Partial Recursive Functions in Martin-Löf Type Theory

In this article we revisit the approach by Bove and Capretta for formulating partial recursive functions in Martin-Löf Type Theory by indexed inductive-recursive definitions. We will show that all inductiverecursive definitions used there can be replaced by inductive definitions. However, this encoding results in an additional technical overhead. In order to obtain directly executable partial recursive functions, we introduce restrictions on the indexed inductive-recursive definitions used. Then we introduce a data type of partial recursive functions. This allows to define higher order partial recursive functions like the map functional, which depend on other partial recursive functions. This data type will be based on the closed formalisation of indexed inductive-recursive definitions introduced by Dybjer and the author. All elements of this data type will represent partial recursive functions, and the set of partial recursive functions will be closed under the standard operations for forming partial recursive functions, and under the total functions.