(** * Polymax Bounding  - The lemma [polymax_bounding] states that the output of an expression [e] with [n] normal variables is bounded by the sum of the polynomial [poly_BC n e] and of the size of its longest safe input. *)
Fixpoint poly_BC n (e : BC) : pol :=
match e with
| zero => pcst 0 0
| proj n s i => if (leb (S i) n) then pproj n i else (pcst n 0)
| succ b => pcst 0 1
| pred => pcst 0 0
| cond => pcst 0 0
| rec g h0 h1 =>
pplus (pshift (poly_BC (n - 1) g))
(pmult (pproj n 0) (pplus (poly_BC n h0) (poly_BC n h1)))
| comp n s h rl tl =>
(pplus (pcst n 0)
(pplus (pcomp (poly_BC (length rl) h) (map (poly_BC n) rl ))
(pplusl (map (poly_BC n) tl))))
end.

Lemma proj_seq_shift_normal : forall n ln ls d,
map (fun x : BC => sem x (d :: ln) ls) (map (proj (S n) 0) (seq 1 n)) =
map (fun x : BC => sem x ln ls) (map (proj n 0) (seq 0 n)).