Pure Type Systems without Explicit Contexts

• Main Authors: Herman Geuvers, Robbert Krebbers, James McKinna, Freek Wiedijk
• Format: Article
• Language: English
• Published: Open Publishing Association 2010-09-01
• Series: Electronic Proceedings in Theoretical Computer Science
• Online Access: http://arxiv.org/pdf/1009.2792v1

We present an approach to type theory in which the typing judgments do not have explicit contexts. Instead of judgments of shape "Gamma |- A : B", our systems just have judgments of shape "A : B". A key feature is that we distinguish free and bound variables even in pseudo-terms. Specifically we give the rules of the "Pure Type System" class of type theories in this style. We prove that the typing judgments of these systems correspond in a natural way with those of Pure Type Systems as traditionally formulated. I.e., our systems have exactly the same well-typed terms as traditional presentations of type theory. Our system can be seen as a type theory in which all type judgments share an identical, infinite, typing context that has infinitely many variables for each possible type. For this reason we call our system "Gamma_infinity". This name means to suggest that our type judgment "A : B" should be read as "Gamma_infinity |- A : B", with a fixed infinite type context called "Gamma_infinity".