Spinal Atomic Lambda-Calculus

We present the spinal atomic [Formula: see text] -calculus, a typed [Formula: see text] -calculus with explicit sharing and atomic duplication that achieves spinal full laziness: duplicating only the direct paths between a binder and bound variables is enough for beta reduction to proceed. We show this calculus is the result of a Curry–Howard style interpretation of a deep-inference proof system, and prove that it has natural properties with respect to the [Formula: see text] -calculus: confluence and preservation of strong normalisation.