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+
+Require Import ZArith ROmega.
+
+(* Submitted by Xavier Urbain 18 Jan 2002 *)
+
+Lemma lem1 :
+ forall x y : Z, (-5 < x < 5)%Z -> (-5 < y)%Z -> (-5 < x + y + 5)%Z.
+Proof.
+intros x y.
+ (*romega.*)
+Admitted.
+
+(* Proposed by Pierre Crégut *)
+
+Lemma lem2 : forall x : Z, (x < 4)%Z -> (x > 2)%Z -> x = 3%Z.
+intro.
+ romega.
+Qed.
+
+(* Proposed by Jean-Christophe Filliâtre *)
+
+Lemma lem3 : forall x y : Z, x = y -> (x + x)%Z = (y + y)%Z.
+Proof.
+intros.
+ (*romega.*)
+Admitted.
+
+(* Proposed by Jean-Christophe Filliâtre: confusion between an Omega *)
+(* internal variable and a section variable (June 2001) *)
+
+Section A.
+Variable x y : Z.
+Hypothesis H : (x > y)%Z.
+Lemma lem4 : (x > y)%Z.
+ romega.
+Qed.
+End A.
+
+(* Proposed by Yves Bertot: because a section var, L was wrongly renamed L0 *)
+(* May 2002 *)
+
+Section B.
+Variable R1 R2 S1 S2 H S : Z.
+Hypothesis I : (R1 < 0)%Z -> R2 = (R1 + (2 * S1 - 1))%Z.
+Hypothesis J : (R1 < 0)%Z -> S2 = (S1 - 1)%Z.
+Hypothesis K : (R1 >= 0)%Z -> R2 = R1.
+Hypothesis L : (R1 >= 0)%Z -> S2 = S1.
+Hypothesis M : (H <= 2 * S)%Z.
+Hypothesis N : (S < H)%Z.
+Lemma lem5 : (H > 0)%Z.
+ romega.
+Qed.
+End B.
+
+(* From Nicolas Oury (bug #180): handling -> on Set (fixed Oct 2002) *)
+Lemma lem6 :
+ forall (A : Set) (i : Z), (i <= 0)%Z -> ((i <= 0)%Z -> A) -> (i <= 0)%Z.
+intros.
+ romega.
+Qed.
+
+(* Adapted from an example in Nijmegen/FTA/ftc/RefSeparating (Oct 2002) *)
+Require Import Omega.
+Section C.
+Parameter g : forall m : nat, m <> 0 -> Prop.
+Parameter f : forall (m : nat) (H : m <> 0), g m H.
+Variable n : nat.
+Variable ap_n : n <> 0.
+Let delta := f n ap_n.
+Lemma lem7 : n = n.
+ (*romega.*) (*ROMEGA CANT DEAL WITH NAT*)
+Admitted.
+End C.
+
+(* Problem of dependencies *)
+Require Import Omega.
+Lemma lem8 : forall H : 0 = 0 -> 0 = 0, H = H -> 0 = 0.
+intros.
+(* romega.*) (*ROMEGA CANT DEAL WITH NAT*)
+Admitted.
+
+(* Bug that what caused by the use of intro_using in Omega *)
+Require Import Omega.
+Lemma lem9 :
+ forall p q : nat, ~ (p <= q /\ p < q \/ q <= p /\ p < q) -> p < p \/ p <= p.
+intros.
+(* romega.*)(*ROMEGA CANT DEAL WITH NAT*) 
+Admitted.
+
+(* Check that the interpretation of mult on nat enforces its positivity *)
+(* Submitted by Hubert Thierry (bug #743) *)
+(* Postponed... problem with goals of the form "(n*m=0)%nat -> (n*m=0)%Z"
+Require Omega.
+Lemma lem10 : (n, m : nat) (le n (plus n (mult n m))).
+Proof.
+Intros; Omega.
+Qed.
+*)