(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/times".
-
include "nat/plus.ma".
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).
theorem times_n_O: \forall n:nat. O = n*O.
apply assoc_plus.
qed.
+theorem times_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
+apply nat_elim2;intros
+ [left.reflexivity
+ |right.reflexivity
+ |apply False_ind.
+ simplify in H1.
+ apply (not_eq_O_S ? (sym_eq ? ? ? H1))
+ ]
+qed.
+
theorem times_n_SO : \forall n:nat. n = n * S O.
intros.
rewrite < times_n_Sm.
reflexivity.
qed.
+theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
+intros.
+simplify.
+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.