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".