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diff --git a/test-suite/modules/Nat.v b/test-suite/modules/Nat.v
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+Definition T:=nat.
+
+Definition le:=Peano.le.
+
+Hints Unfold le.
+
+Lemma le_refl:(n:nat)(le n n).
+  Auto.
+Save.
+
+Lemma le_trans:(n,m,k:nat)(le n m) -> (le m k) -> (le n k).
+  Unfold le.
+  Intros.
+  Generalize H.
+  Clear H.
+  Elim H0.
+  Auto.
+  Auto.  
+Save.
+
+Lemma le_mono_S : (n,m:nat)(le n m) -> (le (S n) (S m)).
+  Unfold le.
+  NewInduction 1.
+  Auto.
+  Auto.
+Save.
+
+Hints Resolve le_mono_S.
+
+Lemma le_pred_mono : (n,m:nat) (le n m) -> (le (pred n) (pred m)).
+  Unfold le.
+  Induction 1.
+  Auto.
+  Intro.
+  Case m0.
+  Auto.
+  Intro.
+  Simpl.
+  Auto.
+Save.
+
+Hints Resolve le_pred_mono.  
+
+Lemma le_S_mono : (m,n:nat)(le (S n) (S m)) -> (le n m).
+  Intros.
+  Change (le (pred (S n)) (pred (S m))).
+  Auto.
+Save.
+
+Hints Resolve le_S_mono.
+
+Lemma le_antis:(n,m:nat)(le n m) -> (le m n) -> n=m.
+  NewInduction n.
+  Intros.
+  Inversion H0.
+  Reflexivity.
+
+  Intros.
+  Inversion H.
+  Auto.
+  Rewrite (IHn m0).
+  Auto.
+  Rewrite <- H2 in H.
+  Auto.
+  Rewrite <- H2 in H0.
+  Auto.
+Save.
+