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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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   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_pred_succ.ma".
  16 include "ground/arith/nat_plus.ma".
  17
  18 (* ADDITION FOR NON-NEGATIVE INTEGERS ***************************************)
  19
  20 (* Inversions with npred ****************************************************)
  21
  22 (*** plus_inv_S3_sn *)
  23 lemma eq_inv_succ_nplus_sn (o) (m) (n):
  24       ↑o = m + n →
  25       ∨∨ ∧∧ 𝟎 = m & n = ↑o
  26        | ∧∧ m = ↑↓m & o = ↓m + n.
  27 #o #m @(nat_ind_succ … m) -m
  28 [ /3 width=1 by or_introl, conj/
  29 | #m #_ #n <nplus_succ_sn
  30   /4 width=1 by eq_inv_nsucc_bi, or_intror, conj/
  31 ]
  32 qed-.
  33
  34 (*** plus_inv_S3_dx *)
  35 lemma eq_inv_succ_nplus_dx (o) (m) (n):
  36       ↑o = m + n →
  37       ∨∨ ∧∧ 𝟎 = n & m = ↑o
  38        | ∧∧ n = ↑↓n & o = m + ↓n.
  39 #o #m #n @(nat_ind_succ … n) -n
  40 [ /3 width=1 by or_introl, conj/
  41 | #n #_ <nplus_succ_sn
  42   /4 width=1 by eq_inv_nsucc_bi, or_intror, conj/
  43 ]
  44 qed-.