Well-Founded Sized Types in the Calculus of (Co)Inductive Constructions

Type-based termination is a mechanism for ensuring termination and productivity of (co)recursive definitions [4]. Its main feature is the use of sized types (i.e. types annotated with size information) to track the size of arguments in (co)recursive calls. Termination of recursive function (and productivity of corecursive functions) is ensured by restricting recursive calls to smaller arguments (as evidenced by their types). Type-based approaches to termination and productivity have several advantages over the syntactic-based approaches currently implemented in Coq and Agda [5, 1, 2]. In particular, they are more expressive and easier to understand. In previous work [6, 5], we studied the metatheory of extensions of the Calculus of (Co)Inductive Constructions (CIC) with a simple notion of sized types. While these systems are more expressive than the guard predicates of Coq, there have two main shortcomings in the case of coinductive types. First, corecursive definitions that use tail are not expressible, e.g.: corec zeroes := cons(0, cons(0, tail zeroes)). Second, Subject Reduction (SR) does not hold (a well-known problem in Coq that was observed by Gimnez [3]). While in practice, the lack of SR does not seem to be a major issue, it is theoretically unsatisfying. In this work, we consider an extension of CIC with a notion of well-founded sized types, inspired by F ω [2], that solves both issues. As in previous work, (co)inductive types are annotated with size expressions taken from the size algebra given by s ::= ı | s+ 1 | ∞, where ı is a size variable. In this approach, the (simplified) typing rule for streams looks like: