On completeness of reducibility candidates as a semantics of strong normalization

Abstract: This paper defines a sound and complete semantic criterion, based on reducibility candidates, for strong normalization of theories expressed in minimal deduction modulo `a la Curry. The use of Curry-style proof-terms allows to build this criterion on the classic notion of pre-Heyting algebras and makes that criterion concern all theories expressed in minimal deduction modulo. Compared to using Church-style proof-terms, this method provides both a simpler definition of the criterion and a simpler proof of its completeness.