S-m-n theorem
= S-m-n theorem
{c}
{wiki=Smn_theorem}
The S-m-n theorem says that fixed inputs can be compiled into program codes by a total computable transformation. For every $m,n\geq1$, there is a total computable $s_n^m$ such that
$$
f_{s_n^m(e,x_1,\ldots,x_m),n}(y_1,\ldots,y_n)
=f_{e,m+n}(x_1,\ldots,x_m,y_1,\ldots,y_n)
$$
whenever either side is defined.