A Semantic Proof that Reducibility Candidates entail Cut Elimination

Two main lines have been adopted to prove the cut elimination theorem: the syntactic one, that studies the process of reducing cuts, and the semantic one, that consists in interpreting a sequent in some algebra and extracting from this interpretation a cut-free proof of this very sequent. A link between those two methods was exhibited by studying in a semantic way, syntactical tools that allow to prove (strong) normalization of proof-terms, namely reducibility candidates. In the case of deduction modulo, a framework combining deduction and rewriting rules in which theories like Zermelo set theory and higher order logic can be expressed, this is obtained by constructing a reducibility candidates valued model. The existence of such a pre-model for a theory entails strong normalization of its proof-terms and, by the usual syntactic argument, the cut elimination property. In this paper, we strengthen this gate between syntactic and semantic methods, by providing a full semantic proof that the existence of a pre-model entails the cut elimination property for the considered theory in deduction modulo. We first define a new simplified variant of reducibility candidates à la Girard, that is sufficient to prove weak normalization of proof-terms (and therefore the cut elimination property). Then we build, from some model valued on the pre-Heyting algebra of those WN reducibility candidates, a regular model valued on a Heyting algebra on which we apply the usual soundness/strong completeness argument. Finally, we discuss further extensions of this new method towards normalization by evaluation techniques that commonly use Kripke semantics.