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1 (**************************************************************************)
2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
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11 (*        v         GNU General Public License Version 2                  *)
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14
15 include "ground/arith/nat_le_pred.ma".
16 include "ground/arith/nat_lt.ma".
17
18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
19
20 (* Destructions with npred **************************************************)
21
22 (*** S_pred lt_succ_pred lt_inv_O1 *)
23 lemma nlt_des_gen (m) (n): m < n → n = ↑↓n.
24 #m #n @(nat_ind_succ … n) -n //
25 #H elim (nlt_inv_zero_dx … H)
26 qed-.
27
28 (* Inversions with npred ****************************************************)
29
30 (*** lt_inv_gen *)
31 lemma nlt_inv_gen (m) (n): m < n → ∧∧ m ≤ ↓n & n = ↑↓n.
32 /2 width=1 by nle_inv_succ_sn/ qed-.
33
34 (*** lt_inv_S1 *)
35 lemma nlt_inv_succ_sn (m) (n): ↑m < n → ∧∧ m < ↓n & n = ↑↓n.
36 /2 width=1 by nle_inv_succ_sn/ qed-.
37
38 lemma nlt_inv_pred_dx (m) (n): m < ↓n → ↑m < n.
39 #m #n #H >(nlt_des_gen (𝟎) n)
40 [ /2 width=1 by nlt_succ_bi/
41 | /3 width=3 by nle_nlt_trans, nlt_nle_trans/
42 ]
43 qed-.
44
45 lemma nlt_inv_pred_bi (m) (n):
46       ↓m < ↓n → m < n.
47 /3 width=3 by nlt_inv_pred_dx, nle_nlt_trans/
48 qed-.
49
50 (* Constructions with npred *************************************************)
51
52 lemma nlt_zero_sn (n): n = ↑↓n → 𝟎 < n.
53 // qed.
54
55 (*** monotonic_lt_pred *)
56 lemma nlt_pred_bi (m) (n): 𝟎 < m → m < n → ↓m < ↓n.
57 #m #n #Hm #Hmn
58 @nle_inv_succ_bi
59 <(nlt_des_gen … Hm) -Hm
60 <(nlt_des_gen … Hmn) //
61 qed.