(**************************************************************************)
-(* ___ *)
+(* __ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
set "baseuri" "cic:/matita/nat/times".
-include "logic/equality.ma".
-include "nat/nat.ma".
include "nat/plus.ma".
let rec times n m \def
match n with
[ O \Rightarrow O
- | (S p) \Rightarrow (m+(times p m)) ].
+ | (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.
qed.
theorem times_n_Sm :
-\forall n,m:nat.n+n*m = n*(S m).
+\forall n,m:nat. n+(n*m) = n*(S m).
intros.elim n.
simplify.reflexivity.
simplify.apply eq_f.rewrite < H.
apply assoc_plus.
qed.
-(* same problem with symmetric: see plus
-theorem symmetric_times : symmetric nat times. *)
+theorem times_n_SO : \forall n:nat. eq nat n (times n (S O)).
+intros.
+rewrite < times_n_Sm.
+rewrite < times_n_O.
+rewrite < plus_n_O.
+reflexivity.
+qed.
-theorem sym_times :
-\forall n,m:nat.n*m = m*n.
-intros.elim n.
+theorem symmetric_times : symmetric nat times.
+simplify.
+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.
simplify.
intros.elim x.
rewrite > assoc_plus.reflexivity.
qed.
-variant times_plus_distr: \forall n,m,p:nat. n*(m+p)=n*m+n*p
+variant times_plus_distr: \forall n,m,p:nat. n*(m+p) = n*m + n*p
\def distributive_times_plus.
+theorem associative_times: associative nat times.
+simplify.intros.
+elim x.simplify.apply refl_eq.
+simplify.rewrite < sym_times.
+rewrite > times_plus_distr.
+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.