1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "ground_2/ynat/ynat_pred.ma".
17 (* NATURAL NUMBERS WITH INFINITY ********************************************)
19 (* the successor function *)
20 definition ysucc: ynat → ynat ≝ λm. match m with
25 interpretation "ynat successor" 'UpArrow m = (ysucc m).
27 lemma ysucc_inj: ∀m:nat. ↑(yinj m) = yinj (↑m).
30 lemma ysucc_Y: ↑(∞) = ∞.
33 (* Properties ***************************************************************)
35 lemma ypred_succ: ∀m. ↓↑m = m.
38 lemma ynat_cases: ∀n:ynat. n = 0 ∨ ∃m:ynat. n = ↑m.
40 [ * /2 width=1 by or_introl/
41 #n @or_intror @(ex_intro … n) // (**) (* explicit constructor *)
42 | @or_intror @(ex_intro … (∞)) // (**) (* explicit constructor *)
46 lemma ysucc_iter_Y: ∀m. ysucc^m (∞) = ∞.
48 #m #IHm whd in ⊢ (??%?); >IHm //
51 (* Inversion lemmas *********************************************************)
53 lemma ysucc_inv_inj: ∀m,n. ↑m = ↑n → m = n.
54 #m #n #H <(ypred_succ m) <(ypred_succ n) //
57 lemma ysucc_inv_refl: ∀m. ↑m = m → m = ∞.
59 #m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
60 #H elim (lt_refl_false m) //
63 lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ↑n1 →
64 ∃∃m1. n1 = yinj m1 & m2 = S m1.
66 [ #n1 #H destruct /2 width=3 by ex2_intro/
71 lemma ysucc_inv_inj_dx: ∀m2,n1. ↑n1 = yinj m2 →
72 ∃∃m1. n1 = yinj m1 & m2 = S m1.
73 /2 width=1 by ysucc_inv_inj_sn/ qed-.
75 lemma ysucc_inv_Y_sn: ∀m. ∞ = ↑m → m = ∞.
80 lemma ysucc_inv_Y_dx: ∀m. ↑m = ∞ → m = ∞.
81 /2 width=1 by ysucc_inv_Y_sn/ qed-.
83 lemma ysucc_inv_O_sn: ∀m. yinj 0 = ↑m → ⊥. (**) (* explicit coercion *)
84 #m #H elim (ysucc_inv_inj_sn … H) -H
88 lemma ysucc_inv_O_dx: ∀m:ynat. ↑m = 0 → ⊥.
89 /2 width=2 by ysucc_inv_O_sn/ qed-.
91 (* Eliminators **************************************************************)
93 lemma ynat_ind: ∀R:predicate ynat.
94 R 0 → (∀n:nat. R n → R (↑n)) → R (∞) →
96 #R #H1 #H2 #H3 * // #n elim n -n /2 width=1 by/