# Inductive Data Type Systems

In a previous work (Abstract Data Type Systems'', TCS 173(2):349--391 (1997)), the last two authors presented a combined language made of a (strongly normalizable) algebraic rewriting system, and a typed $\lambda$-calculus enriched by pattern-matching definitions following a certain format, called the General Schema'', which generalizes the usual recursor definitions for the natural numbers and similar basic inductive types''. This combined language was shown to be strongly normalizable. The purpose of this paper is to extend the General Schema in order to capture a more general class of inductive types, called strictly positive inductive types'', and to prove the strong normalizability of the resulting combined system. This result provides the computation model for the combined language of an algebraic specification language based on abstract data types, and the strongly typed functional language with strictly positive inductive types.