--- /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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/library_autobatch/nat/times".
+
+include "auto/nat/plus.ma".
+
+let rec times n m \def
+ match n with
+ [ 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/library_autobatch/nat/times/times.con x y).
+
+theorem times_n_O: \forall n:nat. O = n*O.
+intros.elim n
+[ autobatch
+ (*simplify.
+ reflexivity.*)
+| simplify. (* qui autobatch non funziona: Uncaught exception: Invalid_argument ("List.map2")*)
+ assumption.
+]
+qed.
+
+theorem times_n_Sm :
+\forall n,m:nat. n+(n*m) = n*(S m).
+intros.elim n
+[ autobatch.
+ (*simplify.reflexivity.*)
+| simplify.
+ autobatch
+ (*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.
+
+(* NOTA:
+ se non avessi semplificato con autobatch tutto il secondo ramo della tattica
+ elim n, avrei comunque potuto risolvere direttamente con autobatch entrambi
+ i rami generati dalla prima applicazione della tattica transitivity
+ (precisamente transitivity ((n1+m)+n1*m)
+ *)
+
+theorem times_n_SO : \forall n:nat. n = n * S O.
+intros.
+rewrite < times_n_Sm.
+autobatch paramodulation. (*termina la dim anche solo con autobatch*)
+(*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.
+autobatch paramodulation. (* termina la dim anche solo con autobatch*)
+(*rewrite < plus_n_O.
+reflexivity.*)
+qed.
+
+theorem symmetric_times : symmetric nat times.
+unfold symmetric.
+intros.elim x
+[ autobatch
+ (*simplify.apply times_n_O.*)
+| simplify.
+ autobatch
+ (*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.
+| autobatch
+ (*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
+| autobatch
+ (*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.