matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_minus1_plus.ma

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  14
  15 include "ground/arith/ynat_plus.ma".
  16 include "ground/arith/ynat_minus1.ma".
  17 include "ground/arith/ynat_lt_pred_succ.ma".
  18
  19 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
  20
  21 (* Constructions with yminus1 and yplus *************************************)
  22
  23 (*** ylt_plus1_to_minus_inj2 *)
  24 lemma ylt_plus_sn_dx_minus1_dx (n) (x) (z):
  25       x + yinj_nat n < z → x < z - n.
  26 #n @(nat_ind_succ … n) -n //
  27 #n #IH #x #z >ysucc_inj <yplus_succ_shift
  28 /3 width=1 by ylt_des_succ_sn/
  29 qed.
  30
  31 (*** ylt_plus1_to_minus_inj1 *)
  32 lemma ylt_plus_sn_sn_minus1_dx (n) (x) (z):
  33       yinj_nat n + x < z → x < z - n.
  34 /2 width=1 by ylt_plus_sn_dx_minus1_dx/ qed.