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Consider the following definitions:

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

The `define-generic`

call defines `G` as a generic function.
The three next lines define methods for `G`. Each method uses a
sequence of *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 `<top>`

. Thus,
the first method definition is equivalent to

(define-method (G (a <integer>) (b <top>)) 'integer)

Now, let’s look at some possible calls to the generic function `G`:

(G 2 3) ⇒ integer (G 2 #t) ⇒ integer (G 1.2 'a) ⇒ real (G #t #f) ⇒ top (G 1 2 3) ⇒ error (since no method exists for 3 parameters)

The methods above use only one specializer per parameter list. But in general, any or all of a method’s parameters may be specialized. Suppose we define now:

(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)

With these definitions:

(G 1 2) ⇒ integer-integer (G 1 1.0) ⇒ integer-real (G 1 #t) ⇒ integer (G 'a 1) ⇒ top-number

As a further example we shall continue to define operations on the
`<my-complex>`

class. Suppose that we want to use it to implement
complex numbers completely. For instance a definition for the addition
of two complex numbers could be

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

To be sure that the `+`

used in the method `new-+`

is the
standard addition we can do:

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

The `define-generic`

ensures here that `new-+`

will be defined
in the global environment. Once this is done, we can add methods to the
generic function `new-+`

which make a closure on the `+`

symbol. A complete writing of the `new-+`

methods is shown in
Figure 8.1.

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

We take advantage here of the fact that generic function are not obliged
to have a fixed number of parameters. The four first methods implement
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 (as is also true for the
primitive `+`

). The last method takes an arbitrary number of
parameters^{34}. This method acts as a kind of `reduce`

: it calls the
dyadic addition on the *car* of the list and on the result of
applying it on its rest. To finish, the `set!`

permits to redefine
the `+`

symbol to our extended addition.

To conclude our implementation (integration?) of complex numbers, we could redefine standard Scheme predicates in the following manner:

(define-method (complex? c <my-complex>) #t) (define-method (complex? c) #f) (define-method (number? n <number>) #t) (define-method (number? n) #f) …

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

The parameter list for a `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