#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
#############################################################################
##
#M Print( <A> ) . . . . . . . . . . . . . . . . . . print an additive magma
##
InstallMethod( PrintObj,
"for an add. magma",
[ IsAdditiveMagma ],
function( A )
Print( "AdditiveMagma( ... )" );
end );
InstallMethod( PrintObj,
"for an add. magma with generators",
[ IsAdditiveMagma and HasGeneratorsOfAdditiveMagma ],
function( A )
Print( "AdditiveMagma( ", GeneratorsOfAdditiveMagma( A ), " )" );
end );
InstallMethod( PrintObj,
"for an add. magma-with-zero with generators",
[ IsAdditiveMagmaWithZero and HasGeneratorsOfAdditiveMagmaWithZero ],
function( A )
if IsEmpty( GeneratorsOfAdditiveMagmaWithZero( A ) ) then
Print( "AdditiveMagmaWithZero( ", Zero( A ), " )" );
else
Print( "AdditiveMagmaWithZero( ",
GeneratorsOfAdditiveMagmaWithZero( A ), " )" );
fi;
end );
InstallMethod( PrintObj,
"for an add. magma-with-inverses with generators",
[ IsAdditiveGroup and HasGeneratorsOfAdditiveGroup ],
function( A )
if IsEmpty( GeneratorsOfAdditiveGroup( A ) ) then
Print( "AdditiveGroup( ", Zero( A ), " )" );
else
Print( "AdditiveGroup( ",
GeneratorsOfAdditiveGroup( A ), " )" );
fi;
end );
#############################################################################
##
#M ViewObj( <A> ) . . . . . . . . . . . . . . . . . . . view an add. magma
##
InstallMethod( ViewObj,
"for an add. magma",
[ IsAdditiveMagma ],
function( A )
Print( "<additive magma>" );
end );
InstallMethod( ViewObj,
"for an add. magma with generators",
[ IsAdditiveMagma and HasGeneratorsOfAdditiveMagma ],
function( A )
local nrgens;
nrgens := Length( GeneratorsOfAdditiveMagma( A ) );
Print( "<additive magma with ", Pluralize( nrgens, "generator" ), ">" );
end );
InstallMethod( ViewObj,
"for an add. magma-with-zero with generators",
[ IsAdditiveMagmaWithZero and HasGeneratorsOfAdditiveMagmaWithZero ],
function( A )
local nrgens;
nrgens := GeneratorsOfAdditiveMagmaWithZero( A );
if nrgens = 0 then
Print( "<trivial additive magma-with-zero>" );
else
Print( "<additive magma-with-zero with ",
Pluralize( nrgens, "generator" ), ">" );
fi;
end );
InstallMethod( ViewObj,
"for an add. magma-with-inverses with generators",
[ IsAdditiveGroup and HasGeneratorsOfAdditiveGroup ],
function( A )
local nrgens;
nrgens := GeneratorsOfAdditiveGroup( A );
if nrgens = 0 then
Print( "<trivial additive magma-with-inverses>" );
else
Print( "<additive magma-with-inverses with ",
Pluralize( nrgens, "generator" ), ">" );
fi;
end );
#############################################################################
##
#M IsTrivial( <A> ) . . . . . . . test whether an additive magma is trivial
##
InstallImmediateMethod( IsTrivial,
IsAdditiveMagmaWithZero and HasGeneratorsOfAdditiveMagmaWithZero, 0,
function( A )
return ForAll( GeneratorsOfAdditiveMagmaWithZero( A ), IsZero );
end );
InstallImmediateMethod( IsTrivial,
IsAdditiveGroup and HasGeneratorsOfAdditiveGroup, 0,
function( A )
return ForAll( GeneratorsOfAdditiveGroup( A ), IsZero );
end );
#############################################################################
##
#F AdditiveMagma( <gens> )
#F AdditiveMagma( <Fam>, <gens> )
##
InstallGlobalFunction( AdditiveMagma, function( arg )
# list of generators
if Length( arg ) = 1 and IsList( arg[1] ) and 0 < Length( arg[1] ) then
return AdditiveMagmaByGenerators( arg[1] );
# family plus list of generators
elif Length( arg ) = 2 and IsFamily( arg[1] ) and IsList( arg[1] ) then
return AdditiveMagmaByGenerators( arg[1], arg[2] );
# generators
elif 0 < Length( arg ) then
return AdditiveMagmaByGenerators( arg );
fi;
# no argument given, error
Error("usage: AdditiveMagma(<gens>), AdditiveMagma(<Fam>,<gens>)");
end );
#############################################################################
##
#F SubadditiveMagma( <M>, <gens> ) add. submagma of <M> generated by <gens>
##
InstallGlobalFunction( SubadditiveMagma, function( M, gens )
if not IsAdditiveMagma( M ) then
Error( "<M> must be an additive magma" );
elif IsEmpty( gens ) then
return SubadditiveMagmaNC( M, gens );
elif not IsHomogeneousList(gens) then
Error( "<gens> must be a homogeneous list of elements" );
elif not IsIdenticalObj( FamilyObj(M), FamilyObj(gens) ) then
Error( "families of <gens> and <M> are different" );
fi;
if not ForAll( gens, x -> x in M ) then
Error( "<gens> must be elements in <M>" );
fi;
return SubadditiveMagmaNC( M, gens );
end );
#############################################################################
##
#F SubadditiveMagmaNC( <M>, <gens> )
##
## Note that `SubadditiveMagmaNC' is allowed to call `Objectify'
## in the case that <gens> is empty.
##
InstallGlobalFunction( SubadditiveMagmaNC, function( M, gens )
local K, S;
if IsEmpty( gens ) then
K:= NewType( FamilyObj(M),
IsAdditiveMagma
and IsEmpty
and IsAttributeStoringRep );
S:= Objectify( K, rec() );
SetGeneratorsOfAdditiveMagma( S, [] );
else
S:= AdditiveMagmaByGenerators(gens);
fi;
SetParent( S, M );
return S;
end );
#############################################################################
##
#F AdditiveMagmaWithZero( <gens> )
#F AdditiveMagmaWithZero( <Fam>, <gens> )
##
InstallGlobalFunction( AdditiveMagmaWithZero, function( arg )
# list of generators
if Length( arg ) = 1 and IsList( arg[1] ) and 0 < Length( arg[1] ) then
return AdditiveMagmaWithZeroByGenerators( arg[1] );
# family plus list of generators
elif Length( arg ) = 2 and IsFamily( arg[1] ) and IsList( arg[1] ) then
return AdditiveMagmaWithZeroByGenerators( arg[1], arg[2] );
# generators
elif 0 < Length( arg ) then
return AdditiveMagmaWithZeroByGenerators( arg );
fi;
# no argument given, error
Error("usage: AdditiveMagmaWithZero(<gens>), ",
"AdditiveMagmaWithZero(<Fam>,<gens>)");
end );
#############################################################################
##
#F SubadditiveMagmaWithZero( <M>, <gens> )
#F . . . add. submagma-with-one of <M> gen. by <gens>
##
InstallGlobalFunction( SubadditiveMagmaWithZero, function( M, gens )
if not IsAdditiveMagmaWithZero( M ) then
Error( "<M> must be an additive magma-with-zero" );
elif IsEmpty( gens ) then
return SubadditiveMagmaWithZeroNC( M, gens );
elif not IsHomogeneousList(gens) then
Error( "<gens> must be a homogeneous list of elements" );
elif not IsIdenticalObj( FamilyObj(M), FamilyObj(gens) ) then
Error( "families of <gens> and <M> are different" );
fi;
if not ForAll( gens, x -> x in M ) then
Error( "<gens> must be elements in <M>" );
fi;
return SubadditiveMagmaWithZeroNC( M, gens );
end );
#############################################################################
##
#F SubadditiveMagmaWithZeroNC( <M>, <gens> )
##
## Note that `SubadditiveMagmaWithZeroNC' is allowed to call `Objectify'
## in the case that <gens> is empty.
##
## Furthermore note that a trivial additive magma with zero is automatically
## an additive group.
##
InstallGlobalFunction( SubadditiveMagmaWithZeroNC, function( M, gens )
local K, S;
if IsEmpty( gens ) then
K:= NewType( FamilyObj(M),
IsAdditiveGroup
and IsTrivial
and IsAttributeStoringRep );
S:= Objectify( K, rec() );
SetGeneratorsOfAdditiveGroup( S, [] );
else
S:= AdditiveMagmaWithZeroByGenerators(gens);
fi;
SetParent( S, M );
return S;
end );
#############################################################################
##
#F AdditiveGroup( <gens> )
#F AdditiveGroup( <Fam>, <gens> )
##
InstallGlobalFunction( AdditiveGroup, function( arg )
# list of generators
if Length( arg ) = 1 and IsList( arg[1] ) and 0 < Length( arg[1] ) then
return AdditiveGroupByGenerators( arg[1] );
# family plus list of generators
elif Length( arg ) = 2 and IsFamily( arg[1] ) and IsList( arg[1] ) then
return AdditiveGroupByGenerators( arg[1], arg[2] );
# generators
elif 0 < Length( arg ) then
return AdditiveGroupByGenerators( arg );
fi;
# no argument given, error
Error("usage: AdditiveGroup(<gens>), ",
"AdditiveGroup(<Fam>,<gens>)");
end );
#############################################################################
##
#F SubadditiveGroup( <M>, <gens> ) . . . add. subgroup of <M> gen. by <gens>
##
InstallGlobalFunction( SubadditiveGroup, function( M, gens )
if not IsAdditiveGroup( M ) then
Error( "<M> must be an additive group" );
elif IsEmpty( gens ) then
return SubadditiveGroupNC( M, gens );
elif not IsHomogeneousList(gens) then
Error( "<gens> must be a homogeneous list of elements" );
elif not IsIdenticalObj( FamilyObj(M), FamilyObj(gens) ) then
Error( "families of <gens> and <M> are different" );
fi;
if not ForAll( gens, x -> x in M ) then
Error( "<gens> must be elements in <M>" );
fi;
return SubadditiveGroupNC( M, gens );
end );
#############################################################################
##
#F SubadditiveGroupNC( <M>, <gens> )
##
## Note that `SubadditiveGroupNC' is allowed to call `Objectify'
## in the case that <gens> is empty.
##
InstallGlobalFunction( SubadditiveGroupNC, function( M, gens )
local K, S;
if IsEmpty( gens ) then
K:= NewType( FamilyObj(M),
IsAdditiveGroup
and IsTrivial
and IsAttributeStoringRep );
S:= Objectify( K, rec() );
SetGeneratorsOfAdditiveGroup( S, [] );
else
S:= AdditiveGroupByGenerators(gens);
fi;
SetParent( S, M );
return S;
end );
#############################################################################
##
#M TrivialSubadditiveMagmaWithZero( <M> ) . . . for an add.-magma-with-zero
##
InstallMethod( TrivialSubadditiveMagmaWithZero,
"for add.-magma-with-zero",
[ IsAdditiveMagmaWithZero ],
M -> SubadditiveMagmaWithZeroNC( M, [] ) );
#############################################################################
##
#M AdditiveMagmaByGenerators( <gens> ) . . . . . . . . . . for a collection
##
InstallMethod( AdditiveMagmaByGenerators,
"for collection",
[ IsCollection ],
function( gens )
local M;
M:= Objectify( NewType( FamilyObj( gens ),
IsAdditiveMagma and IsAttributeStoringRep ),
rec() );
SetGeneratorsOfAdditiveMagma( M, AsList( gens ) );
return M;
end );
#############################################################################
##
#M AdditiveMagmaByGenerators( <Fam>, <gens> ) . . . . . for family and list
##
InstallOtherMethod( AdditiveMagmaByGenerators,
"for family and list",
[ IsFamily, IsList ],
function( family, gens )
local M;
if not ( IsEmpty(gens) or IsIdenticalObj( FamilyObj(gens), family ) ) then
Error( "<family> and family of <gens> do not match" );
fi;
M:= Objectify( NewType( family,
IsAdditiveMagma and IsAttributeStoringRep ),
rec() );
SetGeneratorsOfAdditiveMagma( M, AsList( gens ) );
return M;
end );
#############################################################################
##
#M AdditiveMagmaWithZeroByGenerators( <gens> ) . . . . . . for a collection
##
InstallMethod( AdditiveMagmaWithZeroByGenerators,
"for collection",
[ IsCollection ],
function( gens )
local M;
M:= Objectify( NewType( FamilyObj( gens ),
IsAdditiveMagmaWithZero and IsAttributeStoringRep ),
rec() );
SetGeneratorsOfAdditiveMagmaWithZero( M, AsList( gens ) );
return M;
end );
#############################################################################
##
#M AdditiveMagmaWithZeroByGenerators( <Fam>, <gens> ) . for family and list
##
InstallOtherMethod( AdditiveMagmaWithZeroByGenerators,
"for family and list",
[ IsFamily, IsList ],
function( family, gens )
local M;
if not ( IsEmpty(gens) or IsIdenticalObj( FamilyObj(gens), family ) ) then
Error( "<family> and family of <gens> do not match" );
fi;
M:= Objectify( NewType( family,
IsAdditiveMagmaWithZero and IsAttributeStoringRep ),
rec() );
SetGeneratorsOfAdditiveMagmaWithZero( M, AsList( gens ) );
return M;
end );
#############################################################################
##
#M AdditiveGroupByGenerators( <gens> ) . . . . for a collection
##
InstallMethod( AdditiveGroupByGenerators,
"for collection",
[ IsCollection ],
function( gens )
local M;
M:= Objectify( NewType( FamilyObj( gens ),
IsAdditiveGroup and IsAttributeStoringRep ),
rec() );
SetGeneratorsOfAdditiveGroup( M, AsList( gens ) );
return M;
end );
#############################################################################
##
#M AdditiveGroupByGenerators(<Fam>,<gens>) . for family and list
##
InstallOtherMethod( AdditiveGroupByGenerators,
"for family and list",
[ IsFamily, IsList ],
function( family, gens )
local M;
if not ( IsEmpty(gens) or IsIdenticalObj( FamilyObj(gens), family ) ) then
Error( "<family> and family of <gens> do not match" );
fi;
M:= Objectify( NewType( family,
IsAdditiveGroup and IsAttributeStoringRep ),
rec() );
SetGeneratorsOfAdditiveGroup( M, AsList( gens ) );
return M;
end );
#############################################################################
##
#M GeneratorsOfAdditiveMagma( <A> )
#M GeneratorsOfAdditiveMagmaWithZero( <A> )
#M GeneratorsOfAdditiveGroup( <A> )
##
## If nothing special is known about the additive magma <A> we have
## no chance to get the required generators.
##
## If we know `GeneratorsOfAdditiveMagma',
## they are also `GeneratorsOfAdditiveMagmaWithZero'.
## If we know `GeneratorsOfAdditiveMagmaWithZero',
## they are also `GeneratorsOfAdditiveGroup'.
##
InstallImmediateMethod( GeneratorsOfAdditiveMagmaWithZero,
IsAdditiveMagmaWithZero and HasGeneratorsOfAdditiveMagma
and IsAttributeStoringRep, 0,
A -> GeneratorsOfAdditiveMagma( A ) );
InstallImmediateMethod( GeneratorsOfAdditiveGroup,
IsAdditiveGroup and HasGeneratorsOfAdditiveMagmaWithZero
and IsAttributeStoringRep, 0,
A -> GeneratorsOfAdditiveMagmaWithZero( A ) );
InstallMethod( GeneratorsOfAdditiveMagma,
[ IsAdditiveMagmaWithZero and HasGeneratorsOfAdditiveMagmaWithZero ],
A -> Concatenation( GeneratorsOfAdditiveMagmaWithZero( A ),
[ Zero( A ) ] ) );
InstallMethod( GeneratorsOfAdditiveMagma,
[ IsAdditiveGroup and HasGeneratorsOfAdditiveGroup ],
A -> Concatenation( GeneratorsOfAdditiveGroup( A ),
[ Zero( A ) ],
List( GeneratorsOfAdditiveGroup( A ),
AdditiveInverse ) ) );
InstallMethod( GeneratorsOfAdditiveMagmaWithZero,
[ IsAdditiveGroup and HasGeneratorsOfAdditiveGroup ],
A -> Concatenation( GeneratorsOfAdditiveGroup( A ),
List( GeneratorsOfAdditiveGroup( A ),
AdditiveInverse ) ) );
#############################################################################
##
#M Representative( <A> ) . . . . . . . . . one element of an additive magma
##
InstallMethod( Representative,
"for additive magma with known generators",
[ IsAdditiveMagma and HasGeneratorsOfAdditiveMagma ],
RepresentativeFromGenerators( GeneratorsOfAdditiveMagma ) );
InstallMethod( Representative,
"for additive-magma-with-zero with known generators",
[ IsAdditiveMagmaWithZero and HasGeneratorsOfAdditiveMagmaWithZero ],
RepresentativeFromGenerators( GeneratorsOfAdditiveMagmaWithZero ) );
InstallMethod( Representative,
"for additive-magma-with-inverses with known generators",
[ IsAdditiveGroup and HasGeneratorsOfAdditiveGroup ],
RepresentativeFromGenerators( GeneratorsOfAdditiveGroup ) );
InstallMethod( Representative,
"for additive-magma-with-zero with known zero",
[ IsAdditiveMagmaWithZero and HasZero ], SUM_FLAGS,
Zero );
InstallMethod( Representative,
"for additive-magma-with-zero with stored parent",
[ IsAdditiveMagmaWithZero and HasParentAttr ],
function( A )
if not IsIdenticalObj( A, Parent( A ) ) then
return Zero( Representative( Parent( A ) ) );
fi;
TryNextMethod();
end );
#############################################################################
##
#M AdditiveNeutralElement( <A> ) . . . . . . . . . zero of an additive magma
##
InstallMethod( AdditiveNeutralElement,
[ IsAdditiveMagma ],
function( M )
local m;
if IsFinite( M ) then
for m in M do
if ForAll( M, n -> n + m = n ) then
return m;
fi;
od;
return fail;
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M Zero( <A> ) . . . . . . . . . . . . . . . . . . zero of an additive magma
##
InstallOtherMethod( Zero,
"for additive magma",
[ IsAdditiveMagma ],
function( A )
local zero;
zero:= Zero( Representative( A ) );
if zero <> fail and zero in A then
return zero;
else
return fail;
fi;
end );
InstallOtherMethod( Zero,
"for additive magma with zero (look at family)",
[ IsAdditiveMagmaWithZero ], SUM_FLAGS,
function( A )
A:= ElementsFamily( FamilyObj( A ) );
if HasZero( A ) then
return Zero( A );
else
TryNextMethod();
fi;
end );
#T immediate?
InstallOtherMethod( Zero,
"for an add. magma-with-zero with parent (ask the parent)",
[ IsAdditiveMagmaWithZero and HasParent ],
function( A )
if not IsIdenticalObj( A, Parent( A ) ) then
return Zero( Parent( A ) );
fi;
TryNextMethod();
end );
#T really ask the parent for such information?
InstallOtherMethod( Zero,
"for additive magma with zero",
[ IsAdditiveMagmaWithZero ],
A -> Zero( Representative( A ) ) );
#############################################################################
##
#M Characteristic(<obj>)
##
InstallMethod( Characteristic,
"delegate to family (magma)",
[ IsAdditiveMagmaWithZero ],
function( el )
return Characteristic( FamilyObj(el) );
end );
#############################################################################
##
#M Enumerator( <A> ) . . . . enumerator of trivial additive magma with zero
##
BindGlobal( "EnumeratorOfTrivialAdditiveMagmaWithZero", A -> Immutable( [ Zero( A ) ] ) );
InstallMethod( Enumerator,
"for trivial add. magma-with-zero",
[ IsAdditiveMagmaWithZero and IsTrivial ],
EnumeratorOfTrivialAdditiveMagmaWithZero );
#############################################################################
##
#F ClosureAdditiveMagmaDefault( <A>, <elm> ) closure of add. magma with elm
##
BindGlobal( "ClosureAdditiveMagmaDefault", function( A, elm )
local C, # closure `\< <a>, <obj> \>', result
gens, # generators of <A>
gen, # generator of <A> or <C>
Celements, # intermediate list of elements
len; # current number of elements
gens:= GeneratorsOfAdditiveMagma( A );
# try to avoid adding an element to an add. magma that already contains it
if elm in gens
or ( HasAsSSortedList( A ) and elm in AsSSortedList( A ) )
then
return A;
fi;
# make the closure add. magma
gens:= Concatenation( gens, [ elm ] );
C:= AdditiveMagmaByGenerators( gens );
UseSubsetRelation( C, A );
# if the elements of <A> are known then extend this list
# (multiply each element from the left and right with the new
# generator, and then multiply with all elements until the
# list becomes stable)
if HasAsSSortedList( A ) then
Celements := ShallowCopy( AsSSortedList( A ) );
AddSet( Celements, elm );
UniteSet( Celements, Celements + elm );
UniteSet( Celements, elm + Celements );
repeat
len:= Length( Celements );
for gen in Celements do
UniteSet( Celements, Celements + gen );
UniteSet( Celements, gen + Celements );
od;
until len = Length( Celements );
SetAsSSortedList( C, AsSSortedList( Celements ) );
SetIsFinite( C, true );
SetSize( C, Length( Celements ) );
fi;
# return the closure
return C;
end );
#############################################################################
##
#M Enumerator( <A> ) . . . . . . . . . set of the elements of an add. magma
##
BindGlobal( "EnumeratorOfAdditiveMagma", function( A )
local gens, # add. magma generators of <A>
H, # subadd. magma of the first generators of <A>
gen; # generator of <A>
# handle the case of an empty add. magma
gens:= GeneratorsOfAdditiveMagma( A );
if IsEmpty( gens ) then
return [];
fi;
# start with the empty add. magma and its element list
H:= SubadditiveMagma( A, [] );
SetAsSSortedList( H, Immutable( [ ] ) );
# Add the generators one after the other.
# We use a function that maintains the elements list for the closure.
for gen in gens do
H:= ClosureAdditiveMagmaDefault( H, gen );
od;
# return the list of elements
Assert( 2, HasAsSSortedList( H ) );
return AsSSortedList( H );
end );
InstallMethod( Enumerator,
"generic method for an add. magma",
[ IsAdditiveMagma and IsAttributeStoringRep ],
EnumeratorOfAdditiveMagma );
#############################################################################
##
#M IsSubset( <M>, <N> ) . . . . . . . . . . . . . . for two additive magmas
##
InstallMethod( IsSubset,
"for two additive magmas",
IsIdenticalObj,
[ IsAdditiveMagma, IsAdditiveMagma ],
function( M, N )
return IsSubset( M, GeneratorsOfAdditiveMagma( N ) );
end );
#############################################################################
##
#M IsSubset( <M>, <N> ) . . . . . . . . . for two additive magmas with zero
##
InstallMethod( IsSubset,
"for two additive magmas with zero",
IsIdenticalObj,
[ IsAdditiveMagmaWithZero, IsAdditiveMagmaWithZero ],
function( M, N )
return IsSubset( M, GeneratorsOfAdditiveMagmaWithZero( N ) );
end );
#############################################################################
##
#M IsSubset( <M>, <N> ) . . . . . . . for two additive magmas with inverses
##
InstallMethod( IsSubset,
"for two additive magmas with inverses",
IsIdenticalObj,
[ IsAdditiveGroup, IsAdditiveGroup ],
function( M, N )
return IsSubset( M, GeneratorsOfAdditiveGroup( N ) );
end );
#############################################################################
##
#M ClosureAdditiveGroup( <A>, <a> ) . . . . . . for add. group and element
##
InstallMethod( ClosureAdditiveGroup,
"for add. group and element",
IsCollsElms,
[ IsAdditiveGroup, IsAdditiveElement ],
function( A, a )
# if possible test if the element lies in the add. group already
if a in GeneratorsOfAdditiveGroup( A ) or
( HasAsList( A ) and a in AsList( A ) ) then
return A;
fi;
# Otherwise make a new add. group.
return AdditiveGroupByGenerators(
Concatenation( GeneratorsOfAdditiveGroup( A ), [ a ] ) );
end );
#############################################################################
##
#M ClosureAdditiveGroup( <A>, <B> ) . . . . . . . . . . for two add. groups
##
InstallOtherMethod( ClosureAdditiveGroup,
"for two add. groups",
IsIdenticalObj,
[ IsAdditiveGroup, IsAdditiveGroup ],
function( A, B )
local C, b;
C:= A;
for b in GeneratorsOfAdditiveGroup( B ) do
C:= ClosureAdditiveGroup( C, b );
od;
return C;
end );
