Considered Both as a Paradigmatic Language and as a Meta-language

Intuitionistic Linear Logic (ILL) is a resource-conscious logic. The Curry-Howard Isomorphism (CHI) applied to ILL, generates typed functional-like languages that have primitive constants by means of which the amount of resources (terms), used during the computation, is explicit. ! is an untyped functional-like language inspired from a typed language joined at ILL by CHI. We want to use the resource-aware language ! both as a paradigmatic programming language and as a meta-language for implementing a fragment of the untyped -calculus . For using ! in the first way we give an algorithm for automatically assigning formulas of ILL as types to terms of ! . Concerning the second kind of use, we introduce a one-step translation Tr from the fragment C of that can be typed à la Curry to the typable fragment of ! in ILL. Tr preserves the linear-behaved terms of C and is both correct and complete, in a reasonable sense, w.r.t. the -reductions on .