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.

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

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

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).