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diff --git a/matita/matita/nlibrary/nat/compare.ma b/matita/matita/nlibrary/nat/compare.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "nat/order.ma".
+include "datatypes/bool.ma".
+
+naxiom decompose1: ¬(lt O O).
+naxiom decompose2: ∀n. ¬(lt (S n) O).
+
+ndefinition ltb: nat → nat → bool.
+ #n; nelim n
+  [ #m; ncases m
+    [ napply false
+    | #m'; napply true ]
+##| #n'; #Hn'; #m; ncases m
+    [ napply false
+    | #m'; ncases (Hn' m')
+       [ napply true
+       | napply false]##]
+nqed.