matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_pred.ma

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   3 (*      ||M||                                                             *)
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  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "ground_2A/notation/functions/predecessor_1.ma".
  16 include "ground_2A/lib/arith.ma".
  17 include "ground_2A/ynat/ynat.ma".
  18
  19 (* NATURAL NUMBERS WITH INFINITY ********************************************)
  20
  21 (* the predecessor function *)
  22 definition ypred: ynat → ynat ≝ λm. match m with
  23 [ yinj m ⇒ pred m
  24 | Y      ⇒ Y
  25 ].
  26
  27 interpretation "ynat predecessor" 'Predecessor m = (ypred m).
  28
  29 (* Properties ***************************************************************)
  30
  31 lemma ypred_inj_rew: ∀m:nat. ⫰m = pred m.
  32 // qed.
  33
  34 (* Inversion lemmas *********************************************************)
  35
  36 lemma ypred_inv_refl: ∀m. ⫰m = m → m = 0 ∨ m = ∞.
  37 * // #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
  38 /4 width=1 by pred_inv_refl, or_introl, eq_f/
  39 qed-.