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

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  14
  15 include "ground/ynat/ynat.ma".
  16
  17 (* NATURAL NUMBERS WITH INFINITY ********************************************)
  18
  19 (* the predecessor function *)
  20 definition ypred: ynat → ynat ≝ λm. match m with
  21 [ yinj m ⇒ ↓m
  22 | Y      ⇒ Y
  23 ].
  24
  25 interpretation "ynat predecessor" 'DownArrow m = (ypred m).
  26
  27 lemma ypred_O: ↓(yinj 0) = yinj 0.
  28 // qed.
  29
  30 lemma ypred_S: ∀m:nat. ↓(↑m) = yinj m.
  31 // qed.
  32
  33 lemma ypred_Y: (↓∞) = ∞.
  34 // qed.
  35
  36 (* Inversion lemmas *********************************************************)
  37
  38 lemma ypred_inv_refl: ∀m:ynat. ↓m = m → m = 0 ∨ m = ∞.
  39 * // #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
  40 /4 width=1 by pred_inv_fix_sn, or_introl, eq_f/
  41 qed-.