Abstract
The Barendregt–Geuvers–Klop conjecture states that every weakly normalizing pure type system is also strongly normalizing. We show that this is true for a uniform class of systems which includes, e.g., the left-hand side of Barendregt's λ-cube as well as the system λU. This seems to be the first result giving a positive answer to the conjecture not merely for some concrete systems for which strong normalization is known to hold, but for a uniform class of systems in which not all systems are strongly normalizing.
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