(** * Multiplication by a constant  - with any arities: (n,0) *)
Fixpoint scalar_e (a:nat)(n:nat)(e:BC) : BC :=
match a with
| 0 => zero_e n 0
| S a' => comp n 0 plus_e [e] [scalar_e a' n e]
end.

Lemma scalar_arities :
forall a n e,
arities e = ok_arities n 0 ->
arities (scalar_e a n e) = ok_arities n 0.