--- /dev/null
+(**************************************************************************)
+(* __ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+include "nat/plus.ma".
+
+let rec times n m \def
+ match n with
+ [ O \Rightarrow O
+ | (S p) \Rightarrow m+(times p m) ].
+
+interpretation "natural times" 'times x y = (cic:/matita/nat/times/times.con x y).
+
+theorem times_n_O: \forall n:nat. O = n*O.
+intros.elim n.
+simplify.reflexivity.
+simplify.assumption.
+qed.
+
+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.
+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.
+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.
+rewrite < times_n_O.
+rewrite < plus_n_O.
+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.
+simplify.apply times_n_O.
+simplify.rewrite > H.apply times_n_Sm.
+qed.
+
+variant sym_times : \forall n,m:nat. n*m = m*n \def
+symmetric_times.
+
+theorem distributive_times_plus : distributive nat times plus.
+unfold distributive.
+intros.elim x.
+simplify.reflexivity.
+simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
+apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
+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.
+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.