  A compiler is fully-abstract if the compilation from source language programs
to target language programs reflects and preserves behavioural equivalence.
Such compilers have important security benefits, as they limit the power of an
attacker interacting with the program in the target language to that of an
attacker interacting with the program in the source language. Proving compiler
full-abstraction is, however, rather complicated. A common proof technique is
based on the back-translation of target-level program contexts to
behaviourally-equivalent source-level contexts. However, constructing such a
back- translation is problematic when the source language is not strong enough
to embed an encoding of the target language. For instance, when compiling from
STLC to ULC, the lack of recursive types in the former prevents such a
back-translation.
  We propose a general and elegant solution for this problem. The key insight
is that it suffices to construct an approximate back-translation. The
approximation is only accurate up to a certain number of steps and conservative
beyond that, in the sense that the context generated by the back-translation
may diverge when the original would not, but not vice versa. Based on this
insight, we describe a general technique for proving compiler full-abstraction
and demonstrate it on a compiler from STLC to ULC. The proof uses asymmetric
cross-language logical relations and makes innovative use of step-indexing to
express the relation between a context and its approximate back-translation.
The proof extends easily to common compiler patterns such as modular
compilation and it, to the best of our knowledge, it is the first compiler full
abstraction proof to have been fully mechanised in Coq. We believe this proof
technique can scale to challenging settings and enable simpler, more scalable
proofs of compiler full-abstraction.
