tauto. fold x_i. replace (x_i mod 2 ^ p) with (x' mod 2 ^ p). exact r_i_eq. unfold x'. rewrite mod2p with (c' := 0).
apply f_equal2 with (f := Zmod). ring. trivial. Qed.
End parm_sec.
Section repeat.
Variable M_in_Z_ : Z_ (2 ^ w).
Variable ptM'cM'' : Z_ w * Z_ w * Z_ (2 ^ w) * Z_ (2 ^ w) * Z_ (2 ^ w).
Variable x : Z. (* multi-word *)
Let p_in_Z_w := fst (fst (fst (fst ptM'cM''))).
Let t_in_Z_w := snd (fst (fst (fst ptM'cM''))).
Let M'_in_Z_ := snd (fst (fst ptM'cM'')).
Let c_in_Z_ := snd (fst ptM'cM'').
Let M''_in_Z_ := snd ptM'cM''.
Let M := _Z M_in_Z_.
Let p := _Z p_in_Z_w.
Let t := _Z t_in_Z_w.
Let c := _Z c_in_Z_.
Let M'' := _Z M''_in_Z_.
Let M' := _Z M'_in_Z_.
Hypothesis le_1_M : 1 <= M.
Hypothesis le_M_max : M <= 2 ^ (w - 1).
Hypothesis p_M : p = Zlog_sup M.
Hypothesis t_M : t = w - p \/ M = 1.
Hypothesis c_M : c = 2 ^ p.
Hypothesis M''_M : M'' = (2 ^ t * M) mod 2 ^ w.
Hypothesis M'_M : M' = 2 ^ (p + w) / M mod 2 ^ w.
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.