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In this paper we define and study intersection type assignment systems for first-order rewriting extended with application, -abstraction, and -reduction ( ). One of the main results presented is that, using a suitable notion of approximation of terms, any typeable term of a that satisfies a general scheme for recursive definitions has an approximant of the same type. From this result we deduce, for different classes of typeable terms, a head-normalization and a normalization theorem.

Rewrite systems with abstraction and β-rule: Types, approximants and normalization