 
Summary: Verifying a Semantic ##Conversion Test for
MartinL˜of Type Theory
Andreas Abel, 1 Thierry Coquand, 2 and Peter Dybjer 2#
1 Department of Computer Science, LudwigMaximiliansUniversity, Munich
abel@tcs.ifi.lmu.de
2 Department of Computer Science, Chalmers University of Technology
coquand,peterd@cs.chalmers.se
Abstract. Typechecking algorithms for dependent type theories often
rely on the interpretation of terms in some semantic domain of values
when checking equalities. Here we analyze a version of Coquand's algo
rithm for checking the ##equality of such semantic values in a theory
with a predicative universe hierarchy and large elimination rules. Al
though this algorithm does not rely on normalization by evaluation ex
plicitly, we show that similar ideas can be employed for its verification.
In particular, our proof uses the new notions of contextual reification and
strong semantic equality.
The algorithm is part of a bidirectional type checking algorithm which
checks whether a normal term has a certain semantic type, a technique
used in the proof assistants Agda and Epigram. We work with an abstract
notion of semantic domain in order to accommodate a variety of possible
