Nominal Techniques in Isabelle/HOL

Abstract

This paper describes a formalisation of the lambda-calculus in a HOLbased theorem prover using nominal techniques. Central to the formalisation is an inductive set that is bijective with the alpha-equated lambda-terms. Unlike de-Bruijn indices, however, this inductive set includes names and reasoning about it is very similar to informal reasoning with "pencil and paper". To show this we provide a structural induction principle that requires to prove the lambda-case for fresh binders only. Furthermore, we adapt work by Pitts providing a recursion combinator for the inductive set. The main technical novelty of this work is that it is compatible with the axiom of choice (unlike earlier nominal logic work by Pitts et al); thus we were able to implement all results in Isabelle/HOL and use them to formalise the standard proofs for Church-Rosser, strong-normalisation of beta-reduction, the correctness of the type-inference algorithm W, typical proofs from SOS and much more.

Formalising the pi-calculus using nominal logic

“…For a more thorough presentation of the nominal datatype package in Isabelle the reader is referred to [43], but enough basic definitions will be covered here for the reader to understand the rest of this paper. A nominal datatype definition is like an ordinary data type but it explicitly tags the binding occurrences of names.…”

“…The bound names generated by the rules are guaranteed to be fresh from the context names (just as is guaranteed for induction rules genereated by the nominal package, and for the same reason: avoiding name clashes and α-conversions later in the proof). This idea stems from [43] but was developed independently of similar work in [45]. The logical framework has also been covered in [38].…”

“…Gabbay's thesis [18] uses it to introduce FM set theory, this is the standard ZF set theory but with an extra axiom for freshness of names. Recent work by Urban and Tasson [43] extends this work using ideas from [37] and solves the problem with freshness without introducing new axioms. The techniques have been implemented into the theorem prover Isabelle/HOL, in a nominal datatype package, so that when defining nominal datatypes, Isabelle will automatically generate a type which models the datatype up to α-equivalence as well as induction principles and a recursion combinator allowing the user to create functions on nominal datatypes.…”

Efficiency of a good but not linear nominal unification algorithm

“…Higher-order pattern unification, the foundation of Isabelle [Paulson, 1986], λProlog [Nadathur et al, 1988], and Twelf [Pfenning and Schürmann, 1999], handles a fragment of the βη-rules. Nominal unification, the unification modulo the α-rule, has inspired extensions of logic programming languages such as αProlog [Cheney and Urban, 2004] and αKanren [Byrd and Friedman, 2007], as well as theorem provers such as Nominal Isabelle [Urban and Tasson, 2005] and αLeanTAP [Near et al, 2008]. Although these two theories can be reduced to one another [Cheney, 2005, Levy andVillaret, 2012], implementing higher-order pattern unification is more complicated because it has to deal with β-reduction and capture-avoiding substitution.…”

Implementing Spi Calculus Using Nominal Techniques

“…The Nominal package [22] for Isabelle/HOL aims to provide a framework for reasoning about process calculi and programming languages with binding operators in a convenient way, so that formal proofs should be easy to carry out as informal "pencil-and-paper" proofs. The work is based on the nominal logic [20]; the main technical novelty introduced by Urban et al [22] is that the construction for α-equivalent terms is done without adding any axiom to the Isabelle/HOL logic; therefore the theory is implemented just as a package of Isabelle/HOL, without the need of changing the underlying proof assistant.…”