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Summary: LOGICAL STEP-INDEXED LOGICAL RELATIONS
DEREK DREYER, AMAL AHMED, AND LARS BIRKEDAL
MPI-SWS, Germany
e-mail address: dreyer@mpi-sws.org
Indiana University, USA
e-mail address: amal@cs.indiana.edu
IT University of Copenhagen, Denmark
e-mail address: birkedal@itu.dk
Abstract. Appel and McAllester's "step-indexed" logical relations have proven to be a
simple and effective technique for reasoning about programs in languages with semanti-
cally interesting types, such as general recursive types and general reference types. How-
ever, proofs using step-indexed models typically involve tedious, error-prone, and proof-
obscuring step-index arithmetic, so it is important to develop clean, high-level, equational
proof principles that avoid mention of step indices.
In this paper, we show how to reason about binary step-indexed logical relations in
an abstract and elegant way. Specifically, we define a logic LSLR, which is inspired by
Plotkin and Abadi's logic for parametricity, but also supports recursively defined relations
by means of the modal "later" operator from Appel, Melli`es, Richards, and Vouillon's
"very modal model" paper. We encode in LSLR a logical relation for reasoning relationally
about programs in call-by-value System F extended with general recursive types. Using
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