({\cal M} \; {\cal M})$$

2.  The set of $\l\m$-terms is defined by the following grammar

    $${\cal M}' := {\cal V}  \ | \ \l {\cal V}.  {\cal M}' \ | \ ({\cal
        M}' \; {\cal M}') \ |  \ \m {\cal W}. {\cal M}' \ |  \ ({\cal W} \; {\cal M}')$$

3.  The set of $\l\m^{\f\et\ou}$-terms is defined by the following grammar

    $${\cal M}'' ::= {\cal V} \ | \ \l {\cal V}. {\cal M}'' \ | \ ({\cal
    M}'' \; {\cal E}) \ | \ \<{\cal M}'' , {\cal M}'' \> \ | \ \o_1 {\cal
    M}'' \ | \ \o_2 {\cal M}'' \ | \ \mu {\cal W}. {\cal M}'' \ | \ ({\cal
    W} \; {\cal M}'')$$

    $${\cal E} ::=  {\cal M}'' \ | \  \pi_1 \ | \ \pi_2 \ | \ [{\cal V}.{\cal M}'' ,{\cal V}.{\cal M}'']$$

Note that, for the $\l\m$-calculus, we have adopted here the so-called de Groote calculus which is the extension of Parigot’s calculus where the distinction between named and un-named terms is forgotten. In this calculus, $\m\a$ is not necessarily followed by $[\b]$. We also write $(\a \; M)$ instead of $[\a] M$.

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1.  The reduction rule 