(* n s : arities of e*)
Fixpoint BC_to_Cobham n s (e : BC) : Cobham :=
  match e with
    | zero => Zero
    | proj n s i => Proj (n + s) i
    | succ b => Succ b
    | pred => Pred
    | cond => Cond
    | rec g h0 h1 =>
        Rec (BC_to_Cobham (n - 1) s g)
        (move_arg (S (n+s)) 1 (n-1) (BC_to_Cobham n (S s) h0) )
        (move_arg (S (n+s)) 1 (n-1) (BC_to_Cobham n (S s) h1) )
        (Poly (pplus (psum n s) (poly_BC n e)))
    | comp n s h rl tl =>
      Comp (n + s) (BC_to_Cobham (length rl) (length tl) h)
      (map (fun e => (dummies n s (BC_to_Cobham n 0 e))) rl ++ 
        map (BC_to_Cobham n s) tl)
  end.

Lemma arity_BC_to_Cobham : forall (e : BC) n s, arities e = ok_arities n s -> arity (BC_to_Cobham n s e) = ok_arity (n + s).