X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground%2Farith%2Fnat_le_minus.ma;h=43e56dac4687c51acfc5e012dd56fd62961f97c0;hb=888840f6b3a71d3d686b53b702d362ab90ab0038;hp=607a9135ef31308662bb9003febd8d85bd51a5e5;hpb=ccf5878f2a2ec7f952f140e162391708a740517b;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus.ma
index 607a9135e..43e56dac4 100644
--- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus.ma
+++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus.ma
@@ -21,7 +21,7 @@ include "ground/arith/nat_le_pred.ma".
 
 (*** minus_le *)
 lemma nle_minus_sn_refl_sn (m) (n): m - n ≤ m.
-#m #n elim n -n //
+#m #n @(nat_ind_succ … n) -n //
 #n #IH /2 width=3 by nle_trans/
 qed.
 
@@ -29,27 +29,31 @@ lemma nle_minus_succ_sn (m) (n): ↑n - m ≤ ↑(n - m).
 // qed.
 
 (*** inv_eq_minus_O *)
-lemma nle_eq_minus_O (m) (n): 𝟎 = m - n → m ≤ n.
-#m #n @(nat_ind_2 … m n) //
+lemma nle_eq_zero_minus (m) (n): 𝟎 = m - n → m ≤ n.
+#m #n @(nat_ind_2_succ … m n) //
 /3 width=1 by nle_succ_bi/
 qed.
 
 (*** monotonic_le_minus_l *)
-lemma nle_minus_sn_bi (m) (n) (o): m ≤ n → m-o ≤ n-o.
-#m #n #o elim o -o //
+lemma nle_minus_bi_dx (m) (n) (o): m ≤ n → m-o ≤ n-o.
+#m #n #o @(nat_ind_succ … o) -o //
 #o #IH #Hmn /3 width=1 by nle_pred_bi/
 qed.
 
 (*** monotonic_le_minus_r *)
-lemma nle_minus_dx_bi (m) (n) (o): m ≤ n → o-n ≤ o-m.
+lemma nle_minus_bi_sn (m) (n) (o): m ≤ n → o-n ≤ o-m.
 #m #n #o #H elim H -n //
 #n #_ #IH /2 width=3 by nle_trans/
 qed.
 
+(*** minus_le_trans_sn *)
+lemma nle_minus_sn (o) (m) (n): m ≤ n → m - o ≤ n.
+/2 width=3 by nle_trans/ qed.
+
 (* Inversions with nminus ***************************************************)
 
 (*** eq_minus_O *)
-lemma nle_inv_eq_minus_O (m) (n): m ≤ n → 𝟎 = m - n.
+lemma nle_inv_eq_zero_minus (m) (n): m ≤ n → 𝟎 = m - n.
 #m #n #H elim H -n //
 qed-.