Modular languages support generative type abstraction, ensuring that an abstract type is distinct from its representation, except inside the implementation where the two are synonymous. We show that this well-established feature is in tension with the non-parametric features of newer type systems, such as indexed type families and GADTs. In this paper we solve the problem by using kinds to distinguish between parametric and non-parametric contexts. The result is directly applicable to Haskell, which is rapidly developing support for type-level computation, but the same issues should arise whenever generativity and non-parametric features are combined.

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

We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of Martin-Lof type theory with W-type and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of Martin-Lof type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally we repeat the concept of inductive-recursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.

Instances of a polytypic or generic program for a concrete recursive type often exhibit a recursion scheme that is derived from the recursion scheme of the instantiation type. In practice, the programs obtained from a generic program are usually terminating, but the proof of termination cannot be carried out with traditional methods as term orderings alone, since termination often crucially relies on the program type. In this article, it is demonstrated that type-based termination using sized types handles such programs very well. A framework for sized polytypic programming is developed which ensures (typebased) termination of all instances. 1

Abstract not found

ML5 is a programming language for spatially distributed computing, based on a Curry-Howard correspondence with the modal logic S5. However, the ML5 programming language differs from the logic in several ways. In this paper, we give a semantic embedding of ML5 into the dependently typed programming language Agda, which both explains these discrepancies between ML5 and S5 and suggests some simplifications and generalizations of the language. Our embedding translates ML5 into a slightly different logic: intuitionistic S5 extended with a lax modality that encapsulates effectful computations in a monad. Rather than formalizing lax S5 as a proof theory, we embed it as a universe within the the dependently typed host language, with the universe elimination given by implementing the modal logic’s Kripke semantics. 1

In these lecture notes we give an overview of recent research on the relationship

The UNIX diff program finds the difference between two text files using a classic algorithm for determining the longest common subsequence; however, when working with structured input (e.g. program code), we often want to find the difference between tree-like data (e.g. the abstract syntax tree). In a functional programming language such as Haskell, we can represent this data with a family of (mutually recursive) datatypes. In this paper, we describe a functional, datatype-generic implementation of diff (and the associated program patch). Our approach requires advanced type system features to preserve type safety; therefore, we present the code in Agda, a dependently-typed language well-suited to datatypegeneric programming. In order to establish the usefulness of our work, we show that its efficiency can be improved with memoization and that it can also be defined in Haskell.

We begin by revisiting the idea of using a universe of types to write generic programs in a dependently typed setting by constructing a universe for Strictly Positive Types (SPTs). Here we extend this construction to cover dependent types, i.e. Strictly Positive Families (SPFs), thereby fixing a gap left open in previous work. Using the approach presented here we are able to represent all of Epigram’s datatypes within Epigram including the universe of datatypes itself.

Modular languages support generative type abstraction, ensuring that an abstract type is distinct from its representation, except inside the implementation where the two are synonymous. We show that this well-established feature is in tension with the non-parametric features of newer type systems, such as indexed type families and GADTs. In this paper we solve the problem by using kinds to distinguish between parametric and non-parametric contexts. The result is directly applicable to Haskell, which is rapidly developing support for type-level computation, but the same issues should arise whenever generativity and non-parametric features are combined. 1.