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

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   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       *)
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  13 (**************************************************************************)
  14
  15 include "ground/arith/ynat_lt_succ.ma".
  16 include "ground/arith/ynat_lt_le.ma".
  17
  18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
  19
  20 (* Constructions with ysucc *************************************************)
  21
  22 (*** yle_lt yle_succ1_inj *)
  23 lemma ylt_le_succ_sn (x) (y):
  24       x < ∞ → ↑x ≤ y → x < y.
  25 /3 width=3 by ylt_yle_trans, ylt_succ_dx_refl/ qed.
  26
  27 (* Inversions and destructions on yle ***************************************)
  28
  29 (*** ylt_inv_le *)
  30 lemma ylt_inv_le_succ_sn (x) (y):
  31       x < y → ∧∧ x < ∞ & ↑x ≤ y.
  32 #x #y * -x -y
  33 /3 width=1 by yle_inj, conj/
  34 qed-.
  35
  36 (* Destructions with yle and ysucc ******************************************)
  37
  38 (*** ylt_fwd_le_succ1 *)
  39 lemma ylt_des_le_succ_sn (x) (y): x < y → ↑x ≤ y.
  40 #x #y #H
  41 elim (ylt_inv_le_succ_sn … H) -H #_ //
  42 qed-.
  43
  44 (*** ylt_fwd_succ2 *)
  45 lemma ylt_des_succ_dx (x) (y): x < ↑y → x ≤ y.
  46 #x #y @(ynat_split_nat_inf … y) -y //
  47 #n <ysucc_inj #H
  48 elim (ylt_inv_inj_dx … H) -H #m #Hm #H destruct
  49 /3 width=1 by yle_inj, nlt_inv_succ_dx/
  50 qed-.