include "ground/arith/nat_plus.ma".
include "ground/arith/nat_le.ma".
-(* NON-NEGATIVE INTEGERS ****************************************************)
+(* ORDER FOR NON-NEGATIVE INTEGERS ****************************************************)
-(* Basic constructions with plus ********************************************)
+(* Constructions with nplus ***********************************************************)
(*** monotonic_le_plus_l *)
lemma nle_plus_bi_dx (m) (n1) (n2): n1 ≤ n2 → n1 + m ≤ n2 + m.
lemma nle_plus_dx_sn (m) (n) (o): n ≤ m → n ≤ o + m.
/2 width=3 by nle_trans/ qed.
-(* Main constructions with plus *********************************************)
+(* Main constructions with nplus ********************************************)
(*** le_plus *)
theorem nle_plus_bi (m1) (m2) (n1) (n2):
m1 ≤ m2 → n1 ≤ n2 → m1 + n1 ≤ m2 + n2.
/3 width=3 by nle_plus_bi_dx, nle_plus_bi_sn, nle_trans/ qed.
-(* Basic inversions with plus ***********************************************)
+(* Inversions with nplus ****************************************************)
(*** plus_le_0 *)
lemma nle_inv_plus_zero (m) (n): m + n ≤ 𝟎 → ∧∧ 𝟎 = m & 𝟎 = n.
(*** le_plus_to_le *)
lemma nle_inv_plus_bi_sn (o) (m) (n): o + n ≤ o + m → n ≤ m.
/2 width=2 by nle_inv_plus_bi_dx/ qed-.
-