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14
15 include "ground/arith/nat_le_plus.ma".
16 include "ground/arith/ynat_plus.ma".
17 include "ground/arith/ynat_le_succ.ma".
18
19 (* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
20
21 (* Constructions with yplus *************************************************)
22
23 (*** monotonic_yle_plus *)
24 lemma yle_plus_bi (x1) (x2) (y1) (y2):
25       x1 ≤ y1 → x2 ≤ y2 → x1 + x2 ≤ y1 + y2.
26 #x1 #x2 #y1 #y2 * //
27 #m1 #n1 #Hmn1 * //
28 #m2 #n2 #Hmn2 /3 width=1 by nle_plus_bi, yle_inj/
29 qed.
30
31 (*** monotonic_yle_plus_dx *)
32 lemma yle_plus_bi_dx (z) (x) (y):
33       x ≤ y → x + z ≤ y + z.
34 /2 width=1 by yle_plus_bi/ qed.
35
36 (*** monotonic_yle_plus_sn *)
37 lemma yle_plus_bi_sn (z) (x) (y):
38       x ≤ y → z + x ≤ z + y.
39 /2 width=1 by yle_plus_bi/ qed.
40
41 (*** yle_plus_dx1_trans *)
42 lemma yle_plus_dx_dx (z) (x) (y):
43       z ≤ x → z ≤ x + y.
44 /2 width=1 by yle_plus_bi/ qed.
45
46 (*** yle_plus_dx2_trans *)
47 lemma yle_plus_dx_sn (z) (x) (y):
48       z ≤ y → z ≤ x + y.
49 #z #x #y <yplus_comm
50 /2 width=3 by yle_plus_dx_dx/
51 qed.
52
53 (*** yle_plus_dx1 *)
54 lemma yle_plus_dx_dx_refl (x) (y): x ≤ x + y.
55 /2 width=1 by yle_plus_dx_dx/ qed.
56
57 (*** yle_plus_dx2 *)
58 lemma yle_plus_dx_sn_refl (x) (y): y ≤ x + y.
59 // qed-.
60
61 (* Inversions with yplus ****************************************************)
62
63 (*** yle_inv_monotonic_plus_dx_inj *)
64 lemma yle_inv_plus_bi_dx_inj (o) (x) (y):
65       x + yinj_nat o ≤ y + yinj_nat o → x ≤ y.
66 #o @(nat_ind_succ … o) -o //
67 #o #IH #x #y >ysucc_inj <yplus_succ_dx <yplus_succ_dx #H
68 /3 width=1 by yle_inv_succ_bi/
69 qed-.
70
71 (*** yle_inv_monotonic_plus_sn_inj *)
72 lemma yle_inv_plus_bi_sn_inj (o) (x) (y):
73       yinj_nat o + x ≤ yinj_nat o + y → x ≤ y.
74 /2 width=2 by yle_inv_plus_bi_dx_inj/ qed-.
75
76 (* Destructions with yplus **************************************************)
77
78 (*** yle_fwd_plus_sn2 *)
79 lemma yle_des_plus_sn_sn (x) (y) (z):
80       x + y ≤ z → y ≤ z.
81 /2 width=3 by yle_trans/ qed-.
82
83 (*** yle_fwd_plus_sn1 *)
84 lemma yle_des_plus_sn_dx (x) (y) (z):
85       x + y ≤ z → x ≤ z.
86 /2 width=3 by yle_trans/ qed-.
87
88 (*** yle_fwd_plus_ge *)
89 lemma yle_des_plus_bi_sn_inj_bi (m) (n) (x) (y):
90       n ≤ m → yinj_nat m + x ≤ yinj_nat n + y → x ≤ y.
91 #m #n #x #y #Hnm #H
92 lapply (yle_inj … Hnm) -Hnm #Hnm
93 lapply (yle_plus_bi … Hnm … H) -Hnm -H
94 <yplus_assoc <yplus_inj_bi <yplus_assoc <yplus_inj_bi #H
95 /2 width=2 by yle_inv_plus_bi_sn_inj/
96 qed-.
97
98 (*** yle_fwd_plus_ge_inj *)
99 lemma yle_des_plus_bi_sn_inj_sn (o) (z) (x) (y):
100       z ≤ yinj_nat o → yinj_nat o + x ≤ z + y → x ≤ y.
101 #m #z #x #y #H elim (yle_inv_inj_dx … H) -H
102 #n #Hnm #H destruct
103 /2 width=4 by yle_des_plus_bi_sn_inj_bi/
104 qed-.
105
106 (*** yle_fwd_plus_yge *)
107 lemma yle_des_plus_bi_sn_inj_md (m1) (m2) (y1) (y2):
108       yinj_nat m2 ≤ y1 → y1 + yinj_nat m1 ≤ yinj_nat m2 + y2 → yinj_nat m1 ≤ y2.
109 #m1 #m2 #y1 #y2 @(ynat_split_nat_inf … y2) -y2 //
110 #n2 @(ynat_split_nat_inf … y1) -y1
111 /2 width=4 by yle_des_plus_bi_sn_inj_sn/ 
112 qed-.