 
Summary: Algorithmic equality in Heyting Arithmetic
modulo
Lisa Allali
LogiCal  École polytechnique  Région Ile de France
www.lix.polytechnique.fr/Labo/Lisa.Allali/
1 Introduction
Deduction Modulo is a formalism that aims at distinguish reasoning from com
putation in proofs. A theory modulo is formed with a set of axioms and a con
gruence dened by rewrite rules: the reasoning part of the theory is given by
the axioms, the computational part by the congruence. In deduction modulo, we
can in particular build theories without any axiom, called purely computational
theories. What is interesting in building such theories  purely dened by a set
of rewrite rules  is the possibility, in some cases to simplify the proofs (typically
equality between two closed terms), and also the algorithmic aspect of these
proofs.
The motivation of building a purely computational presentation of Heyting
Arithmetic takes root in La science et l'hypothèse by Henri Poincaré [8] where
the author asks: should the proposition 2 + 2 = 4 be proved or just veried ?
A good way to verify such propositions is to use the formalism of deduction mod
ulo and rewrite rules. In this perspective, Gilles Dowek and Benjamin Werner
