Intuitionistic differential nets and lambda-calculus

We define pure intuitionistic differential proof nets, extending Ehrhard and Regnier’s differential interaction nets with the exponential box of Linear Logic. Normalization of the exponential reduction and confluence of the full one is proved. These results are directed and adjusted to give a translation of Boudol’s untyped λ-calculus with resources extended with a linear-non linear reduction à la Ehrhard and Regnier’s differential λ-calculus. Such reduction comes in two flavours: baby-step and giant-step β-reduction. The translation, based on Girard’s encoding A → B ∼ !A ⊸ B and as such extending the usual one for λ-calculus into proof nets, enjoys bisimulation for giant-step β-reduction. From this result we also derive confluence of both reductions.