The discriminating power of multiplicities in the $\lambda$-calculus

The $\lambda$-calculus with multiplicities is a refinement of the lazy $\lambda$-calculus where the argument in an application comes with a multiplicity, which is an upper bound to the number of its uses. This introduces potential deadlocks in the evaluation. We study the discriminating power of this calculus over the usual $\lambda$-terms. We prove in particular that the observational equivalence induced by contexts with multiplicities coincides with the equality of Lévy-Longo trees associated with $\lambda$-terms. This is a consequence of the characterization we give of the corresponding observational precongruence, as an intensional preorder involving $\eta$-expansion, namely Ong's lazy Plotkin-Scott-Engeler preorder.