Confluence of Pattern-Based Calculi

Different pattern calculi integrate the functional mechanisms from the λ-calculus and the matching capabilities from rewriting. Several approaches are used to obtain the confluence but in practice the proof methods share the same structure and each variation on the way patternabstractions are applied needs another proof of confluence. We propose here a generic confluence proof where the way patternabstractions are applied is axiomatized. Intuitively, the conditions guarantee that the matching is stable by substitution and by reduction. We show that our approach directly applies to different pattern calculi, namely the lambda calculus with patterns, the pure pattern calculus and the rewriting calculus. We also characterize a class of matching algorithms and consequently of pattern-calculi that are not confluent.