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14
15 include "ground/arith/pnat_pred.ma".
16 include "ground/arith/pnat_le.ma".
17
18 (* ORDER FOR POSITIVE INTEGERS **********************************************)
19
20 (* Destructions with ppred **************************************************)
21
22 lemma ple_inv_pred_sn (p) (q): ↓p ≤ q → p ≤ ↑q.
23 #p #q elim p -p
24 /2 width=1 by ple_succ_bi/
25 qed-.
26
27 (* Constructions with ppred *************************************************)
28
29 lemma ple_succ_pred_dx_refl (p): p ≤ ↑↓p.
30 #p @ple_inv_pred_sn // qed.
31
32 lemma ple_pred_sn_refl (p): ↓p ≤ p.
33 #p elim p -p //
34 qed.
35
36 lemma ple_pred_bi (p) (q): p ≤ q → ↓p ≤ ↓q.
37 #p #q #H elim H -q //
38 /2 width=3 by ple_trans/
39 qed.
40
41 lemma ple_pred_sn (p) (q): p ≤ ↑q → ↓p ≤ q.
42 #p #q elim p -p //
43 /2 width=1 by ple_pred_bi/
44 qed-.
45
46 (* Inversions with ppred ****************************************************)
47
48 lemma ple_inv_succ_sn (p) (q):
49       ↑p ≤ q → ∧∧ p ≤ ↓q & q = ↑↓q.
50 #p #q * -q
51 [ /2 width=3 by ple_refl, conj/
52 | #q #Hq /3 width=1 by ple_des_succ_sn, conj/
53 ]
54 qed-.