Definition x_ (i : nat) := exist (in_Z_ (2 ^ w)) (x / 2 ^ (Z_of_nat i * w) mod 2 ^ w) (mod_in_Z__ (x / 2 ^ (Z_of_nat i * w))).Definition f' (d_Sir_Si : Z_ (2 ^ w) * Z_ (2 ^ w)) (xi : Z_ (2 ^ w)) :=
  (d_i M_in_Z_ ptM'cM'' d_Sir_Si xi, r_i M_in_Z_ ptM'cM'' d_Sir_Si xi).Fixpoint foldlz (f : Z_ (2 ^ w) * Z_ (2 ^ w) -> Z_ (2 ^ w) -> Z_ (2 ^ w) * Z_ (2 ^ w)) (d_Sir_Si : Z_ (2 ^ w) * Z_ (2 ^ w)) (a:nat) {struct a} : 
 Z_ (2 ^ w) * Z_ (2 ^ w) :=
  match a with
  | O   => d_Sir_Si
  | S i => foldlz f (f d_Sir_Si (x_ i)) i
  end.

Lemma P'f' : forall (d r : Z_ (2 ^ w)) (h : Z) (l x_i : Z_ (2 ^ w)), P' M p d r h l -> P' M p (fst (f' (d, r) x_i)) (snd (f' (d, r) x_i)) (h * 2 ^ w + l) x_i.