Note that acyclic_term/1 is ISO, but cyclic_term/1 is not.

Consider the evolution of a term from "most uninstantiated" to "ground":

                       uninstantiated
                             |
                             V
              +--------------+-------------+
              |                            |
              V                            |
      nonground noncyclic                  |
              |                            |
              +----------------------------+
              |                            |
              |                            V
              |                     nonground cyclic
              |                            |
              V                            V
        ground acylic                ground cyclic

Try

examine(X) :-
   (ground(X)       -> format("ground,",[])   ; format("non-ground,",[])),
   (cyclic_term(X)  -> format("cyclic,",[])   ; format("not cyclic,",[])),
   (acyclic_term(X) -> format("acyclic~n",[]) ; format("not acyclic~n",[])).
   
X=_      , examine(X). % uninstantiated
X=s(_)   , examine(X). % nonground, no cycles
X=s(a)   , examine(X). % ground, no cycles
X=s(X)   , examine(X). % ground, cycles
X=s(X,_) , examine(X). % nonground, cylces

To see what the predicates say at every point in the evolution of instantiation

                             X=_
                   not "cyclic", "acyclic"
                             |
                             V
              +--------------+-------------+
              |                            |
              V                            |
            X=s(_)                         |
   not "cyclic", "acyclic"                 |
              |                            |
              +----------------------------+
              |                            |
              |                            V
              |                         X=s(X,_)
              |                 "cyclic", not "acyclic"
              |                            |
              V                            V
            X=s(a)                       X=s(X)
    not "cyclic", "acyclic"      "cyclic", not "acyclic"
                                

cyclic_term/1 is a bit optimistic as it says "not cyclic" even though the same term can become cyclic by instantiations later.

The predicates cyclic_term/1 and acyclic_term/1 are in fact "second order": They say something about the current state of computation, not about the term they are examining.

One would like to see these predicates:

	cyclic	cyclic_term/1	acyclic_term/1	acyclic_forever
uninstantiated	throw	false (could change)	true (could change)	false
nonground acyclic	throw	false (could change)	true (could change)	false
ground acylic	false (for sure	false (for sure)	true (for sure)	true
nonground cyclic	true (for sure)	true (for sure)	false (for sure)	false
ground cyclic	true (for sure)	true (for sure)	false (for sure)	false

The above have been implemented in checks.pl