Bounded Natural Functors with Covariance and Contravariance

Bounded natural functors (BNFs) provide a modular framework for the construction of (co)datatypes in higher-order logic. Their functorial operations, the mapper and relator, are restricted to a subset of the parameters, namely those where recursion can take place. For certain applications, such as free theorems, data refinement, quotients, and generalised rewriting, it is desirable that these operations do not ignore the other parameters. In this article, we formalise the generalisation BNF_CC that extends the mapper and relator to covariant and contravariant parameters. We show that

1. BNF_CCs are closed under functor composition and least and greatest fixpoints,
2. subtypes inherit the BNF_CC structure under conditions that generalise those for the BNF case, and
3. BNF_CCs preserve quotients under mild conditions.

These proofs are carried out for abstract BNF_CCs similar to the AFP entry BNF Operations. In addition, we apply the BNF_CC theory to several concrete functors.