*** empty log message ***

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

diff --git a/helm/matita/library/nat/orders.ma b/helm/matita/library/nat/orders.ma
index 811706867e4729f52f57ffd1d7f9183073c122f2..796567fa59ae31483de19312a03f49520b73e96c 100644 (file)
--- a/helm/matita/library/nat/orders.ma
+++ b/helm/matita/library/nat/orders.ma
@@ -14,7 +14,7 @@
 
 set "baseuri" "cic:/matita/nat/orders".
 
-include "nat/plus.ma".
+include "nat/nat.ma".
 include "higher_order_defs/ordering.ma".
 
 (* definitions *)
@@ -104,11 +104,11 @@ intros.elim H.exact I.exact I.
 qed.
 
 (* not le *)
-theorem not_le_Sn_O: \forall n:nat. \lnot (S n \leq O).
+theorem not_le_Sn_O: \forall n:nat. S n \nleq O.
 intros.simplify.intros.apply leS_to_not_zero ? ? H.
 qed.
 
-theorem not_le_Sn_n: \forall n:nat. \lnot (S n \leq n).
+theorem not_le_Sn_n: \forall n:nat. S n \nleq n.
 intros.elim n.apply not_le_Sn_O.simplify.intros.cut S n1 \leq n1.
 apply H.assumption.
 apply le_S_S_to_le.assumption.
@@ -123,7 +123,7 @@ left.simplify.apply le_S_S.assumption.
 qed.
 
 (* not eq *)
-theorem lt_to_not_eq : \forall n,m:nat. n<m \to \lnot (n=m).
+theorem lt_to_not_eq : \forall n,m:nat. n<m \to n \neq m.
 simplify.intros.cut (le (S n) m) \to False.
 apply Hcut.assumption.rewrite < H1.
 apply not_le_Sn_n.
@@ -141,16 +141,16 @@ simplify.intros.
 apply le_S_S_to_le.assumption.
 qed.
 
-theorem not_le_to_lt: \forall n,m:nat. \lnot (n \leq m) \to m<n.
+theorem not_le_to_lt: \forall n,m:nat. n \nleq m \to m<n.
 intros 2.
-apply nat_elim2 (\lambda n,m.\lnot (n \leq m) \to m<n).
+apply nat_elim2 (\lambda n,m.n \nleq m \to m<n).
 intros.apply absurd (O \leq n1).apply le_O_n.assumption.
 simplify.intros.apply le_S_S.apply le_O_n.
 simplify.intros.apply le_S_S.apply H.intros.apply H1.apply le_S_S.
 assumption.
 qed.
 
-theorem lt_to_not_le: \forall n,m:nat. n<m \to \lnot (m \leq n).
+theorem lt_to_not_le: \forall n,m:nat. n<m \to m \nleq n.
 simplify.intros 3.elim H.
 apply not_le_Sn_n n H1.
 apply H2.apply lt_to_le. apply H3.
@@ -173,8 +173,8 @@ qed.
 theorem le_n_O_to_eq : \forall n:nat. n \leq O \to O=n.
 intro.elim n.reflexivity.
 apply False_ind.
-(* non si applica not_le_Sn_O *)
-apply  (not_le_Sn_O ? H1).
+apply not_le_Sn_O.
+goal 17. apply H1.
 qed.
 
 theorem le_n_O_elim: \forall n:nat.n \leq O \to \forall P: nat \to Prop.
@@ -282,7 +282,7 @@ simplify in H. apply H.
 qed.
 
 theorem increasing_to_le: \forall f:nat \to nat. (increasing f) 
-\to \forall m:nat. ex nat (\lambda i. m \le (f i)).
+\to \forall m:nat. \exists i. m \le (f i).
 intros.elim m.
 apply ex_intro ? ? O.apply le_O_n.
 elim H1.
@@ -295,7 +295,7 @@ qed.
 
 theorem increasing_to_le2: \forall f:nat \to nat. (increasing f) 
 \to \forall m:nat. (f O) \le m \to 
-ex nat (\lambda i. (f i) \le m \land m <(f (S i))).
+\exists i. (f i) \le m \land m <(f (S i)).
 intros.elim H1.
 apply ex_intro ? ? O.
 split.apply le_n.apply H.