Confluence of Orthogonal Nominal Rewriting Systems Revisited

Nominal rewriting systems (Fernandez, Gabbay, Mackie, 2004; Fernandez, Gabbay, 2007) have been introduced as a new framework of higher-order rewriting systems based on the nominal approach (Gabbay, Pitts, 2002; Pitts, 2003), which deals with variable binding via permutations and freshness conditions on atoms. Confluence of orthogonal nominal rewriting systems has been shown in (Fernandez, Gabbay, 2007). However, their definition of (non-trivial) critical pairs has a serious weakness so that the orthogonality does not actually hold for most of standard nominal rewriting systems in the presence of binders. To overcome this weakness, we divide the notion of overlaps into the self-rooted and proper ones, and introduce a notion of alpha-stability which guarantees alpha-equivalence of peaks from the self-rooted overlaps. Moreover, we give a sufficient criterion for uniformity and alpha-stability. The new definition of orthogonality and the criterion offer a novel confluence condition effectively applicable to many standard nominal rewriting systems. We also report on an implementation of a confluence prover for orthogonal nominal rewriting systems based on our framework

Constraint Handling Rules with Binders, Patterns and Generic Quantification

Constraint Handling Rules provide descriptions for constraint solvers. However, they fall short when those constraints specify some binding structure, like higher-rank types in a constraint-based type inference algorithm. In this paper, the term syntax of constraints is replaced by $\lambda$-tree syntax, in which binding is explicit; and a new $\nabla$ generic quantifier is introduced, which is used to create new fresh constants.Comment: Paper presented at the 33nd International Conference on Logic Programming (ICLP 2017), Melbourne, Australia, August 28 to September 1, 2017 16 pages, LaTeX, no PDF figure

A Theory of Explicit Substitutions with Safe and Full Composition

Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the first part of this paper. Then, very simple technology in named variable-style notation is used to establish a theory of explicit substitutions for the lambda-calculus which enjoys a whole set of useful properties such as full composition, simulation of one-step beta-reduction, preservation of beta-strong normalisation, strong normalisation of typed terms and confluence on metaterms. Normalisation of related calculi is also discussed.Comment: 29 pages Special Issue: Selected Papers of the Conference "International Colloquium on Automata, Languages and Programming 2008" edited by Giuseppe Castagna and Igor Walukiewic

Proof pearl: abella formalization of lambda-calculus cube property

International audienceIn 1994 Gerard Huet formalized in Coq the cube property of lambda-calculus residuals. His development is based on a clever idea, a beautiful inductive definition of residuals. However, in his formalization there is a lot of noise concerning the representation of terms with binders. We re-interpret his work in Abella, a recent proof assistant based on higher-order abstract syntax and provided with a nominal quantifier. By revisiting Huet's approach and exploiting the features of Abella, we get a strikingly compact and natural development, which makes Huet's idea really shine