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[helm.git] / matita / contribs / RELATIONAL / NPlus / fun.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 "NPlus/inv.ma".
+
+(* Functional properties ****************************************************)
+
+theorem nplus_total: \forall p,q. \exists r. p + q == r.
+ intros 2. elim q; clear q;
+ [ autobatch | decompose. autobatch ].
+qed.
+
+theorem nplus_mono: \forall p,q,r1. (p + q == r1) \to 
+                    \forall r2. (p + q == r2) \to 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: \forall p1, q, r. (p1 + q == r) \to
+                     \forall p2. (p2 + q == r) \to 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.