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 shape Γ ` 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 Γ∞. This name means to suggest that our type judgment A : B should be read as Γ∞ ` A : B, with a fixed infinite type context called Γ∞.