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This paper shows the equivalence of applicative similarity and contextual approximation, and hence also of bisimilarity and contextual equivalence, in the deterministic call-by-need lambda calculus with letrec. Bisimilarity simplifies equivalence proofs in the calculus and opens a way for more convenient correctness proofs for program transformations. Although this property may be a natural one to expect, to the best of our knowledge, this paper is the first one providing a proof. The proof technique is to transfer the contextual approximation into Abramsky’s lazy lambda calculus by a fully abstract and surjective translation. This also shows that the natural embedding of Abramsky’s lazy lambda calculus into the call-by-need lambda calculus with letrec is an isomorphism between the respective term-models. We show that the equivalence property proven in this paper transfers to a call-by-need letrec calculus developed by Ariola and Felleisen. 1998 ACM Subject Classification: F.4.2, F.3.2, F.3.3, F.4.1. Key words and phrases: semantics, contextual equivalence, bisimulation, lambda calculus, call-by-need, letrec.
The diagram-based method to prove correctness of program transformations consists of computing
complete set of (forking and commuting) diagrams, acting on sequences of standard reductions
and program transformations. In many cases, the only missing step for proving correctness of a
program transformation is to show the termination of the rearrangement of the sequences. Therefore
we encode complete sets of diagrams as term rewriting systems and use an automated tool
to show termination, which provides a further step in the automation of the inductive step in
correctness proofs.