Abstract

Pure Type Systems are a good way to factorize the questions of meta-theory about a large family of type systems. They have been introduced as a generalization of Barendregt’s λ-cube, an abstraction of several type systems like the Simply Typed λ-Calculus, System F or the Calculus of Constructions. One critical detail of the Pure Type Systems is their conversion rule that allows to do computation at the level of types. Traditionally, Pure Type Systems are presented in a natural deduction style, and use an untyped notion of conversion. Through the years, several presentations of the Pure Type Systems have been used, with subtle differences like sequent calculus instead of natural deduction, or the use of a typed conversion instead of the untyped original one. The question to know whereas the latter choice leads to equivalent systems has been first asked by Geuvers in the early 90’s, and the answer was only known for particular subclasses of Pure Type Systems. The main contribution of this dissertation is to finally provide a final and positive answer to this question by proving

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