Notation for equality used everywhere.

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

diff --git a/helm/matita/library/nat/compare.ma b/helm/matita/library/nat/compare.ma
index fec46ae2fe1f65f8106020bc4a038553beb1667b..16d689e4cfafb19c34363ccdd6f18e8a6a71abc5 100644 (file)
--- a/helm/matita/library/nat/compare.ma
+++ b/helm/matita/library/nat/compare.ma
@@ -67,26 +67,26 @@ match n with
       [ O \Rightarrow GT
       | (S q) \Rightarrow nat_compare p q]].
 
-theorem nat_compare_n_n: \forall n:nat.(eq compare (nat_compare n n) EQ).
+theorem nat_compare_n_n: \forall n:nat. nat_compare n n = EQ.
 intro.elim n.
 simplify.reflexivity.
 simplify.assumption.
 qed.
 
 theorem nat_compare_S_S: \forall n,m:nat. 
-eq compare (nat_compare n m) (nat_compare (S n) (S m)).
+nat_compare n m = nat_compare (S n) (S m).
 intros.simplify.reflexivity.
 qed.
 
 theorem nat_compare_to_Prop: \forall n,m:nat. 
 match (nat_compare n m) with
   [ LT \Rightarrow (lt n m)
-  | EQ \Rightarrow (eq nat n m)
+  | EQ \Rightarrow n=m
   | GT \Rightarrow (lt m n) ].
 intros.
 apply nat_elim2 (\lambda n,m.match (nat_compare n m) with
   [ LT \Rightarrow (lt n m)
-  | EQ \Rightarrow (eq nat n m)
+  | EQ \Rightarrow n=m
   | GT \Rightarrow (lt m n) ]).
 intro.elim n1.simplify.reflexivity.
 simplify.apply le_S_S.apply le_O_n.
@@ -98,9 +98,9 @@ simplify. apply eq_f. apply H.
 qed.
 
 theorem nat_compare_n_m_m_n: \forall n,m:nat. 
-eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+nat_compare n m = compare_invert (nat_compare m n).
 intros. 
-apply nat_elim2 (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+apply nat_elim2 (\lambda n,m. nat_compare n m = compare_invert (nat_compare m n)).
 intros.elim n1.simplify.reflexivity.
 simplify.reflexivity.
 intro.elim n1.simplify.reflexivity.
@@ -109,12 +109,12 @@ intros.simplify.elim H.reflexivity.
 qed.
      
 theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
-((lt n m) \to (P LT)) \to ((eq nat n m) \to (P EQ)) \to ((lt m n) \to (P GT)) \to 
+((lt n m) \to (P LT)) \to (n=m \to (P EQ)) \to ((lt m n) \to (P GT)) \to 
 (P (nat_compare n m)).
 intros.
 cut match (nat_compare n m) with
 [ LT \Rightarrow (lt n m)
-| EQ \Rightarrow (eq nat n m)
+| EQ \Rightarrow n=m
 | GT \Rightarrow (lt m n)] \to
 (P (nat_compare n m)).
 apply Hcut.apply nat_compare_to_Prop.