matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_pred.ma

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  14
  15 include "ground_2/notation/functions/predecessor_1.ma".
  16 include "ground_2/lib/arith.ma".
  17 include "ground_2/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 ⇒ ⫰m
  24 | Y      ⇒ Y
  25 ].
  26
  27 interpretation "ynat predecessor" 'Predecessor m = (ypred m).
  28
  29 lemma ypred_O: ⫰(yinj 0) = yinj 0.
  30 // qed.
  31
  32 lemma ypred_S: ∀m:nat. ⫰(⫯m) = yinj m.
  33 // qed.
  34
  35 lemma ypred_Y: (⫰∞) = ∞.
  36 // qed.
  37
  38 (* Inversion lemmas *********************************************************)
  39
  40 lemma ypred_inv_refl: ∀m:ynat. ⫰m = m → m = 0 ∨ m = ∞.
  41 * // #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
  42 /4 width=1 by pred_inv_refl, or_introl, eq_f/
  43 qed-.