Specifying programs with propositions and with congruences

Deduction modulo is an extension of first-order predicate logic where axioms are replaced by a congruence, modulo which deduction is performed. Like first-order predicate logic, Deduction modulo is a framework where many theories can be expressed, in particular predicative arithmetic [4] and impredicative arithmetic, usually called first-order and second-order arithmetic (terminology that we try to avoid as both theories are expressed in a first-order setting). Krivine and Parigot’s Second-order functional arithmetic (FA2) [10, 8, 9] is a formulation of impredicative arithmetic tailored to be used to specify and prove programs. In this note, we give a presentation of FA2 in Deduction modulo. Expressing FA2 in Deduction modulo will shed light on an original aspect of the FA2 approach: the fact that programs are specified, not with propositions, but with congruences.