Type theory and rewriting

Towards a unified language based on equational logic, rewriting logic, and type theory

Congruence Types

We introduce a type theoretical framework in which canon ical term rewriting systems can be represented faithfully both from the logical and the computational points of view The framework is based on congruence types a new syntax which combines inductive algebraic and quotient types Congruence types improve on existing work to combine type theories with algebraic rewriting by making explicit th...

Representing the MSR Cryptoprotocol Specification Language in an Extension of Rewriting Logic with Dependent Types

This paper presents a shallow and efficient embedding of the security protocol specification language MSR into an extension of rewriting logic with dependent types. The latter is an instance of the open calculus of constructions which integrates key concepts from equational logic, rewriting logic, and type theory. MSR is based on a form of first-order multiset rewriting extended with existentia...

Extensional Rewriting with Sums

Inspired by recent work on normalisation by evaluation for sums, we propose a normalising and confluent extensional rewriting theory for the simply-typed λ-calculus extended with sum types. As a corollary of confluence we obtain decidability for the extensional equational theory of simply-typed λ-calculus extended with sum types. Unlike previous decidability results, which rely on advanced rewr...

On Modular Properties of Higher Order Extensional Lambda Calculi

We prove that confluence and strong normalisation are both modular properties for the addition of algebraic term rewriting systems to Girard’s F! equipped with either -equality or -equality. The key innovation is the use of -expansions over the more traditional -contractions. We then discuss the difficulties encountered in generalising these results to type theories with dependent types. Here c...

Adjoint Rewriting and the !-type Constructor

This paper provides a sound, complete and decidable equational theory for the terms of the (I; ;?;!)-fragment of intuitionistic linear logic with respect to the class of models known as linear categories. This work uses the natural deduction style presentation of the !-type constructor in DILL so as to be able to apply the ideas of adjoint rewriting. We obtain an expansionary-rewrite rule and c...