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

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  14
  15 include "ground/arith/nat_pred.ma".
  16 include "ground/arith/ynat_nat.ma".
  17
  18 (* PREDECESSOR FOR NON-NEGATIVE INTEGERS WITH INFINITY **********************)
  19
  20 definition ypred_aux (n): ynat ≝
  21            yinj_nat (↓n).
  22
  23 (*** ypred *)
  24 definition ypred: ynat → ynat ≝
  25            ynat_bind_nat ypred_aux (∞).
  26
  27 interpretation
  28   "successor (non-negative integers with infinity)"
  29   'DownArrow x = (ypred x).
  30
  31 (* Constructions ************************************************************)
  32
  33 (*** ypred_O *)
  34 lemma ypred_inj (n): yinj_nat (↓n) = ↓(yinj_nat n).
  35 @(ynat_bind_nat_inj ypred_aux)
  36 qed.
  37
  38 (*** ypred_Y *)
  39 lemma ypred_inf: ∞ = ↓∞.
  40 // qed.
  41
  42 (* Inversions ***************************************************************)
  43
  44 lemma eq_inv_ypred_inj (x) (n):
  45       ↓x = yinj_nat n →
  46       ∃∃m. x = yinj_nat m & n = ↓m.
  47 #x #n @(ynat_split_nat_inf … x) -x
  48 [ #m <ypred_inj #H <(eq_inv_yinj_nat_bi … H) -n
  49   /2 width=3 by ex2_intro/
  50 | #H elim (eq_inv_inf_yinj_nat … H)
  51 ]
  52 qed-.
  53
  54 (*** ypred_inv_refl *)
  55 lemma ypred_inv_refl (x): x = ↓x → ∨∨ 𝟎 = x | ∞ = x.
  56 #x @(ynat_split_nat_inf … x) -x //
  57 #n <ypred_inj #H
  58 lapply (eq_inv_yinj_nat_bi … H) -H #H
  59 lapply (npred_inv_refl … H) -H #H
  60 /2 width=1 by or_introl/
  61 qed-.