ΠΣ: Dependent Types without the Sugar

Abstract

Abstract. The recent success of languages like Agda and Coq demonstrates the potential of using dependent types for programming. These systems rely on many high-level features like datatype definitions, pattern matching and implicit arguments to facilitate the use of the languages. However, these features complicate the metatheoretical study and are a potential source of bugs. To address these issues we introduce ΠΣ, a dependently typed core language. It is small enough for metatheoretical study and the type checker is small enough to be formally verified. In this language there is only one mechanism for recursion-used for types, functions and infinite objectsand an explicit mechanism to control unfolding, based on lifted types. Furthermore structural equality is used consistently for values and types; this is achieved by a new notion of α-equality for recursive definitions. We show, by translating several high-level constructions, that ΠΣ is suitable as a core language for dependently typed programming.

General Bindings and Alpha-Equivalence in Nominal Isabelle

“…In this case they also agree with the alpha-equivalence used in older versions of Nominal Isabelle [26]. 1 In the rest of this section we are going to show that the alpha-equivalences really lead to abstractions where some atoms are bound (or more precisely removed from the support). For this we will consider three abstraction types that are quotients of the relations…”

“…Finally, we apply the permutation operation on booleans. 1 We omit a proof of this fact since the details are hairy and not really important for the purpose of this paper.…”

Truly Modular (Co)datatypes for Isabelle/HOL

“…2 These two definitions reflect the modular, nested view: First α F is defined as an LFP, becoming a BNF in its own right; then α G is defined as an LFP using an equation that nests F. The resulting iterator for α G, iter_ G, has type ((α, γ F) G 0 ⇒ γ) ⇒ α G ⇒ γ, and its characteristic equation recurses through the F components of G using map_ F.…”

“…Such types are interesting in their own right, irrespective of whether they support (co)induction; for example, pattern matching requires only distinctness and injectivity. 2 It may help to think of these types more concretely by taking…”

A typed store-passing translation for general references

“…There exist other calculi that permit productive computation at the type level: ΠΣ [3] and MiniAgda [1] come to mind. Perhaps these calculi could serve as target languages for a type-preserving store-passing translation of general references.…”