+
(**************************************************************************)
(* __ *)
(* ||M|| *)
(* *)
(**************************************************************************)
-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).
+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.
\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.
+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.
+intros. demodulate. reflexivity. (*
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.
+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
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.
+