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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 pattern-abstractions are applied needs another proof of confluence. We propose here a generic confluence proof where the way pattern-abstractions 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. [ABSTRACT FROM AUTHOR]

The rewriting calculus, or Rho Calculus (ρCal), is a simple calculus that uniformly integrates abstraction on patterns and non- determinism. Therefore, it fully integrates rewriting and λ-calculus. The original presentation of the calculus was untyped. In this paper we present a uniform way to decorate the terms of the calculus with types. This gives raise to a new presentation à la Church, together with nine (8+1) type systems which can be placed in a ρ-cube that extends the λ-cube of Barendregt. Due to the matching capabilities of the calculus, the type systems use only one abstraction mechanism and therefore gives an original answer to the identification of the standard "λ" and "∏" abstractors. As a consequence, this brings matching and rewriting as the first class concepts of the Rho-versions of the Logical Framework (LF) of Harper-Honsell-Plotkin, and of the Calculus of Constructions (CC) of Coquand-Huet. [ABSTRACT FROM AUTHOR]