How to define (and name) the corresponding safe term comparison predicates in ISO Prolog?

Question

Standard term order (ISO/IEC 13211-1 7.2 Term order) is defined over all terms — including variables. While there are good uses for this — think of the implement

Answer 1

What the heck! I'll give it a shot, too!

lt(X,Y) :-
   X \== Y,
   (  X \= Y
   -> term_variables(X,Xvars),
      term_variables(Y,Yvars),
      list_vars_excluded(Xvars,Yvars,XonlyVars),
      list_vars_excluded(Yvars,Xvars,YonlyVars),

      _   = s(T_alpha),
      functor(T_omega,zzzzzzzz,255), % HACK!

      copy_term(t(X,Y,XonlyVars,YonlyVars),t(X1,Y1,X1onlyVars,Y1onlyVars)),
      copy_term(t(X,Y,XonlyVars,YonlyVars),t(X2,Y2,X2onlyVars,Y2onlyVars)),
      maplist(=(T_alpha),X1onlyVars), maplist(=(T_omega),Y1onlyVars),
      maplist(=(T_omega),X2onlyVars), maplist(=(T_alpha),Y2onlyVars),

      % do T_alpha and T_omega have an impact on the order?
      (  compare(Cmp,X1,Y1),      
         compare(Cmp,X2,Y2)
      -> Cmp = (<)                % no: demand that X @< Y holds
      ;  throw(error(instantiation_error,lt/2))
      )

   ;  throw(error(instantiation_error,lt/2))
   ).

Some more auxiliary stuff:

listHasMember_identicalTo([X|Xs],Y) :-
   (  X == Y
   -> true
   ;  listHasMember_identicalTo(Xs,Y)
   ).

list_vars_excluded([],_,[]).
list_vars_excluded([X|Xs],Vs,Zs) :-
   (  listHasMember_identicalTo(Vs,X)
   -> Zs = Zs0
   ;  Zs = [X|Zs0]
   ),
   list_vars_excluded(Xs,Vs,Zs0).

Let's have some tests (with GNU Prolog 1.4.4):

?- lt(a(X,Y,c(c),Z1), a(X,Y,b(b,b),Z2)).
yes
?- lt(a(X,Y,b(b,b),Z1), a(X,Y,c(c),Z2)).
no
?- lt(a(X,Y1,c(c),Z1), a(X,Y2,b(b,b),Z2)).
uncaught exception: error(instantiation_error,lt/2)
?- lt(a(X,Y1,b(b,b),Z1), a(X,Y2,c(c),Z2)).
uncaught exception: error(instantiation_error,lt/2)
?- lt(b(b), a(a,a)).
yes
?- lt(a(X), a(Y)).
uncaught exception: error(instantiation_error,lt/2)
?- lt(X, 3).
uncaught exception: error(instantiation_error,lt/2)
?- lt(X+a, X+b).
yes
?- lt(X+a, Y+b).
uncaught exception: error(instantiation_error,lt/2)
?- lt(a(X), b(Y)).
yes
?- lt(a(X), a(Y)).
uncaught exception: error(instantiation_error,lt/2)
?- lt(a(X), a(X)).
no

---

Edit 2015-05-06

Changed the implementation of lt/2 to use T_alpha and T_omega, not two fresh variables.

• lt(X,Y) makes two copies of X (X1 and X2) and two copies of Y (Y1 and Y2).
• Shared variables of X and Y are also shared by X1 and Y1, and by X2 and Y2.
• T_alpha comes before all other terms (in X1, X2, Y1, Y2) w.r.t. the standard order.
• T_omega comes after all other terms in the standard order.
• In the copied terms, the variables that are in X but not in Y (and vice versa) are unified with T_alpha / T_omega.
  • If this has an impact on term ordering, we cannot yet decide the ordering.
  • If it does not, we're done.

Now, the counterexample given by @false works:

?- lt(X+1,1+2).
uncaught exception: error(instantiation_error,lt/2)
?- X=2, lt(X+1,1+2).
no