typo s/\\OF/\\of/

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

diff --git a/helm/matita/library/nat/times.ma b/helm/matita/library/nat/times.ma
index 44f73b71a080b6e253d49b5dfbb2e1b622d45174..d2ce51bfa83dd9605bf5160e227edf3c532aeff0 100644 (file)
--- a/helm/matita/library/nat/times.ma
+++ b/helm/matita/library/nat/times.ma
@@ -21,21 +21,24 @@ include "nat/plus.ma".
 let rec times n m \def 
  match n with 
  [ O \Rightarrow O
- | (S p) \Rightarrow (plus m (times p m)) ].
+ | (S p) \Rightarrow (m+(times p m)) ].
 
-theorem times_n_O: \forall n:nat. eq nat O (times n O).
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "natural times" 'times x y = (cic:/matita/nat/times/times.con x y).
+
+theorem times_n_O: \forall n:nat. O = n*O.
 intros.elim n.
 simplify.reflexivity.
 simplify.assumption.
 qed.
 
 theorem times_n_Sm : 
-\forall n,m:nat. eq nat (plus n (times n  m)) (times n (S m)).
+\forall n,m:nat.n+n*m = n*(S m).
 intros.elim n.
 simplify.reflexivity.
 simplify.apply eq_f.rewrite < H.
-transitivity (plus (plus e1 m) (times e1 m)).symmetry.apply assoc_plus.
-transitivity (plus (plus m e1) (times e1 m)).
+transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
+transitivity ((m+n1)+n1*m).
 apply eq_f2.
 apply sym_plus.
 reflexivity.
@@ -46,17 +49,21 @@ qed.
 theorem symmetric_times : symmetric nat times. *)
 
 theorem sym_times : 
-\forall n,m:nat. eq nat (times n m) (times m n).
+\forall n,m:nat.n*m = m*n.
 intros.elim n.
 simplify.apply times_n_O.
 simplify.rewrite > H.apply times_n_Sm.
 qed.
 
-theorem times_plus_distr: \forall n,m,p:nat.
-eq nat (times n (plus m p)) (plus (times n m) (times n p)).
-intros.elim n.
+theorem distributive_times_plus : distributive nat times plus.
+simplify.
+intros.elim x.
 simplify.reflexivity.
 simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
-apply eq_f.rewrite < assoc_plus. rewrite < sym_plus ? p.
+apply eq_f.rewrite < assoc_plus. rewrite < sym_plus ? z.
 rewrite > assoc_plus.reflexivity.
 qed.
+
+variant times_plus_distr: \forall n,m,p:nat. n*(m+p)=n*m+n*p
+\def distributive_times_plus.
+