Proposition 2.1.13 (Rosser). Define the following $\lambda$-term:

• $A_{+}:=\lambda x.\lambda y.\lambda s.\lambda z.xs(ys(z))$,

• $A_{\ast }:=\lambda x.\lambda y.\lambda s.x(ys)$,

• $A_e:=\lambda x.\lambda y.yx$.

Then for all $n,m\in \mathbb{N}$:

• $A_{+}{c}_n{c}_m{\equiv }_{\beta }{c}_{n+m}$;

• $A_{\ast }{c}_n{c}_m{\equiv }_{\beta }{c}_{nm}$;

• $A_e{c}_n{c}_m{\equiv }_{\beta }{c}_{n^m}$ if $m>0$.