Algorithmic equality in Heyting Arithmetic Lisa Allali 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 Collections: Computer Technologies and Information Sciences