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14
15 include "ground/arith/ynat_succ.ma".
16 include "ground/arith/ynat_le.ma".
17
18 (* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
19
20 (* Constructions with ysucc *************************************************)
21
22 (*** yle_succ *)
23 lemma yle_succ_bi (x) (y): x ≤ y → ↑x ≤ ↑y.
24 #x #y * -x -y
25 /3 width=1 by yle_inj, yle_inf, nle_succ_bi/
26 qed.
27
28 (*** yle_succ_dx *)
29 lemma yle_succ_dx (x) (y): x ≤ y → x ≤ ↑y.
30 #x #y * -x -y
31 /3 width=1 by yle_inj, yle_inf, nle_succ_dx/
32 qed.
33
34 (*** yle_refl_S_dx *)
35 lemma yle_succ_dx_refl (x): x ≤ ↑x.
36 /2 width=1 by yle_succ_dx/ qed.
37
38 (* Inversions with ysucc ****************************************************)
39
40 (*** yle_inv_succ *)
41 lemma yle_inv_succ_bi (x) (y): ↑x ≤ ↑y → x ≤ y.
42 #x #y @(ynat_split_nat_inf … y) -y //
43 #n <ysucc_inj #H
44 elim (yle_inv_inj_dx … H) -H #o #Hmn #H
45 elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct
46 /3 width=1 by yle_inj, nle_inv_succ_bi/
47 qed-.