Section parm_sec.
Variable M_in_Z_ : Z_ (2 ^ w).
Variable ptM'cM'' : Z_ w * Z_ w * Z_ (2 ^ w) * Z_ (2 ^ w) * Z_ (2 ^ w).
Variable d_Sir_Si : Z_ (2 ^ w) * Z_ (2 ^ w).
Variable l_in_Z_ : Z_ (2 ^ w).
Variable x_i_in_Z_ : Z_ (2 ^ w).
Variable h : Z.
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_.
Let l := _Z l_in_Z_.
Let x_i := _Z x_i_in_Z_.
Definition d_Si := fst d_Sir_Si.
Definition r_Si := snd d_Sir_Si.
Hypothesis d_Si_eq_r_Si_eq : P M p d_Si r_Si h l_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.
unfold p. apply le_0__Z. Qed.
apply Zlt_le_weak. exact lt02w. Qed.
Local Hint Resolve le_0_w lt02w le02w : a.
Local Hint Resolve le_0_p : a.
clear x_i_in_Z_ x_i.
rewrite p_M. generalize le_1_M. case M. intro. absurd (2 <= 0); omega. intros q H.
elim (Zlog_sup_correct2 (Zpos q)). omega. omega. intros q H. absurd (2 <= Zneg q). apply Zlt_not_le.
unfold Z.lt. trivial. assumption. Qed.
clear x_i_in_Z_ x_i.
rewrite p_M. elim (Zle_lt_or_eq 1 M). intro H. left. apply Z.le_trans with (Zlog_sup 2). simpl. omega.
apply Zlog_sup_seq. omega. intro H. right. symmetry. trivial. assumption. Qed.
clear x_i_in_Z_ x_i.
elim le_1_p; intro H; omega. Qed.
clear x_i_in_Z_ x_i.
elim lt_0_p; intro H; omega. Qed.
Local Hint Resolve le_M_2pp le_1_p lt_0_p le_0_pm1 : a.
clear x_i_in_Z_ x_i.
elim (Zle_lt_or_eq 1 M). intro H. rewrite p_M. generalize le_1_M. case M. intro. absurd (2 <= 0); omega.
intros q H0. elim (Zlog_sup_correct2 (Zpos q)). omega. omega. intros q H0. absurd (1 <= Zneg q).
apply Zlt_not_le. unfold Z.lt. trivial. assumption. omega. assumption. Qed.
Local Hint Resolve lt_2pm1_M : a.
clear x_i_in_Z_ x_i.
elim (Zle_lt_or_eq 1 M). intro H. replace (2 ^ p) with (1 * M + (2 ^ p - M)). apply Zdiv_mult_plus. omega.
cut (M <= 2 ^ p). omega. auto with a. cut (2 ^ p < M + M). omega.
apply Z.lt_le_trans with (2 ^ (p - 1) + 1 + (2 ^ (p - 1) + 1)).
replace (2 ^ (p - 1) + 1 + (2 ^ (p - 1) + 1)) with (2 ^ (p - 1) + 2 ^ (p - 1) + 2). rewrite Z_pow_plus.
replace (p - 1 + 1) with p. omega. ring. elim le_0_pm1. omega. intro H0. absurd (1 < M). omega.
assumption. ring. cut (2 ^ (p - 1) < M). omega. elim lt_2pm1_M. omega. intro H0. absurd (1 < M). omega.
assumption. ring. intro H0. rewrite p_M. rewrite <- H0. trivial. assumption. Qed.

Lemma large_M'_eq : 2 ^ w + (2 ^ (p + w) / M) mod 2 ^ w = 2 ^ (p + w) / M.