Several proof-assistants rely on the very formal basis of Pure Type Systems (PTS) as their foundations. We are concerned with the issues involved in the development of large proofs in these provers such as namespace management, development of reusable proof libraries and separate verification. Although implementations offer many features to address them, few theoretical foundations have been laid for them up to now. This is a problem as features dealing with modularity may have harmful, unsuspected effects on the reliability or usability of an implementation.

In this paper, we propose an extension of Pure Type Systems with a module system, , adapted from SML-like module systems for programming languages. This system gives a theoretical framework addressing the issues mentioned above in a quite uniform way. It is intended to be a theoretical foundation for the module systems of proof-assistants such as Coq or Agda. We show how reliability and usability can be formalized as metatheoretical properties and prove they hold for .

Safety has become a fundamental requirement in all aspects of computer systems. Object-oriented calculi, such as Castagna's λ&-calculus and its variants (Castagna, 1997) ensure type safety in environments based on the distinguished object-oriented paradigm. Although for safety reasons object invariance and operation specifications are getting widely employed in all stages of the engineering process, they are not supported by these calculi. In this paper, a new calculus is presented which supports term (value) constraints besides the key object-oriented mechanisms (class types, inheritance, overloading with multiple dispatch and late binding). We also show how a type with constraints may realise a role, another useful object-oriented modelling element. The soundness of the type system and the confluence of the notion of reduction of the calculus are considered. The contribution also discusses computability issues partially arising from the use of first-order logic to formalise the constraints.

We examine the ML pattern-matching anomalies of useless clauses and non-exhaustive matches. We state the definition of these anomalies, building upon pattern matching semantics, and propose a simple algorithm to detect them. We have integrated the algorithm in the Objective Caml compiler, but we show that the same algorithm is also usable in a non-strict language such as Haskell. Or-patterns are considered for both strict and non-strict languages.