Abstract. In this paper we use the Epigram language to define the universe of regular tree types—closed under empty, unit, sum, product and least fixpoint. We then present a generic decision procedure for Epigram’s in-built equality at each type, taking a complementary approach to that of Benke, Dybjer and Jansson [7]. We also give a generic definition of map, taking our inspiration from Jansson and Jeuring [21]. Finally, we equip the regular universe with the partial derivative which can be interpreted functionally as Huet’s notion of ‘zipper’, as suggested by McBride in [27] and implemented (without the fixpoint case) in Generic Haskell by Hinze, Jeuring and Löh [18]. We aim to show through these examples that generic programming can be ordinary programming in a dependently typed language. 1

Abstract. We present four constructions for standard equipment which can be generated for every inductive datatype: case analysis, structural recursion, no confusion, acyclicity. Our constructions follow a two-level approach—they require less work than the standard techniques which inspired them [11, 8]. Moreover, given a suitably heterogeneous notion of equality, they extend without difficulty to inductive families of datatypes. These constructions are vital components of the translation from dependently typed programs in pattern matching style [7] to the equivalent programs expressed in terms of induction principles [21] and as such play a crucial behind-the-scenes rôle in Epigram [25]. 1

This paper, submitted as a ‘pearl’, introduces a small but useful generalisation to the ‘derivative ’ operation on datatypes underlying Huet’s notion of ‘zipper ’ (Huet 1997; McBride 2001; Abbott et al. 2005b), giving a concrete representation to one-hole contexts in data which is in mid-transformation. This operator, ‘dissection’, turns a container-like functor into a bifunctor representing a onehole context in which elements to the left of the hole are distinguished in type from elements to its right. I present dissection for polynomial functors, although it is certainly more general, preferring to concentrate here on its diverse applications. For a start, map-like operations over the functor and fold-like operations over the recursive data structure it induces can be expressed by tail recursion alone. Moreover, the derivative is readily recovered from the dissection, along with Huet’s navigation operations. A further special case of dissection, ‘division’, captures the notion of leftmost hole, canonically distinguishing values with no elements from those with at least one. By way of a more practical example, division and dissection are exploited to give a relatively efficient generic algorithm for abstracting all occurrences of one term from another in a first-order syntax. The source code for the paper is available online 1 and compiles with recent extensions to the Glasgow Haskell Compiler. 1.

Abstract. Containers are a semantic way to talk about strictly positive types. In previous work it was shown that containers are closed under various constructions including products, coproducts, initial algebras and terminal coalgebras. In the present paper we show that, surprisingly, the category of containers is cartesian closed, giving rise to a full cartesian closed subcategory of endofunctors. The result has interesting applications syntax. We also show that while the category of containers has finite limits, it is not locally cartesian closed. 1

Abstract. Containers are a semantic way to talk about strictly positive types. In previous work it was shown that containers are closed under various constructions including products, coproducts, initial algebras and terminal coalgebras. In the present paper we show that, surprisingly, the category of containers is cartesian closed, giving rise to a full cartesian closed subcategory of endofunctors. The result has interesting applications syntax. We also show that the category of containers has finite limits, but it is not locally cartesian closed. 1