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. Besides, they allow 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 bisimulation up to context 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 bisimulation up to context validating the eta-law, whereas no theory of normal-form bisimulations for the calculus of abortive control has been presented before. Our results have been fully formalized in the Coq proof assistant.
In @cite_9 , Lassen claims that It is also possible to prove congruence of enf bisimilarity and enf bisimilarity up to @math directly like the congruence proofs for other normal form bisimilarities (tree equivalences) in @cite_28 , although the congruence proofs (...) require non-trivial changes to the relational substitutive context closure operation in op.cit. (...) Moreover, from the direct congruence proofs, we can derive bisimulation “up to context” proof principles like those for other normal form bisimilarities in op.cit.'' To our knowledge, such a proof is not published anywhere; we tried to carry out the congruence proof by following this comment, but we do not know how to conclude in the case of enf bisimilarity up to @math . We discuss what the problem is at the end of the proof of Lemma .
@cite_28 @cite_9
1537995368 1712407203
Abstract On the basis of an operational bisimulation account of Bohm tree equivalence, a novel operationally-based development of the Bohm tree theory is presented, including an elementary congruence proof for Bohm tree equivalence. The approach is also applied to other sensible and lazy tree theories. Finally, a syntactic proof principle, called bisimulation up to context, is derived from the congruence proofs. It is used to give a simple syntactic proof of the least fixed point property of fixed point combinators. The paper surveys notions of bisimulation and trees for sensible λ-theories based on reduction to head normal forms as well as for lazy λ-theories based on weak head normal forms. This paper describes two new bisimulation equivalences for the pure untyped call-by-value spl lambda -calculus, called enf bisimilarity and enf bisimilarity up to spl eta . They are based on eager reduction of terms to eager normal form (enf), analogously to co-inductive bisimulation characterizations of Levy-Longo tree equivalence and Bohm tree equivalence (up to spl eta ). We argue that enf bisimilarity is the call-by-value analogue of Levy-Longo tree equivalence. Enf bisimilarity (up to spl eta ) is the congruence on source terms induced by the call-by-value CPS transform and Bohm tree equivalence (up to spl eta ) on target terms. Enf bisimilarity and enf bisimilarity up to spl eta enjoy powerful bisimulation proof principles which, among other things, can be used to establish a retraction theorem for the call-by-value CPS transform.