 
Summary: Algorithmic equality in Heyting Arithmetic
Modulo
Lisa Allali
LogiCal  Ecole polytechnique  INRIA,
www.lix.polytechnique.fr/Labo/Lisa.Allali/
allali@lix.polytechnique.fr
1 Introduction
We present in this paper a version of Heyting arithmetic where all the axioms
are dropped and replaced by rewrite rules. A previous work has been done by
Gilles Dowek and Benjamin Werner presenting Heyting Arithmetic in such a
way [3], but where equality was dened by a Leibniz rule : a proposition of
the form x = y was rewritten in their system into p (x p y p), that is
provable if x and y are two equal closed terms, but not as simply as it could be
expected. In this paper, in contrary, when x and y are closed terms, we considere
checking equality between terms is just a computation : x = y rewrites directly
to or .
We followed a remark of Schwichtenberg, about how a set of rewrite rules could
be (or not) enough to decide equality in Heyting Arithmetic. In the work we
present here, we answer positively to this question and present a set of rewrite
rules that dene a new Heyting Arithmetic modulo HA, that is
