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

Third try! Developed and tested with GNU Prolog 1.4.4.

---

Exhibit 'A': "as simple as it gets"

lt(X,Y) :-
   X \== Y,
   (  X \= Y
   -> alpha_omega(Alpha,Omega),
      term_variables(X+Y,Vars),                           % A
      \+ \+ (label_vars(Vars,Alpha,Omega), X @< Y),
      (  \+ (label_vars(Vars,Alpha,Omega), X @> Y)
      -> true
      ;  throw(error(instantiation_error,lt/2))
      )
   ;  throw(error(instantiation_error,lt/2))
   ).    

---

Exhibit 'B': "no need to label all vars"

lt(X,Y) :-
   X \== Y,
   (  X \= Y
   -> alpha_omega(Alpha,Omega),
      term_variables(X,Xvars),                            % B
      term_variables(Y,Yvars),                            % B 
      vars_vars_needed(Xvars,Yvars,Vars),                 % B
      \+ \+ (label_vars(Vars,Alpha,Omega), X @< Y),
      (  \+ (label_vars(Vars,Alpha,Omega), X @> Y)
      -> true
      ;  throw(error(instantiation_error,lt/2))
      )
   ;  throw(error(instantiation_error,lt/2))
   ).

vars_vars_needed([],    [],    []).
vars_vars_needed([A|_], [],    [A]).
vars_vars_needed([],    [B|_], [B]).
vars_vars_needed([A|As],[B|Bs],[A|ABs]) :-
   (  A \== B
   -> ABs = [B]
   ;  vars_vars_needed(As,Bs,ABs)
   ).

---

Some shared code:

alpha_omega(Alpha,Omega) :-
    Alpha is -(10.0^1000),    % HACK!
    functor(Omega,z,255).     % HACK!

label_vars([],_,_).
label_vars([Alpha|Vs],Alpha,Omega) :- label_vars(Vs,Alpha,Omega).
label_vars([Omega|Vs],Alpha,Omega) :- label_vars(Vs,Alpha,Omega).