Notation for equality used everywhere.

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

diff --git a/helm/matita/library/nat/plus.ma b/helm/matita/library/nat/plus.ma
index e8750d0cba0f41a0995a23524a8d6f887a83204e..39e48f21e58a85811b1eb99e4666cb0c89dcaea3 100644 (file)
--- a/helm/matita/library/nat/plus.ma
+++ b/helm/matita/library/nat/plus.ma
@@ -21,13 +21,13 @@ let rec plus n m \def
  [ O \Rightarrow m
  | (S p) \Rightarrow S (plus p m) ].
 
-theorem plus_n_O: \forall n:nat. eq nat n (plus n O).
+theorem plus_n_O: \forall n:nat. n = plus n O.
 intros.elim n.
 simplify.reflexivity.
 simplify.apply eq_f.assumption.
 qed.
 
-theorem plus_n_Sm : \forall n,m:nat. eq nat (S (plus n  m)) (plus n (S m)).
+theorem plus_n_Sm : \forall n,m:nat. S (plus n  m) = plus n (S m).
 intros.elim n.
 simplify.reflexivity.
 simplify.apply eq_f.assumption.
@@ -38,7 +38,7 @@ and functions/symmetric; functions symmetric is not in
 functions.moo why?
 theorem symmetric_plus: symmetric nat plus. *)
 
-theorem sym_plus: \forall n,m:nat. eq nat (plus n m) (plus m n).
+theorem sym_plus: \forall n,m:nat. plus n m = plus m n.
 intros.elim n.
 simplify.apply plus_n_O.
 simplify.rewrite > H.apply plus_n_Sm.
@@ -50,7 +50,7 @@ simplify.reflexivity.
 simplify.apply eq_f.assumption.
 qed.
 
-theorem assoc_plus : \forall n,m,p:nat. eq nat (plus (plus n m) p) (plus n (plus m p))
+theorem assoc_plus : \forall n,m,p:nat. plus (plus n m) p = plus n (plus m p)
 \def associative_plus.
 
 theorem injective_plus_r: \forall n:nat.injective nat nat (\lambda m.plus n m).
@@ -59,7 +59,7 @@ exact H.
 apply H.apply inj_S.apply H1.
 qed.
 
-theorem inj_plus_r: \forall p,n,m:nat.eq nat (plus p n) (plus p m) \to (eq nat n m)
+theorem inj_plus_r: \forall p,n,m:nat. plus p n = plus p m \to n=m
 \def injective_plus_r.
 
 theorem injective_plus_l: \forall m:nat.injective nat nat (\lambda n.plus n m).
@@ -71,5 +71,5 @@ rewrite < sym_plus y.
 assumption.
 qed.
 
-theorem inj_plus_l: \forall p,n,m:nat.eq nat (plus n p) (plus m p) \to (eq nat n m)
+theorem inj_plus_l: \forall p,n,m:nat. plus n p = plus m p \to n=m
 \def injective_plus_l.