+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / Matita is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/sigma_and_pi".
+
+include "nat/factorial.ma".
+include "nat/lt_arith.ma".
+include "nat/exp.ma".
+
+let rec sigma n f m \def
+ match n with
+ [ O \Rightarrow (f m)
+ | (S p) \Rightarrow (f (S p+m))+(sigma p f m)].
+
+let rec pi n f m \def
+ match n with
+ [ O \Rightarrow f m
+ | (S p) \Rightarrow (f (S p+m))*(pi p f m)].
+
+theorem eq_sigma: \forall f,g:nat \to nat.
+\forall n,m:nat.
+(\forall i:nat. m \le i \to i \le m+n \to f i = g i) \to
+(sigma n f m) = (sigma n g m).
+intros 3.elim n.
+simplify.apply H.apply le_n.rewrite < plus_n_O.apply le_n.
+simplify.
+apply eq_f2.apply H1.
+change with (m \le (S n1)+m).apply le_plus_n.
+rewrite > (sym_plus m).apply le_n.
+apply H.intros.apply H1.assumption.
+rewrite < plus_n_Sm.
+apply le_S.assumption.
+qed.
+
+theorem eq_pi: \forall f,g:nat \to nat.
+\forall n,m:nat.
+(\forall i:nat. m \le i \to i \le m+n \to f i = g i) \to
+(pi n f m) = (pi n g m).
+intros 3.elim n.
+simplify.apply H.apply le_n.rewrite < plus_n_O.apply le_n.
+simplify.
+apply eq_f2.apply H1.
+change with (m \le (S n1)+m).apply le_plus_n.
+rewrite > (sym_plus m).apply le_n.
+apply H.intros.apply H1.assumption.
+rewrite < plus_n_Sm.
+apply le_S.assumption.
+qed.
+
+theorem eq_fact_pi: \forall n. (S n)! = pi n (\lambda m.m) (S O).
+intro.elim n.
+simplify.reflexivity.
+change with ((S(S n1))*(S n1)! = ((S n1)+(S O))*(pi n1 (\lambda m.m) (S O))).
+rewrite < plus_n_Sm.rewrite < plus_n_O.
+apply eq_f.assumption.
+qed.
+
+theorem exp_pi_l: \forall f:nat\to nat.\forall n,m,a:nat.
+(exp a (S n))*pi n f m= pi n (\lambda p.a*(f p)) m.
+intros.elim n.simplify.rewrite < times_n_SO.reflexivity.
+simplify.
+rewrite < H.
+rewrite > assoc_times.
+rewrite > assoc_times in\vdash (? ? ? %).
+apply eq_f.rewrite < assoc_times.
+rewrite < assoc_times.
+apply eq_f2.apply sym_times.reflexivity.
+qed.