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14
15 include "ground/arith/ynat_lt_succ.ma".
16 include "ground/arith/ynat_lt_le.ma".
17
18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY *********************)
19
20 (* Constructions with ysucc *************************************************)
21
22 (*** yle_lt yle_succ1_inj *)
23 lemma ylt_le_succ_sn (x) (y):
24       x < ∞ → ↑x ≤ y → x < y.
25 /3 width=3 by ylt_yle_trans, ylt_succ_dx_refl/ qed.
26
27 (* Inversions with yle and ysucc ********************************************)
28
29 (*** ylt_inv_le *)
30 lemma ylt_inv_le_succ_sn (x) (y):
31       x < y → ∧∧ x < ∞ & ↑x ≤ y.
32 #x #y * -x -y
33 /3 width=1 by yle_inj, conj/
34 qed-.
35
36 (* Destructions with yle and ysucc ******************************************)
37
38 (*** ylt_fwd_le_succ1 *)
39 lemma ylt_des_le_succ_sn (x) (y): x < y → ↑x ≤ y.
40 #x #y #H
41 elim (ylt_inv_le_succ_sn … H) -H #_ //
42 qed-.
43
44 (*** ylt_fwd_succ2 *)
45 lemma ylt_des_succ_dx (x) (y): x < ↑y → x ≤ y.
46 #x #y @(ynat_split_nat_inf … y) -y //
47 #n <ysucc_inj #H
48 elim (ylt_inv_inj_dx … H) -H #m #Hm #H destruct
49 /3 width=1 by yle_inj, nlt_inv_succ_dx/
50 qed-.