Formalising Inductive and Coinductive Containers

Containers capture the concept of strictly positive data types in programming. The original development of containers is done in the internal language of locally cartesian closed categories (LCCCs) with disjoint coproducts and W-types, and uniqueness of identity proofs (UIP) is implicitly assumed throughout. Although it is claimed that these developments can also be interpreted in extensional Martin-L"of type theory, this interpretation is not made explicit. In this paper, we present a formalisation of the results that `containers preserve least and greatest fixed points' in \cubicalagda, thereby giving a formulation in intensional type theory. Our proofs do not make use of UIP and thereby generalise the original results from talking about container functors on $\mathsf{Set}$ to container functors on the wild category of types. Our main incentive for using \cubicalagda is that its path type restores the equivalence between bisimulation and coinductive equality. Thus, besides developing container theory in a more general setting, we also demonstrate the usefulness of \cubicalagda's path type to coinductive proofs.