 
Summary: Collection Principles in
Dependent Type Theory ?
Peter Aczel 1 and Nicola Gambino 2
1 Departments of Mathematics and Computer Science, University of Manchester,
email: petera@cs.man.ac.uk
2 Department of Computer Science, University of Manchester,
email: ngambino@cs.man.ac.uk
Abstract. We introduce logicenriched intuitionistic type theories, that
extend intuitionistic dependent type theories with primitive judgements
to express logic. By adding type theoretic rules that correspond to the
collection axiom schemes of the constructive set theory CZF we obtain
a generalisation of the type theoretic interpretation of CZF. Suitable
logicenriched type theories allow also the study of reinterpretations of
logic. We end the paper with an application to the doublenegation in
terpretation.
Introduction
In [1] the constructive set theory CZF was given an interpretation in the depen
dent type theory ML 1 V. This type theory is a version of MartinLof's intuition
istic type theory with one universe of small types, but no W types except for the
special W type V which is used to interpret the universe of sets of CZF. In [2]
