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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "ground/arith/nat_le_minus.ma".
  16 include "ground/arith/nat_lt_pred.ma".
  17
  18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
  19
  20 (* Constructions with nminus ************************************************)
  21
  22 (*** monotonic_lt_minus_l *)
  23 lemma nlt_minus_sn_bi (o) (m) (n): o ≤ m → m < n → m - o < n - o.
  24 #o #m #n #Hom #Hmn
  25 lapply (nle_minus_sn_bi … o Hmn) -Hmn
  26 <(nminus_succ_sn … Hom) //
  27 qed.
  28
  29 (* Destructions with nminus *************************************************)
  30
  31 (*** minus_pred_pred *)
  32 lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m.
  33 #m #n #Hm #Hn
  34 >(nlt_inv_zero_sn … Hm) in ⊢ (??%?); -Hm
  35 >(nlt_inv_zero_sn … Hn) in ⊢ (??%?); -Hn
  36 //
  37 qed-.
  38
  39 lemma nlt_des_minus_dx (o) (m) (n): m < n - o → o < n.
  40 #o @(nat_ind_succ … o) -o
  41 [ #m #n <nminus_zero_dx
  42   /2 width=3 by le_nlt_trans/
  43 | #o #IH #m #n <nminus_succ_dx_pred_sn #H
  44   /3 width=2 by nlt_inv_pred_dx/
  45 ]
  46 qed-.