A classical version of system Fω

We present a version of system Fω in which the layer of type constructors is essentially the same as in the traditional presentation of Fω [Gir72], whereas provability of types is classical. The proof-term calculus accounting for the classical reasoning is a variant of Barbanera and Berardi’s symmetric λ-calculus [BB96]. We prove that the whole calculus is strongly normalising. For the layer of type constructors, we use Tait and Girard’s reducibility method combined with orthogonality techniques. For the (classical) layer of terms, we use Barbanera and Berardi’s method based on a symmetric notion of reducibility candidate which does not seem to be captured by orthogonality.