Proving Soundness of Extensional Normal-Form Bisimilarities

TL;DR

This paper introduces a systematic method to prove the soundness of extensional normal-form bisimilarities, including up-to techniques, across various lambda calculus variants, formalized in Coq.

Contribution

It provides a general proof technique for soundness of extensional bisimilarities with up-to methods, applicable to multiple lambda calculi including those with control operators.

Findings

Established soundness of bisimilarities with up-to context in call-by-value lambda calculus.

Extended soundness results to calculi with shift, reset, call/cc, and abort.

Formalized proofs in the Coq proof assistant.

Abstract

Normal-form bisimilarity is a simple, easy-to-use behavioral equivalence that relates terms in $\lambda$-calculi by decomposing their normal forms into bisimilar subterms. Moreover, it typically allows for powerful up-to techniques, such as bisimulation up to context, which simplify bisimulation proofs even further. However, proving soundness of these relations becomes complicated in the presence of $\eta$-expansion and usually relies on ad hoc proof methods which depend on the language. In this paper we propose a more systematic proof method to show that an extensional normal-form bisimilarity along with its corresponding up to context technique are sound. We illustrate our technique with three calculi: the call-by-value $\lambda$-calculus, the call-by-value $\lambda$-calculus with the delimited-control operators shift and reset, and the call-by-value $\lambda$-calculus with the abortive control operators call/cc and abort. In the first two cases, there was previously no sound up to context technique validating the $\eta$-law, whereas no theory of normal-form bisimulations for a calculus with call/cc and abort has been presented before. Our results have been fully formalized in the Coq proof assistant.