  We present a unification problem based on first-order syntactic unification
which ask whether every problem in a schematically-defined sequence of
unification problems is unifiable, so called loop unification. Alternatively,
our problem may be formulated as a recursive procedure calling first-order
syntactic unification on certain bindings occurring in the solved form
resulting from unification. Loop unification is closely related to Narrowing as
the schematic constructions can be seen as a rewrite rule applied during
unification, and primal grammars, as we deal with recursive term constructions.
However, loop unification relaxes the restrictions put on variables as fresh as
well as used extra variables may be introduced by rewriting. In this work we
consider an important special case, so called semiloop unification. We provide
a sufficient condition for unifiability of the entire sequence based on the
structure of a sufficiently long initial segment. It remains an open question
whether this condition is also necessary for semiloop unification and how it
may be extended to loop unification.
