]> matita.cs.unibo.it Git - helm.git/blob - matita/matita/contribs/lambdadelta/ground/arith/ynat_lt_plus.ma
update in delayed_updating
[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / ynat_lt_plus.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/nat_lt_plus.ma".
16 include "ground/arith/ynat_plus.ma".
17 include "ground/arith/ynat_lt_succ.ma".
18
19 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
20
21 (* Constructions with yplus *************************************************)
22
23 (*** ylt_plus_Y *)
24 lemma ylt_plus_inf (x) (y):
25       x < ∞ → y < ∞ → x + y < ∞.
26 #x #y #Hx #Hy
27 elim (ylt_des_gen_sn … Hx) -Hx #m #H destruct
28 elim (ylt_des_gen_sn … Hy) -Hy #n #H destruct
29 //
30 qed.
31
32 (*** ylt_plus_dx1_trans *)
33 lemma ylt_plus_dx_sn (z) (x) (y):
34       z < x → z < x + y.
35 #z #x #y * -z -x //
36 #o #m #Hom @(ynat_split_nat_inf … y) - y //
37 /3 width=1 by ylt_inj, nle_plus_bi/
38 qed.
39
40 (*** ylt_plus_dx2_trans *)
41 lemma ylt_plus_dx_dx (z) (x) (y):
42       z < y → z < x + y.
43 #z #x #y <yplus_comm
44 /2 width=1 by ylt_plus_dx_sn/
45 qed.
46
47 (*** monotonic_ylt_plus_dx_inj *)
48 lemma ylt_plus_bi_dx_inj (o) (x) (y):
49       x < y → x + yinj_nat o < y + yinj_nat o.
50 #o #x #y #Hxy
51 @(nat_ind_succ … o) -o //
52 #n #IH >ysucc_inj <yplus_succ_dx <yplus_succ_dx
53 /2 width=1 by ylt_succ_bi/
54 qed.
55
56 (*** monotonic_ylt_plus_sn_inj *)
57 lemma ylt_plus_bi_sn_inj (o) (x) (y):
58       x < y → yinj_nat o + x < yinj_nat o + y.
59 /2 width=1 by ylt_plus_bi_dx_inj/ qed.
60
61 (*** monotonic_ylt_plus_dx *)
62 lemma ylt_plus_bi_dx (z) (x) (y):
63       x < y → z < ∞ → x + z < y + z.
64 #z #x #y #Hxy #Hz
65 elim (ylt_des_gen_sn … Hz) -Hz #o #H destruct
66 /2 width=1 by ylt_plus_bi_dx_inj/
67 qed.
68
69 (*** monotonic_ylt_plus_sn *)
70 lemma ylt_plus_bi_sn (z) (x) (y):
71       x < y → z < ∞ → z + x < z + y.
72 #z #x #y #Hxy #Hz <yplus_comm <yplus_comm in ⊢ (??%); 
73 /2 width=1 by ylt_plus_bi_dx/
74 qed.
75
76 (* Inversions with yplus ****************************************************)
77
78 (*** yplus_inv_monotonic_dx *)
79 lemma eq_inv_yplus_bi_dx (z) (x) (y):
80       z < ∞ → x + z = y + z → x = y.
81 #z #x #y #H
82 elim (ylt_des_gen_sn … H) -H #o #H destruct
83 /2 width=2 by eq_inv_yplus_bi_dx_inj/
84 qed-.
85
86 (*** yplus_inv_monotonic_23 *)
87 lemma yplus_inv_plus_bi_23 (z) (x1) (x2) (y1) (y2):
88       z < ∞ → x1 + z + y1 = x2 + z + y2 → x1 + y1 = x2 + y2.
89 #z #x1 #x2 #y1 #y2 #Hz
90 <yplus_plus_comm_23 <yplus_plus_comm_23 in ⊢ (???%→?); #H
91 @(eq_inv_yplus_bi_dx … H) // (* * auto does not work *)
92 qed-.
93
94 (*** ylt_inv_plus_Y *)
95 lemma ylt_inv_plus_inf (x) (y):
96       x + y < ∞ → ∧∧ x < ∞ & y < ∞.
97 #x #y #H
98 elim (ylt_des_gen_sn … H) -H #o #H
99 elim (eq_inv_yplus_inj … H) -H
100 /2 width=1 by conj/
101 qed-.
102
103 (* Destructions with yplus **************************************************)
104
105 (*** ylt_inv_monotonic_plus_dx *)
106 lemma ylt_des_plus_bi_dx (z) (x) (y):
107       x + z < y + z → x < y.
108 #z @(ynat_split_nat_inf … z) -z
109 [ #o #x @(ynat_split_nat_inf … x) -x
110   [ #m #y @(ynat_split_nat_inf … y) -y //
111     #n <yplus_inj_bi <yplus_inj_bi #H
112     /4 width=2 by ylt_inv_inj_bi, ylt_inj, nlt_inv_plus_bi_dx/
113   | #y <yplus_inf_sn #H
114     elim (ylt_inv_inf_sn … H)
115   ]
116 | #x #y <yplus_inf_dx #H
117   elim (ylt_inv_inf_sn … H)
118 ]
119 qed-.
120
121 lemma ylt_des_plus_bi_sn (z) (x) (y):
122       z + x < z + y → x < y.
123 /2 width=2 by ylt_des_plus_bi_dx/ qed-.