  In a previous paper (of which this is a prosecution) we investigated the
extraction of proof-theoretic properties of natural deduction derivations from
their impredicative translation into System F. Our key idea was to introduce an
extended equational theory for System F codifying at a syntactic level some
properties found in parametric models. In a recent series of papers a different
approach to extract proof-theoretic properties of natural deduction derivations
was proposed by defining predicative variants of the usual translation,
embedding intuitionistic propositional logic into the atomic fragment of System
F. In this paper we show that this approach finds a general explanation within
our equational study of second-order natural deduction, and a clear semantic
justification provided by parametricity.
