  The general form of safe recursion (or ramified recurrence) can be expressed
by an infinite graph rewrite system including unfolding graph rewrite rules
introduced by Dal Lago, Martini and Zorzi, in which the size of every normal
form by innermost rewriting is polynomially bounded. Every unfolding graph
rewrite rule is precedence terminating in the sense of Middeldorp, Ohsaki and
Zantema. Although precedence terminating infinite rewrite systems cover all the
primitive recursive functions, in this paper we consider graph rewrite systems
precedence terminating with argument separation, which form a subclass of
precedence terminating graph rewrite systems. We show that for any precedence
terminating infinite graph rewrite system G with a specific argument
separation, both the runtime complexity of G and the size of every normal form
in G can be polynomially bounded. As a corollary, we obtain an alternative
proof of the original result by Dal Lago et al.
