Confluence in Non-Left-Linear Untyped Higher-Order Rewrite Theories

Abstract

We develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. We show that confluence is preserved for a large subset of terms that contains all pure lambda terms. Our results are applied to famous Klop’s examples of non-confluent behaviours in presence of convergent rewrite rules and to fragments of various encodings, in a dependent type theory with rewrite rules, of the Calculus of Constructions with polymorphic universes.