Fixpoint unary_preserv (e : BC) : bool :=
match e with
| zero => true
| proj n s j => true
| succ b => b
| pred => true
| cond => true
| rec g h0 h1 => unary_preserv g  &&
unary_preserv h0 &&
unary_preserv h1
| comp _ _ h nl sl => unary_preserv h &&
forallb unary_preserv nl &&
forallb unary_preserv sl
end.
(** * Zero  - with any arities (n, s) *)
(** * One  - with any arities (n, s) *)
Definition one_e (n s:nat) : BC :=
comp n s (comp 0 0 (succ true) nil (zero :: nil)) nil nil.
(** * Successor  - arities: (1, 0) *)
Definition succ_e : BC := succ true.
(** * 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.

Lemma nat2BC_correct :
forall n s x nl sl, length (sem (nat2BC n s x) nl sl) = x.