(* 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.
Opaque Poly.

(* Specifications: *)
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).
Lemma BC_to_Cobham_correct : forall (e : BC) n s,
arities e = ok_arities n s ->
(forall xl yl,
n = length xl ->
s = length yl ->
sem e xl yl = Sem (BC_to_Cobham n s e) (xl ++ yl)).
Lemma app_prop : forall {A} (l2 l1 : list A) (P : A -> Prop),
(fix f (l : list A) : Prop :=
match l with
| nil => True
| e :: l' => P e /\ f l'
end) l1  ->
(fix f (l : list A) : Prop :=
match l with
| nil => True
| e :: l' => P e /\ f l'
end) l2  ->
(fix f (l : list A) : Prop :=
match l with
| nil => True
| e :: l' => P e /\ f l'
end) (l1 ++ l2).
Lemma plusl_monotonic : forall l (f1 f2 : nat -> nat),
(forall x, f1 x <= f2 x) ->
plusl (map f1 l) <= plusl (map f2 l).
Lemma BC_to_Cobham_bounded : forall (e : BC) n s,
arities e = ok_arities n s ->
rec_bounded' (BC_to_Cobham n s e).