Abstract

Toyama’s theorem states that the union of two confluent term rewrite systems with disjoint signatures is again confluent. This is a fundamental result in term rewriting, and several proofs appear in the literature. The underlying proof technique has been adapted to prove further results like persistence of confluence (if a many-sorted term rewrite system is confluent, then the underlying unsorted system is confluent) or the preservation of confluence by currying.

Highlights

  • Toyama’s theorem [13,17,19] states that confluence is modular, i.e., that the union of two confluent term rewrite systems (TRSs) over disjoint signatures is confluent if and only if the two TRSs themselves are confluent

  • Modularity opens up a decomposition approach to proving confluence, which is attractive, because different confluence criteria may apply to the constituent TRSs that do not apply to their union

  • The set of multi-hole contexts over F and V is denoted by C(F, V). (Multi-hole contexts are terms that may contain occurrences of an extra constant, representing their holes.) If C is a multi-hole context with n holes, C[t1, . . . , tn] denotes the term obtained by replacing the i-th hole in C by ti for 1 i n

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Summary

Introduction

Toyama’s theorem [13,17,19] states that confluence is modular, i.e., that the union of two confluent term rewrite systems (TRSs) over disjoint signatures is confluent if and only if the two TRSs themselves are confluent. – The notion of modularity has been generalized as well, by weakening the assumption that the signatures of the two TRSs are disjoint; for example, confluence is modular for layer-preserving composable TRSs [16], and for quasi-ground systems [12]. Isabelle/HOL [15] is an interactive proof assistant based on higher-order logic with a Hindley-Milner type system, extended with type classes. In this paper we describe a formalization of layer systems in Isabelle/HOL as part of IsaFoR. As part of our formalization effort, we have extended CeTA with support for a decomposition technique based on persistence of confluence, allowing CSI and potentially other confluence tools to produce certifiable proofs using this technique.

Preliminaries
Layer Conditions
Persistence
Persistent Decomposition
Currying
Conclusion

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