propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_plus_rplus.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_plus_rplus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_plus_rplus.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 "ground/arith/nat_rplus_succ.ma".
+include "ground/arith/nat_plus.ma".
+
+(* ADDITION FOR NON-NEGATIVE INTEGERS ***************************************)
+
+(* Constructions with rplus *************************************************)
+
+lemma nrplus_inj_sn (p) (n):
+      ninj (p + n) = ninj p + n.
+#p #n @(nat_ind_succ … n) -n //
+#n #IH <nplus_succ_dx <IH //
+qed.
+
+(* Constructions with rplus and npsucc **************************************)
+
+lemma nrplus_npsucc_sn (m) (n):
+      npsucc (m + n) = npsucc m + n.
+#m @(nat_ind_succ … m) -m //
+#m #IH #n <npsucc_succ <nrplus_succ_sn //
+qed.