Refining Inductive Types

TL;DR

This paper presents a generic method for deriving inductive characterisations of refined data types in dependently typed programming, simplifying their creation and use while maintaining compatibility with existing techniques.

Contribution

It introduces a generic approach to derive inductive characterisations of refined types, reducing ad hoc refinements and enhancing their integration with standard programming and reasoning techniques.

Findings

Provides a generic derivation method for inductive refinements.

Ensures refinements are compatible with existing inductive type techniques.

Facilitates further refinement of data types.

Abstract

Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the N-indexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purpose-specific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations must often store redundant information about data and their refinements. In this paper we show how to generically derive inductive characterisations of refinements of inductive types, and argue that these characterisations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. Our characterisations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined.