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Algorithmic equality in Heyting Arithmetic Lisa Allali

Summary: Algorithmic equality in Heyting Arithmetic
Lisa Allali
LogiCal - École polytechnique - Région Ile de France
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
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


Source: Allali, Lisa - Laboratoire d’Informatique, École Polytechnique


Collections: Computer Technologies and Information Sciences