propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_le_plus.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma
index c1dd32661bb90ae09af5e7733b0c3488ac0a0429..7a53e23f745851ddc67afeab505582aa40f4592e 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma
@@ -15,9 +15,9 @@
 include "ground/arith/nat_plus.ma".
 include "ground/arith/nat_le.ma".
 
-(* ORDER FOR NON-NEGATIVE INTEGERS ****************************************************)
+(* ORDER FOR NON-NEGATIVE INTEGERS ******************************************)
 
-(* Constructions with nplus ***********************************************************)
+(* Constructions with nplus *************************************************)
 
 (*** monotonic_le_plus_l *)
 lemma nle_plus_bi_dx (m) (n1) (n2): n1 ≤ n2 → n1 + m ≤ n2 + m.
@@ -38,12 +38,12 @@ lemma nle_plus_dx_dx_refl (m) (n): m ≤ m + n.
 lemma nle_plus_dx_sn_refl (m) (n): m ≤ n + m.
 /2 width=1 by nle_plus_bi_sn/ qed.
 
-(*** le_plus_b *)
-lemma nle_plus_dx_dx (m) (n) (o): n + o ≤ m → n ≤ m.
+(*** le_plus_a *)
+lemma nle_plus_dx_sn (o) (m) (n): n ≤ m → n ≤ o + m.
 /2 width=3 by nle_trans/ qed.
 
-(*** le_plus_a *)
-lemma nle_plus_dx_sn (m) (n) (o): n ≤ m → n ≤ o + m.
+(*** le_plus_b *)
+lemma nle_plus_dx_dx (o) (m) (n): n ≤ m → n ≤ m + o.
 /2 width=3 by nle_trans/ qed.
 
 (* Main constructions with nplus ********************************************)
@@ -55,6 +55,10 @@ theorem nle_plus_bi (m1) (m2) (n1) (n2):
 
 (* Inversions with nplus ****************************************************)
 
+(*** le_plus_b *)
+lemma nle_inv_plus_sn_dx (m) (n) (o): n + o ≤ m → n ≤ m.
+/2 width=3 by nle_trans/ qed.
+
 (*** plus_le_0 *)
 lemma nle_inv_plus_zero (m) (n): m + n ≤ 𝟎 → ∧∧ 𝟎 = m & 𝟎 = n.
 /3 width=1 by nle_inv_zero_dx, eq_inv_zero_nplus/ qed-.