propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_rplus_succ.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_succ_iter.ma".
+include "ground/arith/nat_rplus.ma".
+
+(* RIGHT ADDITION FOR NON-NEGATIVE INTEGERS *********************************)
+
+(* Constructions with nsucc *************************************************)
+
+lemma nrplus_succ_dx (p) (n): ↑(p+n) = p + ↑n.
+#p #n @(niter_succ … psucc)
+qed.
+
+lemma nrplus_succ_shift (p) (n): ↑p + n = p + ↑n.
+// qed.
+
+lemma nrplus_unit_sn (n): ↑n = 𝟏 + n.
+#n @(nat_ind_succ … n) -n //
+qed.
+
+(* Advanced constructions ***************************************************)
+
+lemma nrplus_comm_23 (p) (n1) (n2):
+      p + n1 + n2 = p + n2 + n1.
+#p #n1 @(nat_ind_succ … n1) -n1 //
+qed.