- libraryObjects: new default "natural numbers" with the uri of nat.

[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..f59622218e84c38535f862a7f8917fd7a5b4d87d 100644 (file)
--- a/helm/software/matita/library/nat/times.ma
+++ b/helm/software/matita/library/nat/times.ma
@@ -1,3 +1,4 @@
+
 (**************************************************************************)
 (*       __                                                               *)
 (*      ||M||                                                             *)
@@ -12,8 +13,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/times".
-
 include "nat/plus.ma".
 
 let rec times n m \def 
@@ -21,8 +20,12 @@ 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.
@@ -34,13 +37,17 @@ theorem times_n_Sm :
 \forall n,m:nat. n+(n*m) = n*(S m).
 intros.elim n.
 simplify.reflexivity.
-simplify.apply eq_f.rewrite < H.
+simplify.
+demodulate all.
+(*
+apply eq_f.rewrite < H.
 transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
 transitivity ((m+n1)+n1*m).
 apply eq_f2.
 apply sym_plus.
 reflexivity.
 apply assoc_plus.
+*)
 qed.
 
 theorem times_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
@@ -54,11 +61,11 @@ apply nat_elim2;intros
 qed.
 
 theorem times_n_SO : \forall n:nat. n = n * S O.
-intros.
+intros. demodulate. reflexivity. (*
 rewrite < times_n_Sm.
 rewrite < times_n_O.
 rewrite < plus_n_O.
-reflexivity.
+reflexivity.*)
 qed.
 
 theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
@@ -68,11 +75,34 @@ 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.
-simplify.rewrite > H.apply times_n_Sm.
+intros.elim x;
+ [ simplify. apply times_n_O.
+ | demodulate. reflexivity. (*
+(* applyS times_n_Sm. *) 
+simplify.rewrite > H.apply times_n_Sm.*)]
 qed.
 
 variant sym_times : \forall n,m:nat. n*m = m*n \def
@@ -82,23 +112,38 @@ theorem distributive_times_plus : distributive nat times plus.
 unfold distributive.
 intros.elim x.
 simplify.reflexivity.
-simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
+simplify.
+demodulate all.
+(*
+rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
 apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
-rewrite > assoc_plus.reflexivity.
+rewrite > assoc_plus.reflexivity. *)
 qed.
 
 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.
-simplify.rewrite < sym_times.
+unfold associative.
+intros.
+elim x. simplify.apply refl_eq. 
+simplify.
+demodulate all.
+(*
+rewrite < sym_times.
 rewrite > distr_times_plus.
 rewrite < sym_times.
 rewrite < (sym_times (times n y) z).
 rewrite < H.apply refl_eq.
+*)
 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. 
+demodulate. reflexivity.
+(* autobatch paramodulation. *)
+qed.
+