The intensional lambda calculus

Abstract. We introduce a natural deduction formulation for the Logic of Proofs, a refinement of modal logic S4 in which the assertion 2A is replaced by [[s]]A whose intended reading is “s is a proof of A”. A term calculus for this formulation yields a typed lambda calculus λI that internalises intensional information on how a term is computed. In the same way that the Logic of Proofs internalises its own derivations, λI internalises its own computations. Confluence and strong normalisation of λI is proved. This system serves as the basis for the study of type theories that internalise intensional aspects of computation.