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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
Require Export ArithRing.
Require Export Compare_dec.
Require Export Wf_nat.
Require Export Arith.
Require Export Omega.
intros a; elim a; auto.
intros n' Hrec b; case b; auto.
Qed.
Remark expand_mult2 : forall x : nat, 2 * x = x + x.
intros x; ring.
Qed.
unfold not in |- *; intros x y H H1; elim (lt_irrefl x);
pattern x at 2 in |- *; rewrite H1; auto.
Qed.
Hint Resolve lt_neq.
intros f Hmon x y Hlt; case (le_gt_dec y x); auto.
intros Hle; elim (le_lt_or_eq _ _ Hle).
intros Hlt'; elim (lt_asym _ _ Hlt); apply Hmon; auto.
intros Heq; elim (lt_neq _ _ Hlt); rewrite Heq; auto.
Qed.

Theorem mult_lt : forall a b c : nat, c <> 0 -> a < b -> a * c < b * c.