General Bindings and Alpha-Equivalence in Nominal Isabelle

Abstract

ABSTRACT. Nominal Isabelle is a definitional extension of the Isabelle/HOL theorem prover. It provides a proving infrastructure for reasoning about programming language calculi involving named bound variables (as opposed to de-Bruijn indices). In this paper we present an extension of Nominal Isabelle for dealing with general bindings, that means term constructors where multiple variables are bound at once. Such general bindings are ubiquitous in programming language research and only very poorly supported with single binders, such as lambda-abstractions. Our extension includes new definitions of alpha-equivalence and establishes automatically the reasoning infrastructure for alphaequated terms. We also prove strong induction principles that have the usual variable convention already built in.

Formalizing Bounded Increase

“…As we want to apply Isabelle's code generator [11] on our certification algorithm, we have to formalize conditional constraints via a deep embedding. Therefore, we had the choice between at least two alternatives how to deal with bound variables: we can use a dedicated approach like Nominal Isabelle [21,22], or we perform renamings, α-equivalence, . .…”

Generalised Name Abstraction for Nominal Sets

“…The Nominal Isabelle package for interactive theorem proving over abstract syntax now supports various notions of generalised binding [24]. Perhaps surprisingly, this work is not explicitly based on the established notion of generalised abstraction.…”

“…Perhaps surprisingly, this work is not explicitly based on the established notion of generalised abstraction. Nonetheless it is clear that [24]'s set-binding and list-binding are the abstractions of P fin (A), and the nominal set A * of finite lists of atoms, respectively. It is hoped this paper will bring the concept of generalised abstraction back into view, and that our new Thm.…”

“…This model supports reasoning techniques that closely match pen-and-paper practice, in which we often explicitly manipulate bound names. Nominal sets have inspired a literature too extensive to summarise here, with nominal variants offered of everything from equational logic [4], to interactive theorem proving [24], to game semantics [23].…”

“…Atom-abstraction nominal sets as a semantics for binding one name at a time are now well established; what of applications involving more complex binding? Such applications, such as the functional programming operator letrec that binds lists of names simultaneously, are known to be imperfectly modelled by one-ata-time binding -see [1,21,24] for varied discussions on this point. This paper explores and critiques the fundamental mathematics that could provide a nominal sets semantics for such general binding.…”