Proof irrelevance and subset types
Our aim in this chapter is to present a syntactic model for the extensional constructs of proof irrelevance and subset types. The model we give not only provides these concepts, but also relates two approaches to program development in Martin-Löf type theory  or the Extended Calculus of Constructions (ECC) . Under the first one specifications and types are freely mixed using ∑-types. This methodology underpins a project for development of correct software at the University of Ulm . It is also the most natural approach and is probably employed by most users of type systems like the ones under consideration.
KeywordsType Theory Syntactic Category Equality Rule Equality Judgement Refined Type
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