prolog sum & sub using successors ?

Hi , what is substraction and summation using successors , can any one show me an example for that i know how to do it the normal way .

/* sub(X, Y, Z) ---subtraction */

sub(X, Y, Z) :- add(Y, Z, X).

From stackoverflow

• First, you need to have a predicate succ. This is how SWI-Prolog defines it:

  succ(?Int1, ?Int2)
      True  if  Int2 = Int1+ 1  and  Int1>=0.     At  least  one  of  the
      arguments must be  instantiated to a natural number.  This predicate
      raises the domain-error not_less_than_zero if called with a negative
      integer.   E.g. succ(X, 0)  fails silently and succ(X, -1) raises  a
      domain-error.
  

  Given that, we can define add like this:

  add(0, Y, Y).
  add(X, Y, Z) :-
    succ(PredX, X),
    add(PredX, Y, PredZ),
    succ(PredZ, Z).
  

  And subtract like this:

  subtract(X, 0, X).
  subtract(X, Y, Z) :-
    succ(PredY, Y),
    succ(PredX, X),
    subtract(PredX, PredY, Z).
  

  Note that neither one of these will handle negative numbers (because succ doesn't), and therefore I haven't bothered to make subtract function when Y > X.

  ---

  EDIT:

  Here's a version of add and subtract that work on any instantiation pattern. I still didn't bother with type-checking (as mentioned by Kaarel in the comments), or negative numbers.

  add(0, 0, 0).
  add(0, Y, Y).
  add(X, 0, X).
  add(X, Y, Z) :-
    nonvar(X),
    succ(PredX, X),
    (nonvar(Z) -> 
      succ(PredZ, Z), add(PredX, Y, PredZ) 
    ; 
      add(PredX, Y, PredZ), succ(PredZ, Z)
    ).
  add(X, Y, Z) :-
    nonvar(Y),
    add(Y, X, Z).
  
  subtract(0, 0, 0).
  subtract(X, 0, X).
  subtract(X, X, 0).
  subtract(X, Y, Z) :-
    add(Y, Z, X).
  

  Kaarel : Note that the instantiation patterns add(-, +, +) and add(+, -, +) do not work under this definition. Also, you could cut after "add(0, Y, Y)", and check the input type, currently: ?- add(0, blah, R). R = blah ; false.

  Pesto : Actually, add(+, -, +) does work because succ is never called on Y. However, see my edit.

  Kaarel : ?- add(2, X, 3). ERROR: succ/2: Arguments are not sufficiently instantiated