updating the structures for sorts

[helm.git] / helm / software / matita / library / Z / times.ma

diff --git a/helm/software/matita/library/Z/times.ma b/helm/software/matita/library/Z/times.ma
index e5e1cdb452fa38ce33a8ba73f95b6ca4a5d5f27d..eefe4af7ead7d2f818216c44c6c9174fe298ccad 100644 (file)
--- a/helm/software/matita/library/Z/times.ma
+++ b/helm/software/matita/library/Z/times.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/Z/times".
-
 include "nat/lt_arith.ma".
 include "Z/plus.ma".
 
@@ -32,8 +30,7 @@ definition Ztimes :Z \to Z \to Z \def
          | (pos n) \Rightarrow (neg (pred ((S m) * (S n))))
          | (neg n) \Rightarrow (pos (pred ((S m) * (S n))))]].
          
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer times" 'times x y = (cic:/matita/Z/times/Ztimes.con x y).
+interpretation "integer times" 'times x y = (Ztimes x y).
 
 theorem Ztimes_z_OZ:  \forall z:Z. z*OZ = OZ.
 intro.elim z.
@@ -42,6 +39,8 @@ simplify.reflexivity.
 simplify.reflexivity.
 qed.
 
+definition Zone \def pos O.
+
 theorem Ztimes_neg_Zopp: \forall n:nat.\forall x:Z.
 neg n * x = - (pos n * x).
 intros.elim x.
@@ -49,6 +48,7 @@ simplify.reflexivity.
 simplify.reflexivity.
 simplify.reflexivity.
 qed.
+
 theorem symmetric_Ztimes : symmetric Z Ztimes.
 change with (\forall x,y:Z. x*y = y*x).
 intros.elim x.rewrite > Ztimes_z_OZ.reflexivity.
@@ -67,8 +67,23 @@ qed.
 variant sym_Ztimes : \forall x,y:Z. x*y = y*x
 \def symmetric_Ztimes.
 
+theorem Ztimes_Zone_l: \forall z:Z. Ztimes Zone z = z.
+intro.unfold Zone.simplify.
+elim z;simplify
+  [reflexivity
+  |rewrite < plus_n_O.reflexivity
+  |rewrite < plus_n_O.reflexivity
+  ]
+qed.
+
+theorem Ztimes_Zone_r: \forall z:Z. Ztimes z Zone = z.
+intro.
+rewrite < sym_Ztimes.
+apply Ztimes_Zone_l.
+qed.
+
 theorem associative_Ztimes: associative Z Ztimes.
-change with (\forall x,y,z:Z. (x*y)*z = x*(y*z)).
+unfold associative.
 intros.elim x.
   simplify.reflexivity. 
   elim y.