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Local Open Scope Z_scope.
(** Factorization lemmas *)
(** Other specific variants of theorems dedicated for the Omega tactic *)
Definition fast_Zplus_comm (x y : Z) (P : Z -> Prop)
(H : P (y + x)) := eq_ind_r P H (Z.add_comm x y).
Definition fast_Zplus_assoc_reverse (n m p : Z) (P : Z -> Prop)
(H : P (n + (m + p))) := eq_ind_r P H (Zplus_assoc_reverse n m p).
Definition fast_Zplus_assoc (n m p : Z) (P : Z -> Prop)
(H : P (n + m + p)) := eq_ind_r P H (Z.add_assoc n m p).
Definition fast_Zplus_permute (n m p : Z) (P : Z -> Prop)
(H : P (m + (n + p))) := eq_ind_r P H (Z.add_shuffle3 n m p).
Definition fast_OMEGA10 (v c1 c2 l1 l2 k1 k2 : Z) (P : Z -> Prop)
(H : P (v * (c1 * k1 + c2 * k2) + (l1 * k1 + l2 * k2))) :=
eq_ind_r P H (OMEGA10 v c1 c2 l1 l2 k1 k2).
Definition fast_OMEGA11 (v1 c1 l1 l2 k1 : Z) (P : Z -> Prop)
(H : P (v1 * (c1 * k1) + (l1 * k1 + l2))) :=
eq_ind_r P H (OMEGA11 v1 c1 l1 l2 k1).
Definition fast_OMEGA12 (v2 c2 l1 l2 k2 : Z) (P : Z -> Prop)
(H : P (v2 * (c2 * k2) + (l1 + l2 * k2))) :=
eq_ind_r P H (OMEGA12 v2 c2 l1 l2 k2).
Definition fast_OMEGA15 (v c1 c2 l1 l2 k2 : Z) (P : Z -> Prop)
(H : P (v * (c1 + c2 * k2) + (l1 + l2 * k2))) :=
eq_ind_r P H (OMEGA15 v c1 c2 l1 l2 k2).
Definition fast_OMEGA16 (v c l k : Z) (P : Z -> Prop)
(H : P (v * (c * k) + l * k)) := eq_ind_r P H (OMEGA16 v c l k).
Definition fast_OMEGA13 (v l1 l2 : Z) (x : positive) (P : Z -> Prop)
(H : P (l1 + l2)) := eq_ind_r P H (OMEGA13 v l1 l2 x).
Definition fast_OMEGA14 (v l1 l2 : Z) (x : positive) (P : Z -> Prop)
(H : P (l1 + l2)) := eq_ind_r P H (OMEGA14 v l1 l2 x).
Definition fast_Zred_factor0 (x : Z) (P : Z -> Prop)
(H : P (x * 1)) := eq_ind_r P H (Zred_factor0 x).
Definition fast_Zopp_eq_mult_neg_1 (x : Z) (P : Z -> Prop)
(H : P (x * -1)) := eq_ind_r P H (Z.opp_eq_mul_m1 x).
Definition fast_Zmult_comm (x y : Z) (P : Z -> Prop)
(H : P (y * x)) := eq_ind_r P H (Z.mul_comm x y).
Definition fast_Zopp_plus_distr (x y : Z) (P : Z -> Prop)
(H : P (- x + - y)) := eq_ind_r P H (Z.opp_add_distr x y).
Definition fast_Zopp_mult_distr_r (x y : Z) (P : Z -> Prop)
(H : P (x * - y)) := eq_ind_r P H (Zopp_mult_distr_r x y).
Definition fast_Zmult_plus_distr_l (n m p : Z) (P : Z -> Prop)
(H : P (n * p + m * p)) := eq_ind_r P H (Z.mul_add_distr_r n m p).
Definition fast_Zmult_assoc_reverse (n m p : Z) (P : Z -> Prop)
(H : P (n * (m * p))) := eq_ind_r P H (Zmult_assoc_reverse n m p).
Definition fast_Zred_factor1 (x : Z) (P : Z -> Prop)
(H : P (x * 2)) := eq_ind_r P H (Zred_factor1 x).
Definition fast_Zred_factor2 (x y : Z) (P : Z -> Prop)
(H : P (x * (1 + y))) := eq_ind_r P H (Zred_factor2 x y).
Definition fast_Zred_factor3 (x y : Z) (P : Z -> Prop)
(H : P (x * (1 + y))) := eq_ind_r P H (Zred_factor3 x y).
Definition fast_Zred_factor4 (x y z : Z) (P : Z -> Prop)
(H : P (x * (y + z))) := eq_ind_r P H (Zred_factor4 x y z).
Definition fast_Zred_factor5 (x y : Z) (P : Z -> Prop)
(H : P y) := eq_ind_r P H (Zred_factor5 x y).
Definition fast_Zred_factor6 (x : Z) (P : Z -> Prop)
(H : P (x + 0)) := eq_ind_r P H (Zred_factor6 x).

Theorem intro_Z :
forall n:nat,  exists y : Z, Z.of_nat n = y /\ 0 <= y * 1 + 0.