Order-Sorted Inductive Types

System F ! is an extension of system F ! with subtyping and bounded quantiication. Order-sorted algebra is an extension of many-sorted algebra with overloading and subtyping. We combine both formalisms to obtain IF ! , a higher-order typed-calculus with subtyping, bounded quan-tiication and order-sorted inductive types, i.e. data types with built-in subtyping and overloading. Moreover we show that IF ! enjoys important meta-theoretic properties, including connuence, strong normaliza-tion, subject reduction and decidability of type-checking.