The Intensional Lambda Calculus

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 λ that internalises intensional information on how a term is computed. In the same way that the Logic of Proofs internalises its own derivations, λ internalises its own computations. Confluence and strong normalisation of λ is proved. This system serves as the basis for the study of type theories that internalise intensional aspects of computation.