Find the super category among the known collections categories.
super := IsCollection;
flags := WITH_IMPS_FLAGS( FLAGS_FILTER( elms_filter ) );
for pair in CATEGORIES_COLLECTIONS do
if IS_SUBSET_FLAGS( flags, FLAGS_FILTER( pair[1] ) ) then
super := super and pair[2];
fi;
od;
# Construct the name of the category.
name := "CategoryCollections(";
APPEND_LIST_INTR( name, SHALLOW_COPY_OBJ( NameFunction(elms_filter) ) );
APPEND_LIST_INTR( name, ")" );
CONV_STRING( name );
# Construct the collections category.
coll_filter:= NewCategory( name, super );
ADD_LIST( CATEGORIES_COLLECTIONS, [ elms_filter, coll_filter ] );
return coll_filter;
end );
#############################################################################
##
#f DeclareCategoryCollections( <name> )
##
## binds the collections category of the category that is bound to the
## global variable with name <name> to the global variable associated to the
## name <nname>.
## If <name> is of the form `<initname>Collection' then <nname> is equal to
## `<initname>CollColl',
## if <name> is of the form `<initname>Coll' then <nname> is equal to
## `<initname>CollColl',
## otherwise we have <nname> equal to `<name>Collection'.
##
BIND_GLOBAL( "DeclareCategoryCollections", function( name )
local len, coll_name;
len:= LEN_LIST( name );
if 3 < len and name{ [ len-3 .. len ] } = "Coll" then
coll_name:= SHALLOW_COPY_OBJ( name );
APPEND_LIST_INTR( coll_name, "Coll" );
elif 9 < len and name{ [ len-9 .. len ] } = "Collection" then
coll_name:= name{ [ 1 .. len-6 ] };
APPEND_LIST_INTR( coll_name, "Coll" );
else
coll_name:= SHALLOW_COPY_OBJ( name );
APPEND_LIST_INTR( coll_name, "Collection" );
fi;
BIND_GLOBAL( coll_name, CategoryCollections( VALUE_GLOBAL( name ) ) );
end );
#############################################################################
##
#F DeclareSynonym( <name>, <value> )
#F DeclareSynonymAttr( <name>, <value> )
##
#T Why is this in this file?
##
BIND_GLOBAL( "DeclareSynonym", function( name, value )
BIND_GLOBAL( name, value );
end );
BIND_GLOBAL( "DeclareSynonymAttr", function( name, value )
local nname;
BIND_GLOBAL( name, value );
nname:= "Set";
APPEND_LIST_INTR( nname, name );
BIND_GLOBAL( nname, Setter( value ) );
nname:= "Has";
APPEND_LIST_INTR( nname, name );
BIND_GLOBAL( nname, Tester( value ) );
end );
#############################################################################
##
#V SUBSET_MAINTAINED_INFO
##
## is a list of length two.
## At the first position, a list of lists of the form
## `[ <filtsuper>, <filtsub>, <opr>, <testopr>, <settopr> ]'
## is stored,
## which is used for calls of `UseSubsetRelation( <super>, <sub> )'.
## At the second position, a corresponding list of lists of the form
## `[ <flagsopr>, <flagssub>, <rank> ]'
## is stored, which is used for choosing an appropriate ordering of the
## entries when the lists are enlarged in a call to
## `InstallSubsetMaintenance'.
##
## The meaning of the entries is as follows.
## \beginitems
## <filtsuper> &
## required filter for <super>,
##
## <filtsub> &
## required filter for <sub>,
##
## <opr> &
## operation whose value is inherited from <super> to <sub>,
##
## <testopr> &
## tester filter of <opr>,
##
## <settopr> &
## setter filter of <opr>,
##
## <flagsopr> &
## list of those true flags of <opr>
## that belong neither to categories nor to representations,
##
## <flagssub> &
## list of those true flags of <filtsub>
## that belong neither to categories nor to representations,
##
## <rank> &
## a rational number that denotes the priority of the information
## in the list; `SUBSET_MAINTAINED_INFO' is sorted according to
## decreasing <rank> value.
#T We must be careful to choose the right succession of the methods.
#T Note that one method may require a property that is acquired using
#T another method.
#T For that, we give a method a rank that is lower than that of all methods
#T that may yield some of the requirements and that is higher than that of
#T all methods that require <opr>;
#T if this is not possible then a warning is printed.
#T (Maybe the mechanism has to be changed at some time because of this.
#T Another reason would be the direct installation of methods for
#T `UseSubsetRelation', i.e., the ranks of these methods are not affected
#T by the code in `InstallSubsetMaintenance'.)
## \enditems
##
BIND_GLOBAL( "SUBSET_MAINTAINED_INFO", [ [], [] ] );
#############################################################################
##
#O UseSubsetRelation( <super>, <sub> )
##
## Methods for this operation transfer possibly useful information from the
## domain <super> to its subset <sub>, and vice versa.
##
## `UseSubsetRelation' is designed to be called automatically
## whenever substructures of domains are constructed.
## So the methods must be *cheap*, and the requirements should be as
## sharp as possible!
##
## To achieve that *all* applicable methods are executed, all methods for
## this operation except the default method must end with `TryNextMethod()'.
## This default method deals with the information that is available by
## the calls of `InstallSubsetMaintenance' in the {\GAP} library.
##
DeclareOperation( "UseSubsetRelation", [ IsCollection, IsCollection ] );
InstallMethod( UseSubsetRelation,
"default method that checks maintenances and then returns `true'",
IsIdenticalObj,
[ IsCollection, IsCollection ],
# Make sure that this method is installed with ``real'' rank zero.
- 2 * RankFilter( IsCollection ),
function( super, sub )
local entry;
for entry in SUBSET_MAINTAINED_INFO[1] do
if entry[1]( super ) and entry[2]( sub ) and not entry[4]( sub ) then
entry[5]( sub, entry[3]( super ) );
fi;
od;
return true;
end );
#############################################################################
##
#F InstallSubsetMaintenance( <opr>, <super_req>, <sub_req> )
##
## <opr> must be a property or an attribute.
## The call of `InstallSubsetMaintenance' has the effect that
## for a domain <D> in the filter <super_req>, and a domain <S> in the
## filter <sub_req>,
## the call `UseSubsetRelation( <D>, <S> )' (see~"UseSubsetRelation")
## sets a known value of <opr> for <D> as value of <opr> also for <S>.
## A typical example for which `InstallSubsetMaintenance' is applied
## is given by `<opr> = IsFinite',
## `<super_req> = IsCollection and IsFinite',
## and `<sub_req> = IsCollection'.
##
## If <opr> is a property and the filter <super_req> lies in the filter
## <opr> then we can use also the following inverse implication.
## If $D$ is in the filter whose intersection with <opr> is <super_req>
## and if $S$ is in the filter <sub_req>, $S$ is a subset of $D$, and
## the value of <opr> for $S$ is `false'
## then the value of <opr> for $D$ is also `false'.
#T This is implemented only for the case <super_req> = <opr> and <sub_req>.
##
BIND_GLOBAL( "InstallSubsetMaintenance",
function( operation, super_req, sub_req )
local setter, # setter filter of `operation'
tester, # tester filter of `operation'
upper,
lower,
attrprop, # id `operation' an attribute/property?
rank,
filtssub, # property and attribute flags of `sub_req'
filtsopr, # property and attribute flags of `operation'
triple, # loop over `SUBSET_MAINTAINED_INFO[2]'
req,
flag,
filt1,
filt2,
i;
setter:= Setter( operation );
tester:= Tester( operation );
# Are there methods that may give us some of the requirements?
upper:= SUM_FLAGS;
# (We must not call `SUBTR_SET' here because the lists types may be
# not yet defined.)
filtssub:= [];
for flag in TRUES_FLAGS( FLAGS_FILTER( sub_req ) ) do
if not flag in CATS_AND_REPS then
ADD_LIST_DEFAULT( filtssub, flag );
fi;
od;
for triple in SUBSET_MAINTAINED_INFO[2] do
req:= SHALLOW_COPY_OBJ( filtssub );
INTER_SET( req, triple[1] );
if LEN_LIST( req ) <> 0 and triple[3] < upper then
upper:= triple[3];
fi;
od;
# Are there methods that require `operation'?
lower:= 0;
attrprop:= true;
filt1:= FLAGS_FILTER( operation );
if filt1 = false then
# `operation' is an attribute.
filt1:= FLAGS_FILTER( tester );
else
# Special treatment of categories, representations (makes sense?),
# and filters created by `NewFilter'.
if FLAG2_FILTER( operation ) = 0 then
attrprop:= false;
fi;
fi;
# (We must not call `SUBTR_SET' here because the lists types may be
# not yet defined.)
filtsopr:= [];
for flag in TRUES_FLAGS( filt1 ) do
if not flag in CATS_AND_REPS then
ADD_LIST_DEFAULT( filtsopr, flag );
fi;
od;
for triple in SUBSET_MAINTAINED_INFO[2] do
req:= SHALLOW_COPY_OBJ( filtsopr );
INTER_SET( req, triple[2] );
if LEN_LIST( req ) <> 0 and lower < triple[3] then
lower:= triple[3];
fi;
od;
# Compute the ``rank'' of the maintenance.
# (Do we have a cycle?)
if upper <= lower then
Print( "#W warning: cycle in `InstallSubsetMaintenance'\n" );
rank:= lower;
else
rank:= ( upper + lower ) / 2;
fi;
filt1:= IsCollection and Tester( super_req ) and super_req and tester;
filt2:= IsCollection and Tester( sub_req ) and sub_req;
# Update the info list.
i:= LEN_LIST( SUBSET_MAINTAINED_INFO[2] );
while 0 < i and SUBSET_MAINTAINED_INFO[2][i][3] < rank do
SUBSET_MAINTAINED_INFO[1][ i+1 ]:= SUBSET_MAINTAINED_INFO[1][ i ];
SUBSET_MAINTAINED_INFO[2][ i+1 ]:= SUBSET_MAINTAINED_INFO[2][ i ];
i:= i-1;
od;
SUBSET_MAINTAINED_INFO[2][ i+1 ]:= [ filtsopr, filtssub, rank ];
if attrprop then
SUBSET_MAINTAINED_INFO[1][ i+1 ]:=
[ filt1, filt2, operation, tester, setter ];
else
SUBSET_MAINTAINED_INFO[1][ i+1 ]:=
[ filt1, filt2, operation, operation,
function( sub, val )
SetFeatureObj( sub, operation, val );
end ];
fi;
#T missing in new implementation!
# # Install the method.
# if FLAGS_FILTER( operation ) <> false
# and IS_EQUAL_FLAGS( FLAGS_FILTER( operation and sub_req ),
# FLAGS_FILTER( super_req ) ) then
# InstallMethod( UseSubsetRelation, infostring, IsIdenticalObj,
# [ sub_req, sub_req ], 0,
# function( super, sub )
# if tester( sub ) and not operation( sub ) then
# setter( super, false );
# fi;
# TryNextMethod();
# end );
# fi;
end );
#############################################################################
##
#V ISOMORPHISM_MAINTAINED_INFO
##
## is a list of lists of the form
## `[ <filtsold>, <filtsnew>, <opr>, <testopr>, <settopr>, <old_req>,
## <new_req> ]'
## which is used for calls of `UseIsomorphismRelation( <old>, <new> )'.
## This list is enlarged by calls to `InstallIsomorphismMaintenance'.
##
## The meaning of the entries is as follows.
## \beginitems
## <filtsold> &
## required filter for <old>,
##
## <filtsnew> &
## required filter for <new>,
##
## <opr> &
## operation whose value is inherited from <old> to <new>,
##
## <testopr> &
## tester filter of <opr>,
##
## <settopr> &
## setter filter of <opr>,
##
## <old_req> &
## requirements for <old> in the `InstallIsomorphismMaintenance' call,
##
## <new_req> &
## requirements for <new> in the `InstallIsomorphismMaintenance' call.
## \enditems
##
BIND_GLOBAL( "ISOMORPHISM_MAINTAINED_INFO", [] );
#############################################################################
##
#O UseIsomorphismRelation( <old>, <new> )
##
## Methods for this operation transfer possibly useful information from the
## domain <old> to the isomorphic domain <new>.
##
## `UseIsomorphismRelation' is designed to be called automatically
## whenever isomorphic structures of domains are constructed.
## So the methods must be *cheap*, and the requirements should be as
## sharp as possible!
##
## To achieve that *all* applicable methods are executed, all methods for
## this operation except the default method must end with `TryNextMethod()'.
## This default method deals with the information that is available by
## the calls of `InstallIsomorphismMaintenance' in the {\GAP} library.
##
DeclareOperation( "UseIsomorphismRelation", [ IsCollection, IsCollection ] );
InstallMethod( UseIsomorphismRelation,
"default method that checks maintenances and then returns `true'",
[ IsCollection, IsCollection ],
# Make sure that this method is installed with ``real'' rank zero.
- 2 * RankFilter( IsCollection ),
function( old, new )
local entry;
for entry in ISOMORPHISM_MAINTAINED_INFO do
if entry[1]( old ) and entry[2]( new ) and not entry[4]( new ) then
entry[5]( new, entry[3]( old ) );
fi;
od;
return true;
end );
#############################################################################
##
#F InstallIsomorphismMaintenanceFunction( <func> )
##
## `InstallIsomorphismMaintenanceFunction' installs <func>, so that
## `<func>( <filtsold>, <filtsnew>, <opr>, <testopr>, <settopr>, <old_req>,
## <new_req> )' is called for each isomorphism maintenance.
## More precisely, <func> is called for each entry in the global list
## `ISOMORPHISM_MAINTAINED_INFO', also to those that are entered into this
## list after the installation of <func>.
## (The mechanism is the same as for attributes, which is installed in the
## file `lib/oper.g'.)
##
BIND_GLOBAL( "ISOM_MAINT_FUNCS", [] );
BIND_GLOBAL( "InstallIsomorphismMaintenanceFunction", function( func )
local entry;
for entry in ISOMORPHISM_MAINTAINED_INFO do
CallFuncList( func, entry );
od;
ADD_LIST( ISOM_MAINT_FUNCS, func );
end );
BIND_GLOBAL( "RUN_ISOM_MAINT_FUNCS",
function( arglist )
local func;
for func in ISOM_MAINT_FUNCS do
CallFuncList( func, arglist );
od;
ADD_LIST( ISOMORPHISM_MAINTAINED_INFO, arglist );
end );
#############################################################################
##
#F InstallIsomorphismMaintenance( <opr>, <old_req>, <new_req> )
##
## <opr> must be a property or an attribute.
## The call of `InstallIsomorphismMaintenance' has the effect that
## for a domain <D> in the filter <old_req>, and a domain <E> in the
## filter <new_req>,
## the call `UseIsomorphismRelation( <D>, <E> )'
## (see~"UseIsomorphismRelation")
## sets a known value of <opr> for <D> as value of <opr> also for <E>.
## A typical example for which `InstallIsomorphismMaintenance' is
## applied is given by `<opr> = Size',
## `<old_req> = IsCollection', and `<new_req> = IsCollection'.
#T Up to now, there are no dependencies between the maintenances
#T (contrary to the case of subset maintenances),
#T so we do not take care of the succession.
##
BIND_GLOBAL( "InstallIsomorphismMaintenance",
function( opr, old_req, new_req )
local tester;
tester:= Tester( opr );
RUN_ISOM_MAINT_FUNCS(
[ IsCollection and Tester( old_req ) and old_req and tester,
IsCollection and Tester( new_req ) and new_req,
opr,
tester,
Setter( opr ),
old_req,
new_req ] );
end );
#############################################################################
##
#V FACTOR_MAINTAINED_INFO
##
## is a list of lists of the form
## `[ <filtsnum>, <filtsden>, <filtsfac>, <opr>, <testopr>, <settopr> ]'
## which is used for calls of `UseFactorRelation( <num>, <den>, <fac> )'.
## This list is enlarged by calls to `InstallFactorMaintenance'.
##
## The meaning of the entries is as follows.
## \beginitems
## <filtsnum> &
## required filter for <num>,
##
## <filtsden> &
## required filter for <den>,
##
## <filtsfac> &
## required filter for <fac>,
##
## <opr> &
## operation whose value is inherited from <num> to <fac>,
##
## <testopr> &
## tester filter of <opr>,
##
## <settopr> &
## setter filter of <opr>.
## \enditems
##
BIND_GLOBAL( "FACTOR_MAINTAINED_INFO", [] );
#############################################################################
##
#O UseFactorRelation( <numer>, <denom>, <factor> )
##
## Methods for this operation transfer possibly useful information from the
## domain <numer> or its subset <denom> to the domain <factor> that
## is isomorphic to the factor of <numer> by <denom>, and vice versa.
## <denom> may be `fail', for example if <factor> is just known to be a
## factor of <numer> but <denom> is not available as a {\GAP} object;
## in this case those factor relations are used that are installed without
## special requirements for <denom>.
##
## `UseFactorRelation' is designed to be called automatically
## whenever factor structures of domains are constructed.
## So the methods must be *cheap*, and the requirements should be as
## sharp as possible!
##
## To achieve that *all* applicable methods are executed, all methods for
## this operation except the default method must end with `TryNextMethod()'.
## This default method deals with the information that is available by
## the calls of `InstallFactorMaintenance' in the {\GAP} library.
##
DeclareOperation( "UseFactorRelation",
[ IsCollection, IsObject, IsCollection ] );
IsIdenticalObjObjObjX := function( F1, F2, F3 )
return IsIdenticalObj( F1, F2 );
end;
InstallMethod( UseFactorRelation,
"default method that checks maintenances and then returns `true'",
true,
[ IsCollection, IsObject, IsCollection ],
# Make sure that this method is installed with ``real'' rank zero.
- 2 * RankFilter( IsCollection )-RankFilter(IsObject),
function( num, den, fac )
local entry;
for entry in FACTOR_MAINTAINED_INFO do
if entry[1]( num ) and entry[2]( den ) and entry[3]( fac )
and not entry[5]( fac ) then
entry[6]( fac, entry[4]( num ) );
fi;
od;
return true;
end );
#############################################################################
##
#F InstallFactorMaintenance( <opr>, <numer_req>, <denom_req>, <factor_req> )
##
## <opr> must be a property or an attribute.
## The call of `InstallFactorMaintenance' has the effect that
## for collections <N>, <D>, <F> in the filters <numer_req>, <denom_req>,
## and <factor_req>, respectively,
## the call `UseFactorRelation( <N>, <D>, <F> )'
## (see~"UseFactorRelation")
## sets a known value of <opr> for <N> as value of <opr> also for <F>.
## A typical example for which `InstallFactorMaintenance' is
## applied is given by `<opr> = IsFinite',
## `<numer_req> = IsCollection and IsFinite', `<denom_req> = IsCollection',
## and `<factor_req> = IsCollection'.
##
## For the other direction, if <numer_req> involves the filter <opr>
## then a known `false' value of <opr> for $F$ implies a `false'
## value for $D$ provided that $D$ lies in the filter obtained from
## <numer_req> by removing <opr>.
##
## Note that an implication of a factor relation holds in particular for the
## case of isomorphisms.
## So one need *not* install an isomorphism maintained method when
## a factor maintained method is already installed.
## For example, `UseIsomorphismRelation' (see~"UseIsomorphismRelation")
## will transfer a known `IsFinite' value because of the installed factor
## maintained method.
##
BIND_GLOBAL( "InstallFactorMaintenance",
function( opr, numer_req, denom_req, factor_req )
local tester;
# Information that is maintained under taking factors
# is especially maintained under isomorphisms.
InstallIsomorphismMaintenance( opr, numer_req, factor_req );
tester:= Tester( opr );
ADD_LIST( FACTOR_MAINTAINED_INFO,
[ IsCollection and Tester( numer_req ) and numer_req and tester,
Tester( denom_req ) and denom_req,
IsCollection and Tester( factor_req ) and factor_req,
opr,
tester,
Setter( opr ) ] );
#T not yet available in the new implementation
# if FLAGS_FILTER( opr ) <> false
# and IS_EQUAL_FLAGS( FLAGS_FILTER( opr and factor_req ),
# FLAGS_FILTER( numer_req ) ) then
# InstallMethod( UseFactorRelation, infostring, IsIdenticalObjObjObjX,
# [ factor_req, denom_req, factor_req ], 0,
# function( numer, denom, factor )
# if tester( factor ) and not opr( factor ) then
# setter( numer, false );
# fi;
# TryNextMethod();
# end );
# fi;
end );
#############################################################################
##
#O Iterator( <C> ) . . . . . . . . . . . . iterator for a list or collection
#O Iterator( <list> ) . . . . . . . . . . iterator for a list or collection
##
## Iterators provide a possibility to loop over the elements of a
## (countable) collection <C> or a list <list>, without repetition.
## For many collections <C>,
## an iterator of <C> need not store all elements of <C>,
## for example it is possible to construct an iterator of some infinite
## domains, such as the field of rational numbers.
##
## `Iterator' returns a mutable *iterator* <iter> for its argument.
## If this is a list <list> (which may contain holes),
## then <iter> iterates over the elements (but not the holes) of <list> in
## the same order (see~"IteratorList" for details).
## If this is a collection <C> but not a list then <iter> iterates over the
## elements of <C> in an unspecified order,
## which may change for repeated calls of `Iterator'.
## Because iterators returned by `Iterator' are mutable
## (see~"Mutability and Copyability"),
## each call of `Iterator' for the same argument returns a *new* iterator.
## Therefore `Iterator' is not an attribute (see~"Attributes").
##
## The only operations for iterators are `IsDoneIterator',
## `NextIterator', and `ShallowCopy'.
## In particular, it is only possible to access the next element of the
## iterator with `NextIterator' if there is one, and this can be checked
## with `IsDoneIterator' (see~"NextIterator").
## For an iterator <iter>, `ShallowCopy( <iter> )' is a mutable iterator
## <new> that iterates over the remaining elements independent of <iter>;
## the results of `IsDoneIterator' for <iter> and <new> are equal,
## and if <iter> is mutable then also the results of `NextIterator' for
## <iter> and <new> are equal;
## note that `=' is not defined for iterators,
## so the equality of two iterators cannot be checked with `='.
##
## When `Iterator' is called for a *mutable* collection <C> then it is not
## defined whether <iter> respects changes to <C> occurring after the
## construction of <iter>, except if the documentation explicitly promises
## a certain behaviour. The latter is the case if the argument is a mutable
## list <list> (see~"IteratorList" for subtleties in this case).
##
## It is possible to have `for'-loops run over mutable iterators instead of
## lists.
##
## In some situations, one can construct iterators with a special
## succession of elements,
## see~"IteratorByBasis" for the possibility to loop over the elements
## of a vector space w.r.t.~a given basis.
#T (also for perm. groups, w.r.t. a given stabilizer chain?)
##
## For lists, `Iterator' is implemented by `IteratorList( <list> )'.
## For collections that are not lists, the default method is
## `IteratorList( Enumerator( <C> ) )'.
## Better methods depending on <C> should be provided if possible.
##
## For random access to the elements of a (possibly infinite) collection,
## *enumerators* are used.
## See~"Enumerators" for the facility to compute a list from <C>,
## which provides a (partial) mapping from <C> to the positive integers.
##
#T We wanted to admit an iterator as first argument of `Filtered',
#T `First', `ForAll', `ForAny', `Number'.
#T This is not yet implemented.
#T (Note that the iterator is changed in the call,
#T so the meaning of the operations would be slightly abused,
#T or we must define that these operations first make a shallow copy.)
#T (Additionally, the unspecified order of the elements makes it
#T difficult to define what `First' and `Filtered' means for an iterator.)
##
DeclareOperation( "Iterator", [ IsListOrCollection ] );
#############################################################################
##
#O IteratorSorted( <C> ) . . . . . . . . . . . set iterator for a collection
#O IteratorSorted( <list> ) . . . . . . . . . . . . set iterator for a list
##
## `IteratorSorted' returns a mutable iterator.
## The argument must be a collection <C> or a list <list> that is not
## necessarily dense but whose elements lie in the same family
## (see~"Families").
## It loops over the different elements in sorted order.
##
## For collections <C> that are not lists, the generic method is
## `IteratorList( EnumeratorSorted( <C> ) )'.
##
DeclareOperation( "IteratorSorted", [ IsListOrCollection ] );
#############################################################################
##
#C IsIterator( <obj> ) . . . . . . . . . . test if an object is an iterator
##
## Every iterator lies in the category `IsIterator'.
##
DeclareCategory( "IsIterator", IsObject );
#############################################################################
##
#O IsDoneIterator( <iter> ) . . . . . . . test if an iterator is exhausted
##
## If <iter> is an iterator for the list or collection $C$ then
## `IsDoneIterator( <iter> )' is `true' if all elements of $C$ have been
## returned already by `NextIterator( <iter> )', and `false' otherwise.
##
DeclareOperation( "IsDoneIterator", [ IsIterator ] );
#############################################################################
##
#O NextIterator( <iter> ) . . . . . . . . . . next element from an iterator
##
## Let <iter> be a mutable iterator for the list or collection $C$.
## If `IsDoneIterator( <iter> )' is `false' then `NextIterator' is
## applicable to <iter>, and the result is the next element of $C$,
## according to the succession defined by <iter>.
##
## If `IsDoneIterator( <iter> )' is `true' then it is not defined what
## happens if `NextIterator' is called for <iter>;
## that is, it may happen that an error is signalled or that something
## meaningless is returned, or even that {\GAP} crashes.
##
DeclareOperation( "NextIterator", [ IsIterator and IsMutable ] );
#############################################################################
##
#F TrivialIterator( <elm> )
##
## is a mutable iterator for the collection `[ <elm> ]' that consists of
## exactly one element <elm> (see~"IsTrivial").
##
DeclareGlobalFunction( "TrivialIterator" );
#############################################################################
##
#F IteratorByFunctions( <record> )
##
## `IteratorByFunctions' returns a (mutable) iterator <iter> for which
## `NextIterator', `IsDoneIterator', and `ShallowCopy'
## are computed via prescribed functions.
##
## Let <record> be a record with at least the following components.
## \beginitems
## `NextIterator' &
## a function taking one argument <iter>,
## which returns the next element of <iter> (see~"NextIterator");
## for that, the components of <iter> are changed,
##
## `IsDoneIterator' &
## a function taking one argument <iter>,
## which returns `IsDoneIterator( <iter> )' (see~"IsDoneIterator");
##
## `ShallowCopy' &
## a function taking one argument <iter>,
## which returns a record for which `IteratorByFunctions' can be called
## in order to create a new iterator that is independent of <iter> but
## behaves like <iter> w.r.t. the operations `NextIterator' and
## `IsDoneIterator'.
## \enditems
## Further (data) components may be contained in <record> which can be used
## by these function.
##
## `IteratorByFunctions' does *not* make a shallow copy of <record>,
## this record is changed in place
## (see~"prg:Creating Objects" in ``Programming in {\GAP}'').
##
DeclareGlobalFunction( "IteratorByFunctions" );
#############################################################################
##
#P IsEmpty( <C> ) . . . . . . . . . . . . . . test if a collection is empty
#P IsEmpty( <list> ) . . . . . . . . . . . . . test if a collection is empty
##
## `IsEmpty' returns `true' if the collection <C> resp.~the list <list> is
## *empty* (that is it contains no elements), and `false' otherwise.
##
DeclareProperty( "IsEmpty", IsListOrCollection );
#############################################################################
##
#P IsTrivial( <C> ) . . . . . . . . . . . . test if a collection is trivial
##
## `IsTrivial' returns `true' if the collection <C> consists of exactly one
## element.
##
#T 1996/08/08 M.Schoenert is this a sensible definition?
##
DeclareProperty( "IsTrivial", IsCollection );
InstallFactorMaintenance( IsTrivial,
IsCollection and IsTrivial, IsObject, IsCollection );
#############################################################################
##
#P IsNonTrivial( <C> ) . . . . . . . . . test if a collection is nontrivial
##
## `IsNonTrivial' returns `true' if the collection <C> is empty or consists
## of at least two elements (see~"IsTrivial").
##
#T I need this to distinguish trivial rings-with-one from fields!
#T (indication to introduce antifilters?)
#T 1996/08/08 M.Schoenert is this a sensible definition?
##
DeclareProperty( "IsNonTrivial", IsCollection );
#############################################################################
##
#P IsFinite( <C> ) . . . . . . . . . . . . . test if a collection is finite
##
## `IsFinite' returns `true' if the collection <C> is finite, and `false'
## otherwise.
##
## The default method for `IsFinite' checks the size (see~"Size") of <C>.
##
## Methods for `IsFinite' may call `Size',
## but methods for `Size' must *not* call `IsFinite'.
##
DeclareProperty( "IsFinite", IsCollection );
InstallSubsetMaintenance( IsFinite,
IsCollection and IsFinite, IsCollection );
InstallFactorMaintenance( IsFinite,
IsCollection and IsFinite, IsObject, IsCollection );
InstallTrueMethod( IsFinite, IsTrivial );
#############################################################################
##
#P IsWholeFamily( <C> ) . . test if a collection contains the whole family
##
## `IsWholeFamily' returns `true' if the collection <C> contains the whole
## family (see~"Families") of its elements.
##
DeclareProperty( "IsWholeFamily", IsCollection );
#############################################################################
##
#A Size( <C> ) . . . . . . . . . . . . . . . . . . . . size of a collection
#A Size( <list> ) . . . . . . . . . . . . . . . . . . size of a collection
##
## `Size' returns the size of the collection <C>, which is either an integer
## or `infinity'.
## The argument may also be a list <list>, in which case the result is the
## length of <list> (see~"Length").
##
## The default method for `Size' checks the length of an enumerator of <C>.
##
## Methods for `IsFinite' may call `Size',
## but methods for `Size' must not call `IsFinite'.
##
DeclareAttribute( "Size", IsListOrCollection );
InstallIsomorphismMaintenance( Size, IsCollection, IsCollection );
#############################################################################
##
#A Representative( <C> ) . . . . . . . . . . . . one element of a collection
##
## `Representative' returns a *representative* of the collection <C>.
##
## Note that `Representative' is free in choosing a representative if
## there are several elements in <C>.
## It is not even guaranteed that `Representative' returns the same
## representative if it is called several times for one collection.
## The main difference between `Representative' and `Random'
## (see~"Random") is that `Representative' is free to choose a value that is
## cheap to compute,
## while `Random' must make an effort to randomly distribute its answers.
##
## If <C> is a domain then there are methods for `Representative' that try
## to fetch an element from any known generator list of <C>,
## see~"Domains and their Elements".
## Note that `Representative' does not try to *compute* generators of <C>,
## thus `Representative' may give up and signal an error if <C> has no
## generators stored at all.
##
DeclareAttribute( "Representative", IsListOrCollection );
#############################################################################
##
#A RepresentativeSmallest( <C> ) . . . . . smallest element of a collection
##
## returns the smallest element in the collection <C>, w.r.t.~the ordering
## `\<'.
## While the operation defaults to comparing all elements,
## better methods are installed for some collections.
##
DeclareAttribute( "RepresentativeSmallest", IsListOrCollection );
#############################################################################
##
#O Random( <C> ) . . . . . . . . . . random element of a list or collection
#O Random( <list> ) . . . . . . . . random element of a list or collection
##
## `Random' returns a (pseudo-)random element of the collection <C>
## respectively the list <list>.
##
## The distribution of elements returned by `Random' depends on the
## argument. For a list <list>, all elements are equally likely. The same
## holds usually for finite collections <C> that are not lists. For
## infinite collections <C> some reasonable distribution is used.
##
## See the chapters of the various collections to find out
## which distribution is being used.
##
## For some collections ensuring a reasonable distribution can be
## difficult and require substantial runtime.
## If speed at the cost of equal distribution is desired,
## the operation `PseudoRandom' should be used instead.
##
## Note that `Random' is of course *not* an attribute.
##
DeclareOperation( "Random", [ IsListOrCollection ] );
DeclareOperation( "Random", [ IS_INT, IS_INT ] );
##
#2
## The method used by {\GAP} to obtain random elements may depend on the
## type object.
##
## Many random methods in the library are eventually based on the function
## `RandomList'. As `RandomList' is restricted to lists of up to $2^{28}$
## elements, this may create problems for very large collections. Also note
## that the method used by `RandomList' is intended to provide a fast
## algorithm rather than to produce high quality randomness for
## statistical purposes.
##
## If you implement your own `Random' methods we recommend
## that they initialize their seed to a defined value when they are loaded
## to permit to reproduce calculations even if they involved random
## elements.
#############################################################################
##
#F RandomList( <list> )
##
## \index{random seed}
## For a dense list <list> of up to $2^{28}$ elements,
## `RandomList' returns a (pseudo-)random element with equal distribution.
##
## The algorithm used is an additive number generator (Algorithm A in
## section~3.2.2 of \cite{TACP2} with lag 30)
##
## This random number generator is (deliberately) initialized to the same
## values when {\GAP} is started, so different runs of {\GAP} with the same
## input will always produce the same result, even if random calculations
## are involved.
##
## See `StateRandom' for a description on how to reset the random number
## generator to a previous state.
##
DeclareSynonym( "RandomList", RANDOM_LIST);
#############################################################################
##
#O PseudoRandom( <C> ) . . . . . . . . pseudo random element of a collection
#O PseudoRandom( <list> ) . . . . . . . . . pseudo random element of a list
##
## `PseudoRandom' returns a pseudo random element of the collection <C>
## respectively the list <list>, which can be roughly described as follows.
## For a list <list>, `PseudoRandom' returns the same as `Random'.
## For collections <C> that are not lists,
## the elements returned by `PseudoRandom' are *not* necessarily equally
## distributed, even for finite collections <C>;
## the idea is that `Random' (see~"Random") returns elements according to
## a reasonable distribution, `PseudoRandom' returns elements that are
## cheap to compute but need not satisfy this strong condition, and
## `Representative' (see~"Representative") returns arbitrary elements,
## probably the same element for each call.
##
DeclareOperation( "PseudoRandom", [ IsListOrCollection ] );
#############################################################################
##
#A PseudoRandomSeed( <C> )
##
DeclareAttribute( "PseudoRandomSeed", IsListOrCollection, "mutable" );
#############################################################################
##
#A Enumerator( <C> ) . . . . . . . . . . . list of elements of a collection
#A Enumerator( <list> ) . . . . . . . . . . . . list of elements of a list
##
## `Enumerator' returns an immutable list <enum>.
## If the argument is a list <list> (which may contain holes),
## then `Length( <enum> )' is `Length( <list> )',
## and <enum> contains the elements (and holes) of <list> in the same order.
## If the argument is a collection <C> that is not a list,
## then `Length( <enum> )' is the number of different elements of <C>,
## and <enum> contains the different elements of <C> in an unspecified
## order, which may change for repeated calls of `Enumerator'.
## `<enum>[<pos>]' may not execute in constant time
## (see~"IsConstantTimeAccessList"),
## and the size of <enum> in memory is as small as is feasible.
##
## For lists <list>, the default method is `Immutable'.
## For collections <C> that are not lists, there is no default method.
##
DeclareAttribute( "Enumerator", IsListOrCollection );
#############################################################################
##
#A EnumeratorSorted( <C> ) . . . . . proper set of elements of a collection
#A EnumeratorSorted( <list> ) . . . . . . proper set of elements of a list
##
## `EnumeratorSorted' returns an immutable list <enum>.
## The argument must be a collection <C> or a list <list> which may contain
## holes but whose elements lie in the same family (see~"Families").
## `Length( <enum> )' is the number of different elements of
## <C> resp.~<list>,
## and <enum> contains the different elements in sorted order, w.r.t.~`\<'.
## `<enum>[<pos>]' may not execute in constant time
## (see~"IsConstantTimeAccessList"),
## and the size of <enum> in memory is as small as is feasible.
##
DeclareAttribute( "EnumeratorSorted", IsListOrCollection );
#############################################################################
##
#F EnumeratorOfSubset( <list>, <blist>[, <ishomog>] )
##
## Let <list> be a list, and <blist> a Boolean list of the same length
## (see~"Boolean Lists").
## `EnumeratorOfSubset' returns a list <new> of length equal to the number
## of `true' entries in <blist>,
## such that `<new>[i]', if bound, equals the entry of <list> at the <i>-th
## `true' position in <blist>.
##
## If <list> is homogeneous then also <new> is homogeneous.
## If <list> is *not* homogeneous then the third argument <ishomog> must
## be present and equal to `true' or `false', saying whether or not <new> is
## homogeneous.
##
## This construction is used for example in the situation that <list> is an
## enumerator of a large set, and <blist> describes a union of orbits in an
## action on this set.
##
DeclareGlobalFunction( "EnumeratorOfSubset" );
#############################################################################
##
#F EnumeratorByFunctions( <D>, <record> )
#F EnumeratorByFunctions( <Fam>, <record> )
##
## `EnumeratorByFunctions' returns an immutable, dense, and duplicate-free
## list <enum> for which `IsBound', element access, `Length', and `Position'
## are computed via prescribed functions.
##
## Let <record> be a record with at least the following components.
## \beginitems
## `ElementNumber' &
## a function taking two arguments <enum> and <pos>,
## which returns `<enum>[ <pos> ]' (see~"Basic Operations for Lists");
## it can be assumed that the argument <pos> is a positive integer,
## but <pos> may be larger than the length of <enum> (in which case
## an error must be signalled);
## note that the result must be immutable since <enum> itself is
## immutable,
##
## `NumberElement' &
## a function taking two arguments <enum> and <elm>,
## which returns `Position( <enum>, <elm> )' (see~"Position");
## it cannot be assumed that <elm> is really contained in <enum>
## (and `fail' must be returned if not);
## note that for the three argument version of `Position', the
## method that is available for duplicate-free lists suffices.
## \enditems
## Further (data) components may be contained in <record> which can be used
## by these function.
##
## If the first argument is a domain <D> then <enum> lists the elements of
## <D> (in general <enum> is *not* sorted),
## and methods for `Length', `IsBound', and `PrintObj' may use <D>.
#T is this really true for `Length'?
##
## If one wants to describe the result without creating a domain then the
## elements are given implicitly by the functions in <record>,
## and the first argument must be a family <Fam> which will become the
## family of <enum>;
## if <enum> is not homogeneous then <Fam> must be `ListsFamily',
## otherwise it must be the collections family of any element in <enum>.
## In this case, additionally the following component in <record> is
## needed.
## \beginitems
## `Length' &
## a function taking the argument <enum>,
## which returns the length of <enum> (see~"Length").
## \enditems
##
## The following components are optional; they are used if they are present
## but default methods are installed for the case that they are missing.
## \beginitems
## `IsBound\\[\\]' &
## a function taking two arguments <enum> and <k>,
## which returns `IsBound( <enum>[ <k> ] )'
## (see~"Basic Operations for Lists");
## if this component is missing then `Length' is used for computing the
## result,
##
## `Membership' &
## a function taking two arguments <elm> and <enum>,
## which returns `true' is <elm> is an element of <enum>,
## and `false' otherwise (see~"Basic Operations for Lists");
## if this component is missing then `NumberElement' is used
## for computing the result,
##
## `AsList' &
## a function taking one argument <enum>, which returns a list with the
## property that the access to each of its elements will take roughly
## the same time (see~"IsConstantTimeAccessList");
## if this component is missing then `ConstantTimeAccessList' is used
## for computing the result,
##
## `ViewObj' and `PrintObj' &
## two functions that print what one wants to be printed when
## `View( <enum> )' or `Print( <enum> )' is called
## (see~"View and Print"),
## if the `ViewObj' component is missing then the `PrintObj' method is
## used as a default.
## \enditems
##
## If the result is known to have additional properties such as being
## strictly sorted (see~"IsSSortedList") then it can be useful to set
## these properties after the construction of the enumerator,
## before it is used for the first time.
## And in the case that a new sorted enumerator of a domain is implemented
## via `EnumeratorByFunctions', and this construction is installed as a
## method for the operation `Enumerator' (see~"Enumerator"),
## then it should be installed also as a method for `EnumeratorSorted'
## (see~"EnumeratorSorted").
##
## Note that it is *not* checked that `EnumeratorByFunctions' really returns
## a dense and duplicate-free list.
## `EnumeratorByFunctions' does *not* make a shallow copy of <record>,
## this record is changed in place
## (see~"prg:Creating Objects" in ``Programming in {\GAP}'').
##
## It would be easy to implement a slightly generalized setup for
## enumerators that need not be duplicate-free (where the three argument
## version of `Position' is supported),
## but the resulting overhead for the methods seems not to be justified.
##
DeclareGlobalFunction( "EnumeratorByFunctions" );
#############################################################################
##
#A UnderlyingCollection( <enum> )
##
## An enumerator of a domain can delegate the task to compute its length to
## `Size' for the underlying domain, and `ViewObj' and `PrintObj' methods
## may refer to this domain.
##
DeclareAttribute( "UnderlyingCollection", IsListOrCollection );
#############################################################################
##
#F List( <list> ) . . . . . . . . . . . . list of elements of a collection
#F List( <C> )
#F List( <list>, <func> )
##
## In the first form, where <list> is a list (not necessarily dense or
## homogeneous), `List' returns a new mutable list <new> that contains
## the elements (and the holes) of <list> in the same order;
## thus `List' does the same as `ShallowCopy' (see~"ShallowCopy")
## in this case.
##
## In the second form, where <C> is a collection (see~"Collections")
## that is not a list,
## `List' returns a new mutable list <new> such that `Length( <new> )'
## is the number of different elements of <C>, and <new> contains the
## different elements of <C> in an unspecified order which may change
## for repeated calls.
## `<new>[<pos>]' executes in constant time
## (see~"IsConstantTimeAccessList"),
## and the size of <new> is proportional to its length.
## The generic method for this case is `ShallowCopy( Enumerator( <C> ) )'.
#T this is not reasonable since `ShallowCopy' need not guarantee to return
#T a constant time access list
##
## In the third form, for a dense list <list> and a function <func>,
## which must take exactly one argument, `List' returns a new mutable list
## <new> given by $<new>[i] = <func>( <list>[i] )$.
##
DeclareGlobalFunction( "List" );
DeclareOperation( "ListOp", [ IsListOrCollection ] );
DeclareOperation( "ListOp", [ IsListOrCollection, IsFunction ] );
#############################################################################
##
#O SortedList( <C> )
#O SortedList( <list> )
##
## `SortedList' returns a new mutable and dense list <new>.
## The argument must be a collection <C> or a list <list> which may contain
## holes but whose elements lie in the same family (see~"Families").
## `Length( <new> )' is the number of elements of <C> resp.~<list>,
## and <new> contains the elements in sorted order, w.r.t.~`\<='.
## `<new>[<pos>]' executes in constant time
## (see~"IsConstantTimeAccessList"),
## and the size of <new> in memory is proportional to its length.
##
DeclareOperation( "SortedList", [ IsListOrCollection ] );
#############################################################################
##
#O SSortedList( <C> ) . . . . . . . . . . . set of elements of a collection
#O SSortedList( <list> ) . . . . . . . . . . . . . set of elements of a list
#O Set( <C> )
##
## `SSortedList' (``strictly sorted list'') returns a new dense, mutable,
## and duplicate free list <new>.
## The argument must be a collection <C> or a list <list> which may contain
## holes but whose elements lie in the same family (see~"Families").
## `Length( <new> )' is the number of different elements of <C>
## resp.~<list>,
## and <new> contains the different elements in strictly sorted order,
## w.r.t.~`\<'.
## `<new>[<pos>]' executes in constant time
## (see~"IsConstantTimeAccessList"),
## and the size of <new> in memory is proportional to its length.
##
## `Set' is simply a synonym for `SSortedList'.
##
#T For collections that are not lists, the default method is
#T `ShallowCopy( EnumeratorSorted( <C> ) )'.
##
DeclareOperation( "SSortedList", [ IsListOrCollection ] );
DeclareSynonym( "Set", SSortedList );
#############################################################################
##
#A AsList( <C> ) . . . . . . . . . . . . . list of elements of a collection
#A AsList( <list> ) . . . . . . . . . . . . . . list of elements of a list
##
## `AsList' returns a immutable list <imm>.
## If the argument is a list <list> (which may contain holes),
## then `Length( <imm> )' is `Length( <list> )',
## and <imm> contains the elements (and holes) of <list> in the same order.
## If the argument is a collection <C> that is not a list,
## then `Length( <imm> )' is the number of different elements of <C>,
## and <imm> contains the different elements of <C> in an unspecified
## order, which may change for repeated calls of `AsList'.
## `<imm>[<pos>]' executes in constant time
## (see~"IsConstantTimeAccessList"),
## and the size of <imm> in memory is proportional to its length.
##
## If you expect to do many element tests in the resulting list, it might
## be worth to use a sorted list instead, using `AsSSortedList'.
##
#T For both lists and collections, the default method is
#T `ConstantTimeAccessList( Enumerator( <C> ) )'.
##
DeclareAttribute( "AsList", IsListOrCollection );
#############################################################################
##
#A AsSortedList( <C> )
#A AsSortedList( <list> )
##
## `AsSortedList' returns a dense and immutable list <imm>.
## The argument must be a collection <C> or a list <list> which may contain
## holes but whose elements lie in the same family (see~"Families").
## `Length( <imm> )' is the number of elements of <C> resp.~<list>,
## and <imm> contains the elements in sorted order, w.r.t.~`\<='.
## `<new>[<pos>]' executes in constant time
## (see~"IsConstantTimeAccessList"),
## and the size of <imm> in memory is proportional to its length.
##
## The only difference to the operation `SortedList' (see~"SortedList")
## is that `AsSortedList' returns an *immutable* list.
##
DeclareAttribute( "AsSortedList", IsListOrCollection );
#############################################################################
##
#A AsSSortedList( <C> ) . . . . . . . . . . set of elements of a collection
#A AsSSortedList( <list> ) . . . . . . . . . . . . set of elements of a list
#A AsSet( <C> )
##
## `AsSSortedList' (``as strictly sorted list'') returns a dense, immutable,
## and duplicate free list <imm>.
## The argument must be a collection <C> or a list <list> which may contain
## holes but whose elements lie in the same family (see~"Families").
## `Length( <imm> )' is the number of different elements of <C>
## resp.~<list>,
## and <imm> contains the different elements in strictly sorted order,
## w.r.t.~`\<'.
## `<imm>[<pos>]' executes in constant time
## (see~"IsConstantTimeAccessList"),
## and the size of <imm> in memory is proportional to its length.
##
## Because the comparisons required for sorting can be very expensive for
## some kinds of objects, you should use `AsList' instead if you do not
## require the result to be sorted.
##
## The only difference to the operation `SSortedList' (see~"SSortedList")
## is that `AsSSortedList' returns an *immutable* list.
##
## `AsSet' is simply a synonym for `AsSSortedList'.
##
## In general a function that returns a set of elements is free, in fact
## encouraged, to return a domain instead of the proper set of its elements.
## This allows one to keep a given structure, and moreover the
## representation by a domain object is usually more space efficient.
## `AsSSortedList' must of course *not* do this,
## its only purpose is to create the proper set of elements.
##
#T For both lists and collections, the default method is
#T `ConstantTimeAccessList( EnumeratorSorted( <C> ) )'.
##
DeclareAttribute( "AsSSortedList", IsListOrCollection );
DeclareSynonym( "AsSet", AsSSortedList );
#############################################################################
##
#A AsSSortedListNonstored( <C> )
##
## returns the `AsSSortedList(<C>)' but ensures that this list (nor a
## permutation or substantial subset) will not be
## stored in attributes of <C> unless such a list is already stored.
## This permits to obtain an element list once
## without danger of clogging up memory in the long run.
##
## Because of this guarantee of nonstorage, methods for
## `AsSSortedListNonstored' may not default to `AsSSortedList', but only
## vice versa.
##
DeclareOperation( "AsSSortedListNonstored", [IsListOrCollection] );
#############################################################################
##
#F Elements( <C> )
##
## `Elements' does the same as `AsSSortedList' (see~"AsSSortedList"),
## that is, the return value is a strictly sorted list of the elements in
## the list or collection <C>.
##
## `Elements' is only supported for backwards compatibility.
## In many situations, the sortedness of the ``element list'' for a
## collection is in fact not needed, and one can save a lot of time by
## asking for a list that is *not* necessarily sorted, using `AsList'
## (see~"AsList").
## If one is really interested in the strictly sorted list of elements in
## <C> then one should use `AsSet' or `AsSSortedList' instead.
##
DeclareGlobalFunction( "Elements" );
#############################################################################
##
#F Sum( <list>[, <init>] ) . . . . . . . . . . sum of the elements of a list
#F Sum( <C>[, <init>] ) . . . . . . . . sum of the elements of a collection
#F Sum( <list>, <func>[, <init>] ) . . . . . sum of images under a function
#F Sum( <C>, <func>[, <init>] ) . . . . . . sum of images under a function
##
## In the first two forms `Sum' returns the sum of the elements of the
## dense list <list> resp.~the collection <C> (see~"Collections").
## In the last two forms `Sum' applies the function <func>,
## which must be a function taking one argument,
## to the elements of the dense list <list> resp.~the collection <C>,
## and returns the sum of the results.
## In either case `Sum' returns `0' if the first argument is empty.
##
## The general rules for arithmetic operations apply
## (see~"Mutability Status and List Arithmetic"),
## so the result is immutable if and only if all summands are immutable.
##
## If <list> or <C> contains exactly one element then this element (or its
## image under <func> if applicable) itself is returned, not a shallow copy
## of this element.
##
## If an additional initial value <init> is given,
## `Sum' returns the sum of <init> and the elements of the first argument
## resp.~of their images under the function <func>.
## This is useful for example if the first argument is empty and a different
## zero than `0' is desired, in which case <init> is returned.
##
DeclareGlobalFunction( "Sum" );
#############################################################################
##
#O SumOp( <C> )
#O SumOp( <C>, <func> )
#O SumOp( <C>, <init> )
#O SumOp( <C>, <func>, <init> )
##
## `SumOp' is the operation called by `Sum' if <C> is not an internal list.
##
DeclareOperation( "SumOp", [ IsListOrCollection ] );
#############################################################################
##
#F Product( <list>[, <init>] ) . . . . . . product of the elements of a list
#F Product( <C>[, <init>] ) . . . . product of the elements of a collection
#F Product( <list>, <func>[, <init>] ) . product of images under a function
#F Product( <C>, <func>[, <init>] ) . . product of images under a function
##
## In the first two forms `Product' returns the product of the elements of
## the dense list <list> resp.~the collection <C> (see~"Collections").
## In the last two forms `Product' applies the function <func>,
## which must be a function taking one argument,
## to the elements of the dense list <list> resp.~the collection <C>,
## and returns the product of the results.
## In either case `Product' returns `1' if the first argument is empty.
##
## The general rules for arithmetic operations apply
## (see~"Mutability Status and List Arithmetic"),
## so the result is immutable if and only if all summands are immutable.
##
## If <list> or <C> contains exactly one element then this element (or its
## image under <func> if applicable) itself is returned, not a shallow copy
## of this element.
##
## If an additional initial value <init> is given,
## `Product' returns the product of <init> and the elements of the first
## argument resp.~of their images under the function <func>.
## This is useful for example if the first argument is empty and a different
## identity than `1' is desired, in which case <init> is returned.
##
DeclareGlobalFunction( "Product" );
#############################################################################
##
#O ProductOp( <C> )
#O ProductOp( <C>, <func> )
#O ProductOp( <C>, <init> )
#O ProductOp( <C>, <func>, <init> )
##
## `ProductOp' is the operation called by `Product' if <C> is not
## an internal list.
##
DeclareOperation( "ProductOp", [ IsListOrCollection ] );
#############################################################################
##
#F Filtered( <list>, <func> ) . . . . extract elements that have a property
#F Filtered( <C>, <func> ) . . . . . . extract elements that have a property
##
## returns a new list that contains those elements of the list <list> or
## collection <C> (see~"Collections"), respectively,
## for which the unary function <func> returns `true'.
##
## If the first argument is a list, the order of the elements in the result
## is the same as the order of the corresponding elements of <list>.
## If an element for which <func> returns `true' appears several times in
## <list> it will also appear the same number of times in the result.
## <list> may contain holes, they are ignored by `Filtered'.
##
## For each element of <list> resp.~<C>, <func> must return either `true' or
## `false', otherwise an error is signalled.
##
## The result is 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 argument list (see~"Identical Lists").
##
## List assignment using the operator `{}' (see~"List Assignment") can be
## used to extract elements of a list according to indices given in another
## list.
##
DeclareGlobalFunction( "Filtered" );
#############################################################################
##
#O FilteredOp( <C>, <func> )
##
## `FilteredOp' is the operation called by `Filtered' if <C> is not
## an internal list.
##
DeclareOperation( "FilteredOp", [ IsListOrCollection, IsFunction ] );
#############################################################################
##
#F Number( <list> )
#F Number( <list>, <func> ) . . . . . . count elements that have a property
#F Number( <C>, <func> ) . . . . . . . . count elements that have a property
##
## In the first form, `Number' returns the number of bound entries in the
## list <list>.
## For dense lists `Number', `Length' (see~"Length"),
## and `Size' (see~"Size") return the same value;
## for lists with holes `Number' returns the number of bound entries,
## `Length' returns the largest index of a bound entry,
## and `Size' signals an error.
##
## In the last two forms, `Number' returns the number of elements of the
## list <list> resp.~the collection <C> for which the unary function <func>
## returns `true'.
## If an element for which <func> returns `true' appears several times in
## <list> it will also be counted the same number of times.
##
## For each element of <list> resp.~<C>, <func> must return either `true' or
## `false', otherwise an error is signalled.
##
## `Filtered' (see~"Filtered") allows you to extract the elements of a list
## that have a certain property.
##
DeclareGlobalFunction( "Number" );
#############################################################################
##
#O NumberOp( <C>, <func> )
##
## `NumberOp' is the operation called by `Number' if <C> is not
## an internal list.
##
DeclareOperation( "NumberOp", [ IsListOrCollection, IsFunction ] );
#############################################################################
##
#F ForAll( <list>, <func> )
#F ForAll( <C>, <func> )
##
## tests whether the unary function <func> returns `true' for all elements
## in the list <list> resp.~the collection <C>.
##
DeclareGlobalFunction( "ForAll" );
#############################################################################
##
#O ForAllOp( <C>, <func> )
##
## `ForAllOp' is the operation called by `ForAll' if <C> is not
## an internal list.
##
DeclareOperation( "ForAllOp", [ IsListOrCollection, IsFunction ] );
#############################################################################
##
#F ForAny( <list>, <func> )
#F ForAny( <C>, <func> )
##
## tests whether the unary function <func> returns `true' for at least one
## element in the list <list> resp.~the collection <C>.
##
DeclareGlobalFunction( "ForAny" );
#############################################################################
##
#O ForAnyOp( <C>, <func> )
##
## `ForAnyOp' is the operation called by `ForAny' if <C> is not
## an internal list.
##
DeclareOperation( "ForAnyOp", [ IsListOrCollection, IsFunction ] );
#############################################################################
##
#O ListX( <arg1>, <arg2>, ... <argn>, <func> )
##
## `ListX' returns a new list constructed from the arguments.
##
## Each of the arguments `<arg1>, <arg2>, ... <argn>' must be one of the
## following:
## \beginitems
## a list or collection &
## this introduces a new for-loop in the sequence of nested
## for-loops and if-statements;
##
## a function returning a list or collection &
## this introduces a new for-loop in the sequence of nested
## for-loops and if-statements, where the loop-range depends on
## the values of the outer loop-variables; or
##
## a function returning `true' or `false' &
## this introduces a new if-statement in the sequence of nested
## for-loops and if-statements.
## \enditems
##
## The last argument <func> must be a function,
## it is applied to the values of the loop-variables
## and the results are collected.
##
## Thus `ListX( <list>, <func> )' is the same as `List( <list>, <func> )',
## and `ListX( <list>, <func>, x -> x )' is the same as
## `Filtered( <list>, <func> )'.
##
## As a more elaborate example, assume <arg1> is a list or collection,
## <arg2> is a function returning `true' or `false',
## <arg3> is a function returning a list or collection, and
## <arg4> is another function returning `true' or `false',
## then
##
## \)\kernttindent<result> := ListX( <arg1>, <arg2>, <arg3>, <arg4>, <func> );
##
## is equivalent to
##
## \){\kernttindent}<result> := [];
## \){\kernttindent}for v1 in <arg1> do
## \){\kernttindent\quad}if <arg2>( v1 ) then
## \){\kernttindent\quad\quad}for v2 in <arg3>( v1 ) do
## \){\kernttindent\quad\quad\quad}if <arg4>( v1, v2 ) then
## \){\kernttindent\quad\quad\quad\quad}Add( <result>, <func>( v1, v2 ) );
## \){\kernttindent\quad\quad\quad}fi;
## \){\kernttindent\quad\quad}od;
## \){\kernttindent\quad}fi;
## \){\kernttindent}od;
##
## \goodbreak%
## The following example shows how `ListX' can be used to compute all pairs
## and all strictly sorted pairs of elements in a list.
## \beginexample
## gap> l:= [ 1, 2, 3, 4 ];;
## gap> pair:= function( x, y ) return [ x, y ]; end;;
## gap> ListX( l, l, pair );
## [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ],
## [ 2, 4 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ], [ 3, 4 ], [ 4, 1 ], [ 4, 2 ],
## [ 4, 3 ], [ 4, 4 ] ]
## \endexample
## In the following example, `\<' is the comparison operation:
## \beginexample
## gap> ListX( l, l, \<, pair );
## [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ]
## \endexample
##
DeclareGlobalFunction( "ListX" );
#############################################################################
##
#O SetX( <arg1>, <arg2>, ... <func> )
##
## The only difference between `SetX' and `ListX' is that the result list of
## `SetX' is strictly sorted.
##
DeclareGlobalFunction( "SetX" );
#############################################################################
##
#O SumX( <arg1>, <arg2>, ... <func> )
##
## `SumX' returns the sum of the elements in the list obtained by
## `ListX' when this is called with the same arguments.
##
DeclareGlobalFunction( "SumX" );
#############################################################################
##
#O ProductX( <arg1>, <arg2>, ... <func> )
##
## `ProductX' returns the product of the elements in the list obtained by
## `ListX' when this is called with the same arguments.
##
DeclareGlobalFunction( "ProductX" );
#############################################################################
##
#O Perform( <list>, <func>)
##
## `Perform( <list>, <func> )' applies func to every element of
## <list>, discarding any return values. It does not return a value.
##
DeclareGlobalFunction( "Perform" );
#############################################################################
##
#O IsSubset( <C1>, <C2> ) . . . . . . . . . test for subset of collections
##
## `IsSubset' returns `true' if <C2>, which must be a collection, is a
## *subset* of <C1>, which also must be a collection, and `false' otherwise.
##
## <C2> is considered a subset of <C1> if and only if each element of <C2>
## is also an element of <C1>.
## That is `IsSubset' behaves as if implemented as
## `IsSubsetSet( AsSSortedList( <C1> ), AsSSortedList( <C2> ) )',
## except that it will also sometimes, but not always,
## work for infinite collections,
## and that it will usually work much faster than the above definition.
## Either argument may also be a proper set (see~"Sorted Lists and Sets").
##
DeclareOperation( "IsSubset", [ IsListOrCollection, IsListOrCollection ] );
#############################################################################
##
#F Intersection( <C1>, <C2> ... ) . . . . . . . intersection of collections
#F Intersection( <list> ) . . . . . . . . . . . intersection of collections
#O Intersection2( <C1>, <C2> ) . . . . . . . . . intersection of collections
##
## In the first form `Intersection' returns the intersection of the
## collections <C1>, <C2>, etc.
## In the second form <list> must be a *nonempty* list of collections
## and `Intersection' returns the intersection of those collections.
## Each argument or element of <list> respectively may also be a
## homogeneous list that is not a proper set,
## in which case `Intersection' silently applies `Set' (see~"Set") to it
## first.
##
## The result of `Intersection' is the set of elements that lie in every of
## the collections <C1>, <C2>, etc.
## If the result is a list then it is mutable and new, i.e., not identical
## to any of <C1>, <C2>, etc.
##
## Methods can be installed for the operation `Intersection2' that takes
## only two arguments.
## `Intersection' calls `Intersection2'.
##
## Methods for `Intersection2' should try to maintain as much structure as
## possible, for example the intersection of two permutation groups is
## again a permutation group.
##
DeclareGlobalFunction( "Intersection" );
DeclareOperation( "Intersection2",
[ IsListOrCollection, IsListOrCollection ] );
#############################################################################
##
#F Union( <C1>, <C2> ... ) . . . . . . . . . . . . . . union of collections
#F Union( <list> ) . . . . . . . . . . . . . . . . . . union of collections
#O Union2( <C1>, <C2> ) . . . . . . . . . . . . . . . union of collections
##
## In the first form `Union' returns the union of the
## collections <C1>, <C2>, etc.
## In the second form <list> must be a list of collections
## and `Union' returns the union of those collections.
## Each argument or element of <list> respectively may also be a
## homogeneous list that is not a proper set,
## in which case `Union' silently applies `Set' (see~"Set") to it first.
##
## The result of `Union' is the set of elements that lie in any of the
## collections <C1>, <C2>, etc.
## If the result is a list then it is mutable and new, i.e., not identical
## to any of <C1>, <C2>, etc.
##
## Methods can be installed for the operation `Union2' that takes only two
## arguments.
## `Union' calls `Union2'.
##
DeclareGlobalFunction( "Union" );
DeclareOperation( "Union2", [ IsListOrCollection, IsListOrCollection ] );
#############################################################################
##
#O Difference( <C1>, <C2> ) . . . . . . . . . . . difference of collections
##
## `Difference' returns the set difference of the collections <C1> and <C2>.
## Either argument may also be a homogeneous list that is not a proper set,
## in which case `Difference' silently applies `Set' (see~"Set") to it
## first.
##
## The result of `Difference' is the set of elements that lie in <C1> but
## not in <C2>.
## Note that <C2> need not be a subset of <C1>.
## The elements of <C2>, however, that are not elements of <C1> play no role
## for the result.
## If the result is a list then it is mutable and new, i.e., not identical
## to <C1> or <C2>.
##
DeclareOperation( "Difference", [ IsListOrCollection, IsListOrCollection ] );
#############################################################################
##
#P CanEasilyCompareElements( <obj> )
#F CanEasilyCompareElementsFamily( <fam> )
#P CanEasilySortElements( <obj> )
#F CanEasilySortElementsFamily( <fam> )
##
## `CanEasilyCompareElements' indicates whether the elements in the family
## <fam> of <obj> can be easily compared with `='.
## (In some cases element comparisons are very hard, for example in cases
## where no normal forms for the elements exist.)
##
## The default method for this property is to ask the family of <obj>,
## the default method for the family is to return `false'.
##
## The ability to compare elements may depend on the successful computation
## of certain information. (For example for finitely presented groups it
## might depend on the knowledge of a faithful permutation representation.)
## This information might change over time and thus it might not be a good
## idea to store a value `false' too early in a family. Instead the
## function `CanEasilyCompareElementsFamily' should be called for the
## family of <obj> which returns `false' if the value of
## `CanEasilyCompareElements' is not known for the family without computing
## it. (This is in fact what the above mentioned family dispatch does.)
##
## If a family knows ab initio that it can compare elements this property
## should be set as implied filter *and* filter for the family (the 3rd and
## 4th argument of `NewFamily' respectively). This guarantees that code
## which directly asks the family gets a right answer.
##
## The property `CanEasilySortElements' and the function
## `CanEasilySortElementsFamily' behave exactly in the same way, except
## that they indicate that objects can be compared via `\<'. This property
## implies `CanEasilyCompareElements', as the ordering must be total.
##
DeclareProperty( "CanEasilyCompareElements", IsObject );
DeclareGlobalFunction( "CanEasilyCompareElementsFamily" );
DeclareProperty( "CanEasilySortElements", IsObject );
DeclareGlobalFunction( "CanEasilySortElementsFamily" );
InstallTrueMethod(CanEasilyCompareElements,CanEasilySortElements);
#############################################################################
##
#O CanComputeIsSubset( <A>, <B> )
##
## This filter indicates that {\GAP} can test (via `IsSubset') whether <B>
## is a subset of <A>.
DeclareOperation( "CanComputeIsSubset", [IsObject,IsObject] );
#############################################################################
##
#F CanComputeSize( <dom> )
##
## This filter indicates whether the size of the domain <dom> (which might
## be `infinity') can be computed.
DeclareFilter( "CanComputeSize" );
InstallTrueMethod( CanComputeSize, HasSize );
#############################################################################
##
#E
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