]> matita.cs.unibo.it Git - helm.git/blob - matita/matita/contribs/lambdadelta/ground/arith/pnat_lt_pred.ma
update in delayed_updating
[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / pnat_lt_pred.ma
1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
5 (*      ||T||                                                             *)
6 (*      ||I||       Developers:                                           *)
7 (*      ||T||         The HELM team.                                      *)
8 (*      ||A||         http://helm.cs.unibo.it                             *)
9 (*      \   /                                                             *)
10 (*       \ /        This file is distributed under the terms of the       *)
11 (*        v         GNU General Public License Version 2                  *)
12 (*                                                                        *)
13 (**************************************************************************)
14
15 include "ground/arith/pnat_le_pred.ma".
16 include "ground/arith/pnat_lt.ma".
17
18 (* STRICT ORDER FOR POSITIVE INTEGERS ***************************************)
19
20 (* Destructions with ppred **************************************************)
21
22 lemma plt_des_gen (p) (q): p < q → q = ↑↓q.
23 #p #q elim q -q //
24 #H elim (plt_inv_unit_dx … H)
25 qed-.
26
27 (* Inversions with ppred ****************************************************)
28
29 lemma plt_inv_gen (p) (q): p < q → ∧∧ p ≤ ↓q & q = ↑↓q.
30 /2 width=1 by ple_inv_succ_sn/ qed-.
31
32 lemma plt_inv_succ_sn (p) (q): ↑p < q → ∧∧ p < ↓q & q = ↑↓q.
33 /2 width=1 by ple_inv_succ_sn/ qed-.
34
35 lemma plt_inv_pred_dx (p) (q): p < ↓q → ↑p < q.
36 #p #q #H >(plt_des_gen (𝟏) q)
37 [ /2 width=1 by plt_succ_bi/
38 | /3 width=3 by ple_plt_trans, plt_ple_trans/
39 ]
40 qed-.
41
42 lemma plt_inv_pred_bi (p) (q):
43       ↓p < ↓q → p < q.
44 /3 width=3 by plt_inv_pred_dx, ple_plt_trans/
45 qed-.
46
47 (* Constructions with ppred *************************************************)
48
49 lemma plt_unit_sn (q): q = ↑↓q → 𝟏 < q.
50 // qed.
51
52 lemma plt_pred_bi (p) (q): 𝟏 < p → p < q → ↓p < ↓q.
53 #p #q #Hp #Hpq
54 @ple_inv_succ_bi
55 <(plt_des_gen … Hp) -Hp
56 <(plt_des_gen … Hpq) //
57 qed.