Deciding Properties of Lists using Containers

We exploit the ability to represent data types as container functors [2,1,3] to develop a novel approach to proving properties of lists using arithmetic decision procedures. Containers capture the idea that concrete data types can be characterised by specifying the shape values take and for every possible shape, explaining where positions within that shape are stored. More importantly, a representation theorem guarantees that polymorphic functions between container data types are given by container morphisms, which are characterised by mappings between shapes and positions. The key to our approach is to restrict the shape maps of container morphisms to functions that have decidable equality, but which allow for a large class of functions. We also capture the behaviour of position mappings of container morphisms as functions on the natural numbers. The shape maps which we consider are given by piecewise-linear functions, of type Nn → N. Such functions are decidable, and this enables us to implement decision procedures for lists.