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14
15 include "ground/arith/nat_le_minus_plus.ma".
16 include "ground/arith/nat_lt_minus.ma".
17
18 (* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
19
20 (* Constructions with nminus and nplus **************************************)
21
22 (*** lt_plus_to_minus *)
23 lemma nlt_minus_sn (o) (m) (n): m ≤ n → n < o + m → n - m < o.
24 #o #m #n #Hmn #Ho
25 lapply (nle_minus_sn_sn … Ho) -Ho
26 <nminus_succ_sn //
27 qed.
28
29 (*** lt_plus_to_minus_r *)
30 lemma nlt_minus_dx (o) (m) (n): m + o < n → m < n - o.
31 /2 width=1 by nle_minus_dx_sn/ qed.
32
33 (*** lt_inv_plus_l *)
34 lemma nlt_minus_dx_full (o) (m) (n): m + o < n → ∧∧ o < n & m < n - o.
35 /3 width=3 by nlt_minus_dx, nle_nlt_trans, conj/ qed-.
36
37 (* Inversions with nminus and nplus *****************************************)
38
39 (*** lt_minus_to_plus *)
40 lemma nlt_inv_minus_sn (o) (m) (n): m - o < n → m < n + o.
41 #o #m #n #Ho
42 @nle_inv_minus_sn
43 @(nle_trans … Ho) -Ho //
44 qed-.
45
46 (*** lt_minus_to_plus_r *)
47 lemma nlt_inv_minus_dx (o) (m) (n): m < n - o → m + o < n.
48 #o #m #n #Ho
49 lapply (nle_inv_minus_dx ???? Ho) //
50 /3 width=2 by nlt_des_minus_dx, nlt_des_le/
51 qed-.