We describe a sequent calculus μLJ with primitives for inductive and coinductive datatypes and equip it with reduction rules allowing a sound translation of Gödel’s system T. We introduce the notion of a μ-closed category, relying on a uniform interpretation of open μLJ formulas as strong functors. We show that any μ-closed category is a sound model for μLJ. We then turn to the construction of a concrete μ-closed category based on Hyland-Ong game semantics. The model relies on three main ingredients:...

We define an equivalent variant $L{K}_{sp}$ of the Gentzen sequent calculus $LK$. In $L{K}_{sp}$ weakenings or contractions can be performed in parallel. This modification allows us to interpret a symmetrical system of mix elimination rules ${}_{L{K}_{sp}}$ by a finite rewriting system; the termination of this rewriting system can be machine checked. We give also a self-contained strong normalization proof by structural induction. We give another strong normalization proof by a strictly monotone subrecursive interpretation; this interpretation...