In the simply-typed lambda-calculus, a hereditary substitution replaces a free variable in a normal form r by another normal form s of type a, removing freshly created redexes on the fly. It can be defined by lexicographic induction on a and r, thus, giving rise to a structurally recursive normalizer for the simply-typed lambda-calculus. We generalize this scheme to simultaneous substitutions, preserving its simple termination argument. We further implement hereditary simultaneous substitutions in a functional programming language with sized heterogeneous inductive types, Fωb, arriving at an interpreter whose termination can be tracked by the type system of its host programming language.

Abstract. A type-based approach to termination uses sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is only sound if the type of the recursive function is admissible, i.e., depends on the size index in a certain way. To explore the space of admissible functions in the presence of higher-kinded data types and impredicative polymorphism, a semantics is developed where sized types are interpreted as functions from ordinals into sets of strongly normalizing terms. It is shown that upper semi-continuity of such functions is a sufficient semantical criterion for admissibility. To provide a syntactical criterion, a calculus for semi-continuous function is developed. 1

Instances of a polytypic or generic program for a concrete recursive type often exhibit a recursion scheme that is derived from the recursion scheme of the instantiation type. In practice, the programs obtained from a generic program are usually terminating, but the proof of termination cannot be carried out with traditional methods as term orderings alone, since termination often crucially relies on the program type. In this article, it is demonstrated that type-based termination using sized types handles such programs very well. A framework for sized polytypic programming is developed which ensures (typebased) termination of all instances. 1