- transcript: bugfix

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

diff --git a/helm/software/matita/library/nat/times.ma b/helm/software/matita/library/nat/times.ma
index 24a2846d7a684eb337f932d502fcc6fb9dab206f..6bbdcec4d2083e0200a3bdb7b431d9d577f00997 100644 (file)
--- a/helm/software/matita/library/nat/times.ma
+++ b/helm/software/matita/library/nat/times.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/times".
-
 include "nat/plus.ma".
 
 let rec times n m \def 
@@ -21,8 +19,12 @@ let rec times n m \def
  [ O \Rightarrow O
  | (S p) \Rightarrow m+(times p m) ].
 
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "natural times" 'times x y = (cic:/matita/nat/times/times.con x y).
+interpretation "natural times" 'times x y = (times x y).
+
+theorem times_Sn_m:
+\forall n,m:nat. m+n*m = S n*m.
+intros. reflexivity.
+qed.
 
 theorem times_n_O: \forall n:nat. O = n*O.
 intros.elim n.
@@ -68,10 +70,32 @@ rewrite < plus_n_O.
 reflexivity.
 qed.
 
+alias num (instance 0) = "natural number".
+lemma times_SSO: \forall n.2*(S n) = S(S(2*n)).
+intro.simplify.rewrite < plus_n_Sm.reflexivity.
+qed.
+
+theorem or_eq_eq_S: \forall n.\exists m. 
+n = (S(S O))*m \lor n = S ((S(S O))*m).
+intro.elim n
+  [apply (ex_intro ? ? O).
+   left.reflexivity
+  |elim H.elim H1
+    [apply (ex_intro ? ? a).
+     right.apply eq_f.assumption
+    |apply (ex_intro ? ? (S a)).
+     left.rewrite > H2.
+     apply sym_eq.
+     apply times_SSO
+    ]
+  ]
+qed.
+
 theorem symmetric_times : symmetric nat times. 
 unfold symmetric.
 intros.elim x.
 simplify.apply times_n_O.
+(* applyS times_n_Sm. *) 
 simplify.rewrite > H.apply times_n_Sm.
 qed.
 
@@ -91,8 +115,9 @@ variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
 \def distributive_times_plus.
 
 theorem associative_times: associative nat times.
-unfold associative.intros.
-elim x.simplify.apply refl_eq.
+unfold associative.
+intros.
+elim x. simplify.apply refl_eq. 
 simplify.rewrite < sym_times.
 rewrite > distr_times_plus.
 rewrite < sym_times.
@@ -102,3 +127,8 @@ qed.
 
 variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
 associative_times.
+
+lemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
+intros.autobatch paramodulation.
+qed.
+