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+(* Omega being smarter on recognizing nat and Z *)
+
+Require Import Omega.
+
+Definition nat' := nat.
+
+Theorem le_not_eq_lt : forall (n m:nat),
+    n <= m ->
+    n <> m :> nat' ->
+    n < m.
+Proof.
+  intros.
+  omega.
+Qed.
+
+Goal forall (x n : nat'), x = x + n - n.
+Proof.
+  intros.
+  omega.
+Qed.
+
+Require Import ZArith ROmega.
+
+Open Scope Z_scope.
+
+Definition Z' := Z.
+
+Theorem Zle_not_eq_lt : forall n m,
+    n <= m ->
+    n <> m :> Z' ->
+    n < m.
+Proof.
+  intros.
+  omega.
+  Undo.
+  romega.
+Qed.