Notation for equality used everywhere.

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

diff --git a/helm/matita/library/nat/nat.ma b/helm/matita/library/nat/nat.ma
index f84ffef44cd44bb8da092f5011c48b483cb8e8e0..100a101f95e1fa27797ea2e92e1be7eb456a578b 100644 (file)
--- a/helm/matita/library/nat/nat.ma
+++ b/helm/matita/library/nat/nat.ma
@@ -27,8 +27,7 @@ definition pred: nat \to nat \def
 [ O \Rightarrow  O
 | (S p) \Rightarrow p ].
 
-theorem pred_Sn : \forall n:nat.
-(eq nat n (pred (S n))).
+theorem pred_Sn : \forall n:nat.n=(pred (S n)).
 intros; reflexivity.
 qed.
 
@@ -40,12 +39,11 @@ rewrite > pred_Sn y.
 apply eq_f. assumption.
 qed.
 
-theorem inj_S : \forall n,m:nat. 
-(eq nat (S n) (S m)) \to (eq nat n m)
+theorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m
 \def injective_S.
 
 theorem not_eq_S  : \forall n,m:nat. 
-Not (eq nat n m) \to Not (eq nat (S n) (S m)).
+Not (n=m) \to Not (S n = S m).
 intros. simplify. intros.
 apply H. apply injective_S. assumption.
 qed.
@@ -56,14 +54,14 @@ definition not_zero : nat \to Prop \def
   [ O \Rightarrow False
   | (S p) \Rightarrow True ].
 
-theorem not_eq_O_S : \forall n:nat. Not (eq nat O (S n)).
+theorem not_eq_O_S : \forall n:nat. Not (O=S n).
 intros. simplify. intros.
 cut (not_zero O).
 exact Hcut.
 rewrite > H.exact I.
 qed.
 
-theorem not_eq_n_Sn : \forall n:nat. Not (eq nat n (S n)).
+theorem not_eq_n_Sn : \forall n:nat. Not (n=S n).
 intros.elim n.
 apply not_eq_O_S.
 apply not_eq_S.assumption.