Confluence by Decreasing Diagrams -- Formalized

This paper presents a formalization of decreasing diagrams in the theorem prover Isabelle. It discusses mechanical proofs showing that any locally decreasing abstract rewrite system is confluent. The valley and the conversion version of decreasing diagrams are considered.Comment: 17 pages; valley and conversion version; RTA 201

Proof Orders for Decreasing Diagrams

We present and compare some well-founded proof orders for decreasing diagrams. These proof orders order a conversion above another conversion if the latter is obtained by filling any peak in the former by a (locally) decreasing diagram. Therefore each such proof order entails the decreasing diagrams technique for proving confluence. The proof orders differ with respect to monotonicity and complexity. Our results are developed in the setting of involutive monoids. We extend these results to obtain a decreasing diagrams technique for confluence modulo

Confluence in UnTyped Higher-Order Theories by means of Critical Pairs

User-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. We develop here techniques that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. The present paper concentrates on the case where rewrite rules are left-linear and critical pairs can be joined without using beta-rewrite steps. The other two cases will be addressed in forthcoming papers

Diagrammatic Confluence and Completion

The work was started while the second author was invited at LIX, Ecole Polytechnique, by Ecole Polytechnique and Digiteo Labs.International audienceWe give a new elegant proof that decreasing diagrams imply confluence based on a proof reduction technique, which is then the basis of a novel completion method which proof-reduction relation transforms arbitrary proofs into rewrite proofs even in presence of non-terminating reductions. Unlike previous methods, no ordering of the set of terms is required, but can be used if available. Unlike ordered completion, rewrite proofs are closed under instantiation. Examples are presented, including Kleene's and Huet's classical examples showing that non-terminating local-confluent relations may not be confluent