--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "ground_2/lib/arith.ma".
+
+(* ARITHMETICAL PROPERTIES FOR λδ-2B ****************************************)
+
+lemma arith_l4 (m11) (m12) (m21) (m22):
+ m21+m22-(m11+m12) = m21-m11-m12+(m22-(m11-m21)-(m12-(m21-m11))).
+#m11 #m12 #m21 #m22 >minus_plus
+elim (le_or_ge (m11+m12) m21) #Hm1121
+[ lapply (transitive_le m11 ??? Hm1121) // #Hm121
+ lapply (le_plus_to_minus_l … Hm1121) #Hm12211
+ <plus_minus // @eq_f2 // >(eq_minus_O m11 ?) // >(eq_minus_O m12 ?) //
+| >(eq_minus_O m21 ?) // <plus_O_n <minus_plus <commutative_plus
+ elim (le_or_ge m11 m21) #Hm121
+ [ lapply (le_plus_to_minus_comm … Hm1121) #Hm2112
+ >(eq_minus_O m11 ?) // <plus_minus_associative // <minus_le_minus_minus_comm //
+ | >(eq_minus_O m21 ?) // <minus_le_minus_minus_comm //
+ ]
+]
+qed.
+
+lemma arith_l2 (n1) (n2): ↑n2-n1 = 1-n1+(n2-(n1-1)).
+* // qed.
+
+lemma arith_l1: ∀x. 1 = 1-x+(x-(x-1)).
+// qed.