I think the title is a bit weird but, for example, is there a way to redefine the addition in Scheme so that (+ arg1 arg2) gives arg1 + arg2 + 1?
I can create a new function @:
(define @ (lambda args (+ (apply + args) 1)))
such that (@ 42 13) gives (42 + 13 + 1). But is there a way to do something which would look like:
(define + (lambda args (+ (apply + args) 1)))
but of course without the error since, here, it says that + is undefined?
In R7RS version of Scheme we have imports so here we can do it:
#!r7rs
(import (except (scheme base) +)
(rename (scheme base) (+ base:+)))
(define (+ . args)
(apply base:+ 1 args))
(+ 1 1) ; ==> 3
In R6RS version its the same except (scheme base) is (rnrs base)
Racket has something very similar:
#lang racket
(require (only-in racket/base [+ base:+]))
(define (+ . args)
(apply base:+ 1 args))
(+ 1 1) ; ==> 3
In R5RS report Scheme and older the report allows a user to redefine a procedure to extend the features of it, but it is a violation that it for the defined types would result in different results than the original scheme report version would. Thus redefining + like this is a violation and not portable Scheme. It might work in some implementation, but the wording is done such that a smart Scheme compiler can do constant folding and expect the correct result from supported types.
#!r5rs
(define +
(let ((r5rs:+ +))
(lambda args
(apply r5rs:+ 1 args))))
(+ 1 1) ; ==> 2 or 3, perhaps an error