New naming policy for local variables.

[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..953b3a1b630321187f46f9c2199e8b3dde12a928 100644 (file)
--- a/helm/matita/library/nat/times.ma
+++ b/helm/matita/library/nat/times.ma
@@ -34,8 +34,8 @@ theorem times_n_Sm :
 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 (plus (plus n1 m) (times n1 m)).symmetry.apply assoc_plus.
+transitivity (plus (plus m n1) (times n1 m)).
 apply eq_f2.
 apply sym_plus.
 reflexivity.
@@ -52,11 +52,16 @@ 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.
+eq nat (times n (plus m p)) (plus (times n m) (times n p))
+\def distributive_times_plus.
+