Pure Type Systems without Explicit Contexts

We present an approach to type theory in which the typing judgments do not have explicit contexts. Instead of judgments of the shape A : B, our systems just have judgments of the shape A : B. An essential feature of our systems is that already in pseudo-terms we distinguish the free from the bound variables. Specifically we give the rules of the `Pure Type System' class of type theories in this style. We prove that the type judgments of these systems corresponds in a natural way with the judgments of the traditionally formulated Pure Type Systems. 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 . This name means to suggest that our type judgment A : B should be read as A : B, with a fixed infinite type context called .

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Herman Geuvers, Robbert Krebbers, James McKinna, Freek Wiedijk
• Keywords: CORR 2010, Education, Judgments, Pure Type Systems, Type Judgment