matita/matita/contribs/lambdadelta/ground/arith/nat_rplus_succ.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/arith/nat_succ_iter.ma".
  16 include "ground/arith/nat_rplus.ma".
  17
  18 (* RIGHT ADDITION FOR NON-NEGATIVE INTEGERS *********************************)
  19
  20 (* Constructions with nsucc *************************************************)
  21
  22 lemma nrplus_succ_dx (p) (n): ↑(p+n) = p + ↑n.
  23 #p #n @(niter_succ … psucc)
  24 qed.
  25
  26 lemma nrplus_succ_shift (p) (n): ↑p + n = p + ↑n.
  27 // qed.
  28
  29 lemma nrplus_unit_sn (n): ↑n = 𝟏 + n.
  30 #n @(nat_ind_succ … n) -n //
  31 qed.
  32
  33 (* Advanced constructions ***************************************************)
  34
  35 lemma nrplus_comm_23 (p) (n1) (n2):
  36       p + n1 + n2 = p + n2 + n1.
  37 #p #n1 @(nat_ind_succ … n1) -n1 //
  38 qed.