The standard formulation of bounded quantification, system F , is difficult to work with and lacks important syntactic properties, such as decidability. More tractable variants have been studied, but those studied so far either exclude significant classes of useful programs or lack a compelling semantics. We propose

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

In this paper, we show how to manipulate syntax with binding using a mixed representation of names for free variables (with respect to the task in hand) and de Bruijn indices [dB72] for bound variables. By doing so, we retain the advantages of both representations: naming supports easy, arithmetic-free manipulation of terms; de Bruijn indices eliminate the need for α-conversion. Further, we have ensure that not only the user but also the implementation need never deal with de Bruijn indices, except within key basic operations. Moreover, we give a representation for names which readily supports a power structure naturally reflecting the structure of the implementation. Name choice is safe and straightforward. Our technology combines easily with an approach to syntax manipulation inspired by Huet’s ‘zippers’[Hue97]. Without the technology in this paper, we could not have implemented Epigram [McB04]. Our example—constructing inductive elimination operators for datatype families—is but one of many where it proves invaluable. Prologue In conversation, we like to have names for the people we’re talking about. If we had to say things like ‘the person three to the left of me ’ rather than ‘Fred’, things would get complicated whenever anyone went to the lavatory. You don’t need to have formalized the strengthening property for Pure Type Systems [MP99] to appreciate this basic phenomenon of social interaction. It is in the company of strangers that more primitive pointing-based modes of reference acquire a useful rôle as a way of indicating unambiguously an individual with no socially agreed name. Even so, once a stranger enters the context of the conversation, he or she typically acquires a name. What this name is and who chooses it depends on the power relationships between those involved, as we learned in the playground at school. Moreover, if we are having a conversation about hypothetical individuals—say, Alice, Bob and Unscrupulous Charlie—we have a tendency to name them locally to the discussion. We do not worry about whether Unscrupulous Charlie might actually turn out to be called Shameless David whenever he turns up. That is, we exploit naming locally to assist the construction of explanations which apply to individuals regardless of what they are called. 1 1