some simplifications.

[helm.git] / matita / library / nat / orders.ma

diff --git a/matita/library/nat/orders.ma b/matita/library/nat/orders.ma
index 8c6ce942e33b0d272a8cadbbaabbc48e93eb6e49..8053d50de55bfe8afc7d434243b17ceca34b9536 100644 (file)
--- a/matita/library/nat/orders.ma
+++ b/matita/library/nat/orders.ma
@@ -139,6 +139,19 @@ apply Hcut.assumption.rewrite < H1.
 apply not_le_Sn_n.
 qed.
 
+(*not lt*)
+theorem eq_to_not_lt: \forall a,b:nat.
+a = b \to a \nlt b.
+intros.
+unfold Not.
+intros.
+rewrite > H in H1.
+apply (lt_to_not_eq b b)
+[ assumption
+| reflexivity
+]
+qed.
+
 (* le vs. lt *)
 theorem lt_to_le : \forall n,m:nat. n<m \to n \leq m.
 simplify.intros.unfold lt in H.elim H.
@@ -182,8 +195,8 @@ qed.
 theorem le_n_O_to_eq : \forall n:nat. n \leq O \to O=n.
 intro.elim n.reflexivity.
 apply False_ind.
-apply not_le_Sn_O.
-goal 17. apply H1.
+apply not_le_Sn_O;
+[2: apply H1 | skip].
 qed.
 
 theorem le_n_O_elim: \forall n:nat.n \leq O \to \forall P: nat \to Prop.
@@ -201,7 +214,7 @@ apply H2.reflexivity.
 apply H3. apply le_S_S. assumption.
 qed.
 
-(* le to eq *)
+(* le and eq *)
 lemma le_to_le_to_eq: \forall n,m. n \le m \to m \le n \to n = m.
 apply nat_elim2
   [intros.apply le_n_O_to_eq.assumption