Pure Type Systems, PTSs, were introduced as a generalization of
the type systems of Barendregt's lambda cube and were designed to
provide a foundation for actual proof assistants which will
verify proofs. Systems of illative combinatory logic or lambda
calculus, ICLs, were introduced by Curry and Church as a foundation
for logic and mathematics. In an earlier paper we considered two
changes to the rules of the PTSs which made these rules more like
ICL rules. This led to four kinds of PTSs. Most importantly PTSs
are about statements of the form M:A, where M is a term and
A a type. In ICLs there are no explicit types and the statements
are terms. In this paper we show that for each of the four forms
of PTS there is an equivalent form of ICL, sometimes if certain
conditions hold.
References
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