(* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************)
-(* Rewrites with nminus *****************************************************)
-
-(*** minus_pred_pred *)
-lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m.
-#m #n #Hm #Hn
->(nlt_inv_zero_sn … Hm) in ⊢ (??%?); -Hm
->(nlt_inv_zero_sn … Hn) in ⊢ (??%?); -Hn
-//
-qed-.
-
(* Constructions with nminus ************************************************)
(*** monotonic_lt_minus_l *)
(* Destructions with nminus *************************************************)
+(*** minus_pred_pred *)
+lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m.
+#m #n #Hm #Hn
+>(nlt_inv_zero_sn … Hm) in ⊢ (??%?); -Hm
+>(nlt_inv_zero_sn … Hn) in ⊢ (??%?); -Hn
+//
+qed-.
+
lemma nlt_des_minus_dx (o) (m) (n): m < n - o → o < n.
-#o elim o -o
+#o @(nat_ind_succ … o) -o
[ #m #n <nminus_zero_dx
/2 width=3 by le_nlt_trans/
| #o #IH #m #n <nminus_succ_dx_pred_sn #H