  We provide a type-theoretical characterization of weakly-normalizing terms in
an infinitary lambda-calculus. We adapt for this purpose the standard
quantitative (with non-idempotent intersections) type assignment system of the
lambda-calculus to our infinite calculus. Our work provides a new answer to
Klop's HHN-problem, namely, finding out if there is a type system
characterizing the hereditary head-normalizing (HHN) lambda-terms. Tatsuta
showed that HHN could not be characterized by a finite type system. We prove
that an infinitary type system endowed with a validity condition called
approximability can achieve it.
