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14
15 include "ground/arith/ynat_succ.ma".
16 include "ground/arith/ynat_le_pred.ma".
17
18 (* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
19
20 (* Constructions with ypred and ysucc ***************************************)
21
22 (*** yle_refl_SP_dx *)
23 lemma yle_succ_pred_dx_refl (x): x ≤ ↑↓x.
24 #x @(ynat_split_nat_inf … x) -x
25 /2 width=1 by yle_inj/
26 qed.
27
28 (*** yle_inv_succ2 *)
29 lemma yle_pred_sn (x) (y): x ≤ ↑y → ↓x ≤ y.
30 #x #y0 @(insert_eq_1 … (↑y0))
31 #y * -x -y
32 [ #m #n0 #Hmn #H
33   elim (eq_inv_ysucc_inj … H) -H #n #H1 #H2 destruct
34   /3 width=1 by yle_inj, nle_pred_sn/
35 | #x0 #H <(eq_inv_ysucc_inf … H) -y0 //
36 ]
37 qed.
38
39 (* Inversions with ypred and ysucc ******************************************)
40
41 (*** yle_succ2 *)
42 lemma yle_inv_pred_sn (x) (y): ↓x ≤ y → x ≤ ↑y.
43 #x0 #y @(insert_eq_1 … (↓x0))
44 #x * -x -y // #m0 #n #Hmn #H
45 elim (eq_inv_ypred_inj … H) -H #m #H1 #H2 destruct
46 /3 width=1 by yle_inj, nle_inv_pred_sn/
47 qed-.
48
49 (*** yle_inv_succ1 *)
50 lemma yle_inv_succ_sn (x) (y):
51       ↑x ≤ y → ∧∧ x ≤ ↓y & y = ↑↓y.
52 #x0 #y @(insert_eq_1 … (↑x0))
53 #x * -x -y
54 [ #m0 #n #Hmn #H
55   elim (eq_inv_ysucc_inj … H) -H #m #H1 #H2 destruct
56   elim (nle_inv_succ_sn … Hmn) -Hmn #Hmn #Hn
57   /3 width=1 by yle_inj, conj/
58 | /2 width=1 by yle_inf, conj/
59 ]
60 qed-.