@c Original attribution:

@c
@c STk Reference manual (Appendix: An Introduction to STklos)
@c
@c Copyright (c) 1993-1999 Erick Gallesio - I3S-CNRS/ESSI <eg@unice.fr>
@c Permission to use, copy, modify, distribute,and license this
@c software and its documentation for any purpose is hereby granted,
@c provided that existing copyright notices are retained in all
@c copies and that this notice is included verbatim in any
@c distributions.  No written agreement, license, or royalty fee is
@c required for any of the authorized uses.
@c This software is provided ``AS IS'' without express or implied
@c warranty.
@c

@c Adapted for use in Guile with the authors permission

@c @macro goops                    @c was {\stklos}
@c GOOPS
@c @end macro

@c @macro guile                    @c was {\stk}
@c Guile
@c @end macro

This is chapter was originally written by Erick Gallesio as an appendix
for the STk reference manual, and subsequently adapted to @goops{}.

@menu
* Copyright::
* Intro::                
* Class definition and instantiation::  
* Inheritance::                 
* Generic functions::           
@end menu

@node Copyright, Intro, Tutorial, Tutorial
@section Copyright

Original attribution:

STk Reference manual (Appendix: An Introduction to STklos)

Copyright (c) 1993-1999 Erick Gallesio - I3S-CNRS/ESSI <eg@@unice.fr>
Permission to use, copy, modify, distribute,and license this
software and its documentation for any purpose is hereby granted,
provided that existing copyright notices are retained in all
copies and that this notice is included verbatim in any
distributions.  No written agreement, license, or royalty fee is
required for any of the authorized uses.
This software is provided ``AS IS'' without express or implied
warranty.

Adapted for use in Guile with the authors permission

@node Intro, Class definition and instantiation, Copyright, Tutorial
@section Introduction

@goops{} is the object oriented extension to @guile{}. Its
implementation is derived from @w{STk-3.99.3} by Erick Gallesio and
version 1.3 of the Gregor Kiczales @cite{Tiny-Clos}.  It is very close
to CLOS, the Common Lisp Object System (@cite{CLtL2}) but is adapted for
the Scheme language.

Briefly stated, the @goops{} extension gives the user a full object
oriented system with multiple inheritance and generic functions with
multi-method dispatch.  Furthermore, the implementation relies on a true
meta object protocol, in the spirit of the one defined for CLOS
(@cite{Gregor Kiczales: A Metaobject Protocol}).

The purpose of this tutorial is to introduce briefly the @goops{}
package and in no case will it replace the @goops{} reference manual
(which needs to be urgently written now@ @dots{}).

Note that the operations described in this tutorial resides in modules
that may need to be imported before being available.  The main module is
imported by evaluating:

@lisp
(use-modules (oop goops))
@end lisp
@findex (oop goops)
@cindex main module
@cindex loading
@cindex preparing

@node Class definition and instantiation, Inheritance, Intro, Tutorial
@section Class definition and instantiation

@menu
* Class definition::            
@end menu

@node Class definition,  , Class definition and instantiation, Class definition and instantiation
@subsection Class definition

A new class is defined with the @code{define-class}@footnote{Don't
forget to import the @code{(oop goops)} module} macro. The syntax of
@code{define-class} is close to CLOS @code{defclass}:

@findex define-class
@cindex class
@lisp
(define-class @var{class} (@var{superclass} @dots{})
   @var{slot-description} @dots{}
   @var{class-option} @dots{})
@end lisp

Class options will not be discussed in this tutorial.  The list of
@var{superclass}es specifies which classes to inherit properties from
@var{class} (see @ref{Inheritance} for more details).  A
@var{slot-description} gives the name of a slot and, eventually, some
``properties'' of this slot (such as its initial value, the function
which permit to access its value, @dots{}). Slot descriptions will be
discussed in @ref{Slot description}.
@cindex slot

As an example, let us define a type for representation of complex
numbers in terms of real numbers. This can be done with the following
class definition:

@lisp
(define-class  <complex> (<number>)
   r i)
@end lisp

This binds the variable @code{<complex>}@footnote{@code{<complex>} is in
fact a builtin class in GOOPS.  Because of this, GOOPS will create a new
class.  The old class will still serve as the type for Guile's native
complex numbers.} to a new class whose instances contain two
slots. These slots are called @code{r} an @code{i} and we suppose here
that they contain respectively the real part and the imaginary part of a
complex number. Note that this class inherits from @code{<number>} which
is a pre-defined class.  (@code{<number>} is the direct super class of
the pre-defined class @code{<complex>} which, in turn, is the super
class of @code{<real>} which is the super of
@code{<integer>}.)@footnote{With the new definition of @code{<complex>},
a @code{<real>} is not a @code{<complex>} since @code{<real>} inherits
from @code{ <number>} rather than @code{<complex>}. In practice,
inheritance could be modified @emph{a posteriori}, if needed. However,
this necessitates some knowledge of the meta object protocol and it will
not be shown in this document}.

@node Inheritance, Generic functions, Class definition and instantiation, Tutorial
@section Inheritance
@c \label{inheritance}

@menu
* Class hierarchy and inheritance of slots::  
* Instance creation and slot access::  
* Slot description::            
* Class precedence list::       
@end menu

@node Class hierarchy and inheritance of slots, Instance creation and slot access, Inheritance, Inheritance
@subsection Class hierarchy and inheritance of slots
Inheritance is specified upon class definition. As said in the
introduction, @goops{} supports multiple inheritance.  Here are some
class definitions:

@lisp
(define-class A () a)
(define-class B () b)
(define-class C () c)
(define-class D (A B) d a)
(define-class E (A C) e c)
(define-class F (D E) f)
@end lisp

@code{A}, @code{B}, @code{C} have a null list of super classes. In this
case, the system will replace it by the list which only contains
@code{<object>}, the root of all the classes defined by
@code{define-class}. @code{D}, @code{E}, @code{F} use multiple
inheritance: each class inherits from two previously defined classes.
Those class definitions define a hierarchy which is shown in Figure@ 1.
In this figure, the class @code{<top>} is also shown; this class is the
super class of all Scheme objects. In particular, @code{<top>} is the
super class of all standard Scheme types.

@example
@group
@image{hierarchy}
@center @emph{Fig 1: A class hierarchy}
@iftex
@emph{(@code{<complex>} which is the direct subclass of @code{<number>}
and the direct superclass of @code{<real>} has been omitted in this
figure.)}
@end iftex
@end group
@end example

The set of slots of a given class is calculated by taking the union of the
slots of all its super class. For instance, each instance of the class
D, defined before will have three slots (@code{a}, @code{b} and
@code{d}). The slots of a class can be obtained by the @code{class-slots}
primitive.  For instance,

@lisp
(class-slots A) @result{} ((a))
(class-slots E) @result{} ((a) (e) (c))
(class-slots F) @result{} ((e) (c) (b) (d) (a) (f))
@c used to be ((d) (a) (b) (c) (f))
@end lisp

@emph{Note: } The order of slots is not significant.

@node Instance creation and slot access, Slot description, Class hierarchy and inheritance of slots, Inheritance
@subsection Instance creation and slot access

Creation of an instance of a previously defined
class can be done with the @code{make} procedure. This
procedure takes one mandatory parameter which is the class of the
instance which must be created and a list of optional
arguments. Optional arguments are generally used to initialize some
slots of the newly created instance. For instance, the following form

@findex make
@cindex instance
@lisp
(define c (make <complex>))
@end lisp

will create a new @code{<complex>} object and will bind it to the @code{c}
Scheme variable.

Accessing the slots of the new complex number can be done with the
@code{slot-ref} and the @code{slot-set!}  primitives. @code{Slot-set!}
primitive permits to set the value of an object slot and @code{slot-ref}
permits to get its value.

@findex slot-set!
@findex slot-ref
@lisp
@group
(slot-set! c 'r 10)
(slot-set! c 'i 3)
(slot-ref c 'r) @result{} 10
(slot-ref c 'i) @result{} 3
@end group
@end lisp

Using the @code{describe} function is a simple way to see all the
slots of an object at one time: this function prints all the slots of an
object on the standard output.

First load the module @code{(oop goops describe)}:

@example
@code{(use-modules (oop goops describe))}
@end example

The expression

@smalllisp
(describe c)
@end smalllisp

will now print the following information on the standard output:

@lisp
#<<complex> 401d8638> is an instance of class <complex>
Slots are: 
     r = 10
     i = 3
@end lisp

@node Slot description, Class precedence list, Instance creation and slot access, Inheritance
@subsection Slot description
@c \label{slot-description}

When specifying a slot, a set of options can be given to the
system. Each option is specified with a keyword. The list of authorized
keywords is given below:

@cindex keyword
@itemize @bullet
@item
@code{#:init-value} permits to supply a default value for the slot. This
default value is obtained by evaluating the form given after the
@code{#:init-form} in the global environment, at class definition time.
@cindex default slot value
@findex #:init-value
@cindex top level environment

@item
@code{#:init-thunk} permits to supply a thunk that will provide a
default value for the slot. The value is obtained by evaluating the
thunk a instance creation time.
@c CHECKME: in the global environment?
@findex default slot value
@findex #:init-thunk
@cindex top level environment

@item
@code{#:init-keyword} permits to specify the keyword for initializing a
slot. The init-keyword may be provided during instance creation (i.e. in
the @code{make} optional parameter list). Specifying such a keyword
during instance initialization will supersede the default slot
initialization possibly given with @code{#:init-form}.
@findex #:init-keyword

@item
@code{#:getter} permits to supply the name for the 
slot getter. The name binding is done in the
environment of the @code{define-class} macro.
@findex #:getter
@cindex top level environment
@cindex getter

@item
@code{#:setter} permits to supply the name for the 
slot setter. The name binding is done in the
environment of the @code{define-class} macro.
@findex #:setter
@cindex top level environment
@cindex setter

@item
@code{#:accessor} permits to supply the name for the 
slot accessor. The name binding is done in the global
environment. An accessor permits to get and
set the value of a slot. Setting the value of a slot is done with the extended
version of @code{set!}.
@findex set!
@findex #:accessor
@cindex top level environment
@cindex accessor

@item
@code{#:allocation} permits to specify how storage for
the slot is allocated. Three kinds of allocation are provided. 
They are described below:

@itemize @minus
@item
@code{#:instance} indicates that each instance gets its own storage for
the slot. This is the default.
@item
@code{#:class} indicates that there is one storage location used by all
the direct and indirect instances of the class. This permits to define a
kind of global variable which can be accessed only by (in)direct
instances of the class which defines this slot.
@item
@code{#:each-subclass} indicates that there is one storage location used
by all the direct instances of the class. In other words, if two classes
are not siblings in the class hierarchy, they will not see the same
value.
@item
@code{#:virtual} indicates that no storage will be allocated for this
slot.  It is up to the user to define a getter and a setter function for
this slot. Those functions must be defined with the @code{#:slot-ref}
and @code{#:slot-set!} options. See the example below.
@findex #:slot-set!
@findex #:slot-ref
@findex #:virtual
@findex #:class
@findex #:each-subclass
@findex #:instance
@findex #:allocation
@end itemize
@end itemize

To illustrate slot description, we shall redefine the @code{<complex>} class 
seen before. A definition could be:

@lisp
(define-class <complex> (<number>) 
   (r #:init-value 0 #:getter get-r #:setter set-r! #:init-keyword #:r)
   (i #:init-value 0 #:getter get-i #:setter set-i! #:init-keyword #:i))
@end lisp

With this definition, the @code{r} and @code{i} slot are set to 0 by
default.  Value of a slot can also be specified by calling @code{make}
with the @code{#:r} and @code{#:i} keywords. Furthermore, the generic
functions @code{get-r} and @code{set-r!} (resp. @code{get-i} and
@code{set-i!}) are automatically defined by the system to read and write
the @code{r} (resp. @code{i}) slot.

@lisp
(define c1 (make <complex> #:r 1 #:i 2))
(get-r c1) @result{} 1
(set-r! c1 12)
(get-r c1) @result{} 12
(define c2 (make <complex> #:r 2))
(get-r c2) @result{} 2
(get-i c2) @result{} 0
@end lisp

Accessors provide an uniform access for reading and writing an object
slot.  Writing a slot is done with an extended form of @code{set!}
which is close to the Common Lisp @code{setf} macro. So, another
definition of the previous @code{<complex>} class, using the
@code{#:accessor} option, could be:

@findex set!
@lisp
(define-class <complex> (<number>) 
   (r #:init-value 0 #:accessor real-part #:init-keyword #:r)
   (i #:init-value 0 #:accessor imag-part #:init-keyword #:i))
@end lisp

Using this class definition, reading the real part of the @code{c}
complex can be done with:
@lisp
(real-part c)
@end lisp
and setting it to the value contained in the @code{new-value} variable
can be done using the extended form of @code{set!}.
@lisp
(set! (real-part c) new-value)
@end lisp

Suppose now that we have to manipulate complex numbers with rectangular
coordinates as well as with polar coordinates. One solution could be to
have a definition of complex numbers which uses one particular
representation and some conversion functions to pass from one
representation to the other.  A better solution uses virtual slots. A
complete definition of the @code{<complex>} class using virtual slots is
given in Figure@ 2.

@example
@group
@lisp
(define-class <complex> (<number>)
   ;; True slots use rectangular coordinates
   (r #:init-value 0 #:accessor real-part #:init-keyword #:r)
   (i #:init-value 0 #:accessor imag-part #:init-keyword #:i)
   ;; Virtual slots access do the conversion
   (m #:accessor magnitude #:init-keyword #:magn  
      #:allocation #:virtual
      #:slot-ref (lambda (o)
                  (let ((r (slot-ref o 'r)) (i (slot-ref o 'i)))
                    (sqrt (+ (* r r) (* i i)))))
      #:slot-set! (lambda (o m)
                    (let ((a (slot-ref o 'a)))
                      (slot-set! o 'r (* m (cos a)))
                      (slot-set! o 'i (* m (sin a))))))
   (a #:accessor angle #:init-keyword #:angle
      #:allocation #:virtual
      #:slot-ref (lambda (o)
                  (atan (slot-ref o 'i) (slot-ref o 'r)))
      #:slot-set! (lambda(o a)
                   (let ((m (slot-ref o 'm)))
                      (slot-set! o 'r (* m (cos a)))
                      (slot-set! o 'i (* m (sin a)))))))

@end lisp
@center @emph{Fig 2: A @code{<complex>} number class definition using virtual slots}
@end group
@end example

@sp 3
This class definition implements two real slots (@code{r} and
@code{i}). Values of the @code{m} and @code{a} virtual slots are
calculated from real slot values. Reading a virtual slot leads to the
application of the function defined in the @code{#:slot-ref}
option. Writing such a slot leads to the application of the function
defined in the @code{#:slot-set!} option.  For instance, the following
expression

@findex #:slot-set!
@findex #:slot-ref
@lisp
(slot-set! c 'a 3)
@end lisp

permits to set the angle of the @code{c} complex number. This expression
conducts, in fact, to the evaluation of the following expression

@lisp
((lambda o m)
    (let ((m (slot-ref o 'm)))
       (slot-set! o 'r (* m (cos a)))
       (slot-set! o 'i (* m (sin a))))
  c 3)
@end lisp

A more complete example is given below:

@example
@group
@lisp
(define c (make <complex> #:r 12 #:i 20))
(real-part c) @result{} 12
(angle c) @result{} 1.03037682652431
(slot-set! c 'i 10)
(set! (real-part c) 1)
(describe c) @result{}
          #<<complex> 401e9b58> is an instance of class <complex>
          Slots are: 
               r = 1
               i = 10
               m = 10.0498756211209
               a = 1.47112767430373
@end lisp
@end group
@end example

Since initialization keywords have been defined for the four slots, we
can now define the @code{make-rectangular} and @code{make-polar} standard
Scheme primitives.

@lisp
(define make-rectangular 
   (lambda (x y) (make <complex> #:r x #:i y)))

(define make-polar
   (lambda (x y) (make <complex> #:magn x #:angle y)))
@end lisp

@node Class precedence list,  , Slot description, Inheritance
@subsection Class precedence list

A class may have more than one superclass.  @footnote{This section is an
adaptation of Jeff Dalton's (J.Dalton@@ed.ac.uk) @cite{Brief
introduction to CLOS}} With single inheritance (one superclass), it is
easy to order the super classes from most to least specific. This is the
rule:

@display
@cartouche
Rule 1: Each class is more specific than its superclasses.@c was \bf
@end cartouche
@end display

With multiple inheritance, ordering is harder. Suppose we have

@lisp
(define-class X ()
   (x #:init-value 1))

(define-class Y ()
   (x #:init-value 2))

(define-class Z (X Y)
   (@dots{}))
@end lisp

In this case, the @code{Z} class is more specific than the @code{X} or
@code{Y} class for instances of @code{Z}. However, the @code{#:init-value}
specified in @code{X} and @code{Y} leads to a problem: which one
overrides the other?  The rule in @goops{}, as in CLOS, is that the
superclasses listed earlier are more specific than those listed later.
So:

@display
@cartouche
Rule 2: For a given class, superclasses listed earlier are more
        specific than those listed later.
@end cartouche
@end display

These rules are used to compute a linear order for a class and all its
superclasses, from most specific to least specific.  This order is
called the ``class precedence list'' of the class. Given these two
rules, we can claim that the initial form for the @code{x} slot of
previous example is 1 since the class @code{X} is placed before @code{Y}
in class precedence list of @code{Z}.

These two rules are not always enough to determine a unique order,
however, but they give an idea of how things work.  Taking the @code{F}
class shown in Figure@ 1, the class precedence list is

@example
(f d e a c b <object> <top>)
@end example

However, it is usually considered a bad idea for programmers to rely on
exactly what the order is.  If the order for some superclasses is important,
it can be expressed directly in the class definition.

The precedence list of a class can be obtained by the function 
@code{class-precedence-list}. This function returns a ordered 
list whose first element is the most specific class. For instance,

@lisp
(class-precedence-list B) @result{} (#<<class> B 401b97c8> 
                                     #<<class> <object> 401e4a10> 
                                     #<<class> <top> 4026a9d8>)
@end lisp

However, this result is not too much readable; using the function
@code{class-name} yields a clearer result:

@lisp
(map class-name (class-precedence-list B)) @result{} (B <object> <top>) 
@end lisp

@node Generic functions,  , Inheritance, Tutorial
@section Generic functions

@menu
* Generic functions and methods::  
* Next-method::                 
* Example::                     
@end menu

@node Generic functions and methods, Next-method, Generic functions, Generic functions
@subsection Generic functions and methods

@c \label{gf-n-methods}
Neither @goops{} nor CLOS use the message mechanism for methods as most
Object Oriented language do. Instead, they use the notion of
@dfn{generic functions}.  A generic function can be seen as a methods
``tanker''. When the evaluator requested the application of a generic
function, all the methods of this generic function will be grabbed and
the most specific among them will be applied. We say that a method
@var{M} is @emph{more specific} than a method @var{M'} if the class of
its parameters are more specific than the @var{M'} ones.  To be more
precise, when a generic function must be ``called'' the system will:

@cindex generic function
@enumerate
@item
search among all the generic function those which are applicable
@item
sort the list of applicable methods in the ``most specific'' order
@item
call the most specific method of this list (i.e. the first method of 
the sorted methods list).
@end enumerate

The definition of a generic function is done with the
@code{define-generic} macro. Definition of a new method is done with the
@code{define-method} macro.  Note that @code{define-method} automatically
defines the generic function if it has not been defined
before. Consequently, most of the time, the @code{define-generic} needs
not be used.
@findex define-generic
@findex define-method
Consider the following definitions:

@lisp
(define-generic G)
(define-method  (G (a <integer>) b) 'integer)
(define-method  (G (a <real>) b) 'real)
(define-method  (G a b) 'top)
@end lisp

The @code{define-generic} call defines @var{G} as a generic
function. Note that the signature of the generic function is not given
upon definition, contrarily to CLOS. This will permit methods with
different signatures for a given generic function, as we shall see
later. The three next lines define methods for the @var{G} generic
function. Each method uses a sequence of @dfn{parameter specializers}
that specify when the given method is applicable. A specializer permits
to indicate the class a parameter must belong to (directly or
indirectly) to be applicable. If no specializer is given, the system
defaults it to @code{<top>}. Thus, the first method definition is
equivalent to

@cindex parameter specializers
@lisp
(define-method (G (a <integer>) (b <top>)) 'integer)
@end lisp

Now, let us look at some possible calls to generic function @var{G}:

@lisp
(G 2 3)    @result{} integer
(G 2 #t)   @result{} integer
(G 1.2 'a) @result{} real
@c (G #3 'a) @result{} real       @c was {\sharpsign}
(G #t #f)  @result{} top
(G 1 2 3)  @result{} error (since no method exists for 3 parameters)
@end lisp

The preceding methods use only one specializer per parameter list. Of
course, each parameter can use a specializer. In this case, the
parameter list is scanned from left to right to determine the
applicability of a method. Suppose we declare now

@lisp
(define-method (G (a <integer>) (b <number>))  'integer-number)
(define-method (G (a <integer>) (b <real>))    'integer-real)
(define-method (G (a <integer>) (b <integer>)) 'integer-integer)
(define-method (G a (b <number>))              'top-number)
@end lisp

In this case,

@lisp
(G 1 2)   @result{} integer-integer
(G 1 1.0) @result{} integer-real
(G 1 #t)  @result{} integer
(G 'a 1)  @result{} top-number
@end lisp

@node Next-method, Example, Generic functions and methods, Generic functions
@subsection Next-method

When you call a generic function, with a particular set of arguments,
GOOPS builds a list of all the methods that are applicable to those
arguments and orders them by how closely the method definitions match
the actual argument types.  It then calls the method at the top of this
list.  If the selected method's code wants to call on to the next method
in this list, it can do so by using @code{next-method}.

@lisp
(define-method (Test (a <integer>)) (cons 'integer (next-method)))
(define-method (Test (a <number>))  (cons 'number  (next-method)))
(define-method (Test a)             (list 'top))
@end lisp

With these definitions,

@lisp
(Test 1)   @result{} (integer number top)
(Test 1.0) @result{} (number top)
(Test #t)  @result{} (top)
@end lisp

@code{next-method} is always called as just @code{(next-method)}.  The
arguments for the next method call are always implicit, and always the
same as for the original method call.

If you want to call on to a method with the same name but with a
different set of arguments (as you might with overloaded methods in C++,
for example), you do not use @code{next-method}, but instead simply
write the new call as usual:

@lisp
(define-method (Test (a <number>) min max)
  (if (and (>= a min) (<= a max))
      (display "Number is in range\n"))
  (Test a))

(Test 2 1 10)
@print{}
Number is in range
@result{}
(integer number top)
@end lisp

(You should be careful in this case that the @code{Test} calls do not
lead to an infinite recursion, but this consideration is just the same
as in Scheme code in general.)

@node Example,  , Next-method, Generic functions
@subsection Example

In this section we shall continue to define operations on the @code{<complex>}
class defined in Figure@ 2. Suppose that we want to use it to implement 
complex numbers completely. For instance a definition for the addition of 
two complexes could be

@lisp
(define-method (new-+ (a <complex>) (b <complex>))
  (make-rectangular (+ (real-part a) (real-part b))
                    (+ (imag-part a) (imag-part b))))
@end lisp

To be sure that the @code{+} used in the method @code{new-+} is the standard
addition we can do:

@lisp
(define-generic new-+)

(let ((+ +))
  (define-method (new-+ (a <complex>) (b <complex>))
    (make-rectangular (+ (real-part a) (real-part b))
                      (+ (imag-part a) (imag-part b)))))
@end lisp

The @code{define-generic} ensures here that @code{new-+} will be defined
in the global environment. Once this is done, we can add methods to the
generic function @code{new-+} which make a closure on the @code{+}
symbol.  A complete writing of the @code{new-+} methods is shown in
Figure@ 3.

@example
@group
@lisp
(define-generic new-+)

(let ((+ +))

  (define-method (new-+ (a <real>) (b <real>)) (+ a b))

  (define-method (new-+ (a <real>) (b <complex>)) 
    (make-rectangular (+ a (real-part b)) (imag-part b)))

  (define-method (new-+ (a <complex>) (b <real>))
    (make-rectangular (+ (real-part a) b) (imag-part a)))

  (define-method (new-+ (a <complex>) (b <complex>))
    (make-rectangular (+ (real-part a) (real-part b))
                      (+ (imag-part a) (imag-part b))))

  (define-method (new-+ (a <number>))  a)
  
  (define-method (new-+) 0)

  (define-method (new-+ . args)
    (new-+ (car args) 
      (apply new-+ (cdr args)))))

(set! + new-+)
@end lisp

@center @emph{Fig 3: Extending @code{+} for dealing with complex numbers}
@end group
@end example

@sp 3
We use here the fact that generic function are not obliged to have the
same number of parameters, contrarily to CLOS.  The four first methods
implement the dyadic addition. The fifth method says that the addition
of a single element is this element itself. The sixth method says that
using the addition with no parameter always return 0. The last method
takes an arbitrary number of parameters@footnote{The parameter list for
a @code{define-method} follows the conventions used for Scheme
procedures. In particular it can use the dot notation or a symbol to
denote an arbitrary number of parameters}.  This method acts as a kind
of @code{reduce}: it calls the dyadic addition on the @emph{car} of the
list and on the result of applying it on its rest.  To finish, the
@code{set!} permits to redefine the @code{+} symbol to our extended
addition.

@sp 3
To terminate our implementation (integration?) of  complex numbers, we can 
redefine standard Scheme predicates in the following manner:

@lisp
(define-method (complex? c <complex>) #t)
(define-method (complex? c)           #f)

(define-method (number? n <number>) #t)
(define-method (number? n)          #f)
@dots{}
@dots{}
@end lisp

Standard primitives in which complex numbers are involved could also be
redefined in the same manner.

