# Library Coq.omega.OmegaLemmas

Require Import BinInt Znat.
Local Open Scope Z_scope.

Factorization lemmas

Theorem Zred_factor0 n : n = n * 1.

Theorem Zred_factor1 n : n + n = n * 2.

Theorem Zred_factor2 n m : n + n * m = n * (1 + m).

Theorem Zred_factor3 n m : n * m + n = n * (1 + m).

Theorem Zred_factor4 n m p : n * m + n * p = n * (m + p).

Theorem Zred_factor5 n m : n * 0 + m = m.

Theorem Zred_factor6 n : n = n + 0.

Other specific variants of theorems dedicated for the Omega tactic

Lemma new_var : forall x : Z, exists y : Z, x = y.

Lemma OMEGA1 x y : x = y -> 0 <= x -> 0 <= y.

Lemma OMEGA2 x y : 0 <= x -> 0 <= y -> 0 <= x + y.

Lemma OMEGA3 x y k : k > 0 -> x = y * k -> x = 0 -> y = 0.

Lemma OMEGA4 x y z : x > 0 -> y > x -> z * y + x <> 0.

Lemma OMEGA5 x y z : x = 0 -> y = 0 -> x + y * z = 0.

Lemma OMEGA6 x y z : 0 <= x -> y = 0 -> 0 <= x + y * z.

Lemma OMEGA7 x y z t :
z > 0 -> t > 0 -> 0 <= x -> 0 <= y -> 0 <= x * z + y * t.

Lemma OMEGA8 x y : 0 <= x -> 0 <= y -> x = - y -> x = 0.

Lemma OMEGA9 x y z t : y = 0 -> x = z -> y + (- x + z) * t = 0.

Lemma OMEGA10 v c1 c2 l1 l2 k1 k2 :
(v * c1 + l1) * k1 + (v * c2 + l2) * k2 =
v * (c1 * k1 + c2 * k2) + (l1 * k1 + l2 * k2).

Lemma OMEGA11 v1 c1 l1 l2 k1 :
(v1 * c1 + l1) * k1 + l2 = v1 * (c1 * k1) + (l1 * k1 + l2).

Lemma OMEGA12 v2 c2 l1 l2 k2 :
l1 + (v2 * c2 + l2) * k2 = v2 * (c2 * k2) + (l1 + l2 * k2).

Lemma OMEGA13 (v l1 l2 : Z) (x : positive) :
v * Zpos x + l1 + (v * Zneg x + l2) = l1 + l2.

Lemma OMEGA14 (v l1 l2 : Z) (x : positive) :
v * Zneg x + l1 + (v * Zpos x + l2) = l1 + l2.

Lemma OMEGA15 v c1 c2 l1 l2 k2 :
v * c1 + l1 + (v * c2 + l2) * k2 = v * (c1 + c2 * k2) + (l1 + l2 * k2).

Lemma OMEGA16 v c l k : (v * c + l) * k = v * (c * k) + l * k.

Lemma OMEGA17 x y z : Zne x 0 -> y = 0 -> Zne (x + y * z) 0.

Lemma OMEGA18 x y k : x = y * k -> Zne x 0 -> Zne y 0.

Lemma OMEGA19 x : Zne x 0 -> 0 <= x + -1 \/ 0 <= x * -1 + -1.

Lemma OMEGA20 x y z : Zne x 0 -> y = 0 -> Zne (x + y * z) 0.

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.