(**************************************************************************)
-(* ___ *)
+(* __ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/times.ma".
+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 (plus m (times p m)) ].
+ | (S p) \Rightarrow m+(times p m) ].
-theorem times_n_O: \forall n:nat. eq nat O (times n O).
+(*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.
intros.elim n.
simplify.reflexivity.
simplify.assumption.
qed.
theorem times_n_Sm :
-\forall n,m:nat. eq nat (plus n (times n m)) (times n (S m)).
+\forall n,m:nat. n+(n*m) = n*(S m).
intros.elim n.
simplify.reflexivity.
simplify.apply eq_f.rewrite < H.
-transitivity (plus (plus e1 m) (times e1 m)).symmetry.apply assoc_plus.
-transitivity (plus (plus m e1) (times e1 m)).
+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.
-(* same problem with symmetric: see plus
-theorem symmetric_times : symmetric nat times. *)
+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 sym_times :
-\forall n,m:nat. eq nat (times n m) (times m n).
-intros.elim n.
+theorem symmetric_times : symmetric nat times.
+unfold symmetric.
+intros.elim x.
simplify.apply times_n_O.
simplify.rewrite > H.apply times_n_Sm.
qed.
-theorem times_plus_distr: \forall n,m,p:nat.
-eq nat (times n (plus m p)) (plus (times n m) (times n p)).
-intros.elim n.
+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 ? p.
+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.