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.

Confluence of left-linear higher-order rewrite theories by checking their nested critical pairs

“…We have therefore shown that the theory of global states preserves the confluence of -reductions on all untyped terms, hence improving over (Ferey and Jouannaud 2019), by using the order on rules’ labels inherited from the decision to have ll and uu as small rules. This example could actually not be shown confluent with an empty set of small rules, as can be checked by the reader, because, in that case, there would be additional infinite families whose joinability diagram would require to be decreasing, while the existing infinite families require just the contrary.…”

“… is a mix of the annotated lambda-calculus and Klop’s combinatory reduction systems (Klop 1980) which extends the calculus introduced in (Ferey and Jouannaud 2019) by having annotated abstractions to faithfully abstract dependently typed calculi whose confluence properties are our real target.…”

“… is terminating, and its higher-order critical pairs are joinable with -steps (Ferey and Jouannaud 2019);…”

“… is introduced with a unary abstraction in (Ferey and Jouannaud 2019). Our abstraction operator has arity 2, the first argument being the annotation and the second the body .…”

“…Our main theoretical result is then that higher-order rewriting in combination with beta reduction is confluent on untyped terms if all its nested higher-order critical pairs admit decreasing rewrite diagrams. It is, however, possible to stick to non-nested higher-order critical pairs for rules belonging to a terminating subset, as in (Ferey and Jouannaud 2019). The technique is illustrated with practical examples showing the strength and limits of the result.…”