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14
15 include "ground/arith/ynat_succ.ma".
16 include "ground/arith/ynat_lt.ma".
17
18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
19
20 (* Constructions with ysucc *************************************************)
21
22 (*** ylt_O_succ *)
23 lemma ylt_zero_succ (y): 𝟎 < ↑y.
24 #y @(ynat_split_nat_inf … y) -y
25 /2 width=1 by ylt_inj/
26 qed.
27
28 (*** ylt_succ *)
29 lemma ylt_succ_bi (x) (y): x < y → ↑x < ↑y.
30 #x #y * -x -y
31 /3 width=1 by ylt_inj, ylt_inf, nlt_succ_bi/
32 qed.
33
34 (*** ylt_succ_Y *)
35 lemma ylt_succ_inf (x): x < ∞ → ↑x < ∞.
36 #x @(ynat_split_nat_inf … x) -x //
37 qed.
38
39 (*** ylt_succ2_refl *)
40 lemma ylt_succ_dx_refl (x) (y): x < y → x < ↑x.
41 #x #y #H
42 elim (ylt_des_gen_sn … H) -y #n #H destruct
43 /2 width=1 by ylt_inj/
44 qed.
45
46 (* Inversions with ysucc ****************************************************)
47
48 lemma ylt_inv_succ_inf (x): ↑x < ∞ → x < ∞.
49 #x #H
50 elim (ylt_des_gen_sn … H) -H #m0 #H
51 elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct //
52 qed-.
53
54 (*** ylt_inv_succ *)
55 lemma ylt_inv_succ_bi (x) (y): ↑x < ↑y → x < y.
56 #x #y @(ynat_split_nat_inf … y) -y
57 [ #n <ysucc_inj #H
58   elim (ylt_inv_inj_dx … H) -H #m0 #Hmn #H
59   elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct
60   /3 width=1 by ylt_inj, nlt_inv_succ_bi/
61 | /2 width=1 by ylt_inv_succ_inf/
62 ]
63 qed-.