(** * Conversion from [nat] to [BC]  - with any arities (n, s) *)
Fixpoint nat2BC (n s x:nat) : BC :=
match x with
| 0 => zero_e n s
| S x' => comp n s succ_e nil [nat2BC n s x']
end.

(* Specifications: *)
Lemma nat2BC_arities : forall n s x, arities (nat2BC n s x) = ok_arities n s.
Lemma nat2BC_correct :
forall n s x nl sl, length (sem (nat2BC n s x) nl sl) = x.