--- /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 "NPlus/inv.ma".
+
+(* Functional properties ****************************************************)
+
+theorem nplus_total: ∀p,q. ∃r. p ⊕ q ≍ r.
+ intros; elim q; clear q;
+ [ autobatch | decompose; autobatch ].
+qed.
+
+theorem nplus_mono: ∀p,q,r1. p ⊕ q ≍ r1 →
+ ∀r2. p ⊕ q ≍ r2 → r1 = r2.
+ intros 4; elim H; clear H q r1;
+ [ lapply linear nplus_inv_zero_2 to H1
+ | lapply linear nplus_inv_succ_2 to H3. decompose
+ ]; destruct; autobatch.
+qed.
+
+theorem nplus_inj_1: ∀p1, q, r. p1 ⊕ q ≍ r →
+ ∀p2. p2 ⊕ q ≍ r → p2 = p1.
+ intros 4; elim H; clear H q r;
+ [ lapply linear nplus_inv_zero_2 to H1
+ | lapply linear nplus_inv_succ_2_3 to H3
+ ]; autobatch.
+qed.