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14
15 include "ground/arith/nat_le_pred.ma".
16 include "ground/arith/ynat_pred.ma".
17 include "ground/arith/ynat_le.ma".
18
19 (* ORDER FOR NON-NEGATIVE INTEGERS WITH INFINITY ****************************)
20
21 (* Constructions with ypred *************************************************)
22
23 (*** yle_pred_sn *)
24 lemma yle_pred_sn_trans (x) (y): x ≤ y → ↓x ≤ y.
25 #x #y * -x -y
26 /3 width=3 by yle_inj, nle_trans/
27 qed.
28
29 (*** yle_refl_pred_sn *)
30 lemma yle_pred_sn_refl (x): ↓x ≤ x.
31 /2 width=1 by yle_pred_sn_trans, yle_refl/ qed.
32
33 (*** yle_pred *)
34 lemma yle_pred_bi (x) (y): x ≤ y → ↓x ≤ ↓y.
35 #x #y * -x -y
36 /3 width=1 by yle_inj, nle_pred_bi/
37 qed.