How about defining path/4 like this?
path(R_2, Xs, A,Z) :- % A path `Xs` from `A` to `Z` is ...
walk(R_2, Xs, A,Z), % ... a walk `Xs` from `A` to `Z` ...
all_dif(Xs). % ... with no duplicates in `Xs`.
To aid universal termination, we swap the two goals in above conjunction …
path(R_2, Xs, A,Z) :-
all_dif(Xs), % enforce disequality ASAP
walk(R_2, Xs, A,Z).
… and use the following lazy implementation of all_dif/1:
all_dif(Xs) :- % enforce pairwise term inequality freeze(Xs, all_dif_aux(Xs,[])). % (may be delayed) all_dif_aux([], _). all_dif_aux([E|Es], Vs) :- maplist(dif(E), Vs), % is never delayed freeze(Es, all_dif_aux(Es,[E|Vs])). % (may be delayed)
walk/4 is defined like path/4 and path/5 given by the OP:
:- meta_predicate walk(2, ?, ?, ?).
walk(R_2, [X0|Xs], X0,X) :-
walk_from_to_step(Xs, X0,X, R_2).
:- meta_predicate walk_from_to_step(?, ?, ?, 2).
walk_from_to_step([], X,X, _).
walk_from_to_step([X1|Xs], X0,X, R_2) :-
call(R_2, X0,X1),
walk_from_to_step(Xs, X1,X, R_2).
IMO above path/4 is simpler and more approachable, particularly for novices. Would you concur?