  A logic-enriched type theory (LTT) is a type theory extended with a primitive
mechanism for forming and proving propositions. We construct two LTTs, named
LTTO and LTTO*, which we claim correspond closely to the classical predicative
systems of second order arithmetic ACAO and ACA. We justify this claim by
translating each second-order system into the corresponding LTT, and proving
that these translations are conservative. This is part of an ongoing research
project to investigate how LTTs may be used to formalise different approaches
to the foundations of mathematics.
  The two LTTs we construct are subsystems of the logic-enriched type theory
LTTW, which is intended to formalise the classical predicative foundation
presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has
also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA
as LTTs, we are able to compare them with LTTW. It is a consequence of the work
in this paper that LTTW is strictly stronger than ACAO.
  The conservativity proof makes use of a novel technique for proving one LTT
conservative over another, involving defining an interpretation of the stronger
system out of the expressions of the weaker. This technique should be
applicable in a wide variety of different cases outside the present work.
