+(**************************************************************************)
+(* ___ *)
+(* ||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/library_autobatch/nat/sigma_and_pi".
+
+include "auto/nat/factorial.ma".
+include "auto/nat/exp.ma".
+include "auto/nat/lt_arith.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.
+ autobatch
+ (*apply H
+ [ apply le_n
+ | rewrite < plus_n_O.
+ apply le_n
+ ]*)
+| simplify.
+ apply eq_f2
+ [ apply H1
+ [ autobatch
+ (*change with (m \le (S n1)+m).
+ apply le_plus_n*)
+ | autobatch
+ (*rewrite > (sym_plus m).
+ apply le_n*)
+ ]
+ | apply H.
+ intros.
+ apply H1
+ [ assumption
+ | autobatch
+ (*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.
+ autobatch
+ (*apply H
+ [ apply le_n
+ | rewrite < plus_n_O.
+ apply le_n
+ ] *)
+| simplify.
+ apply eq_f2
+ [ apply H1
+ [ autobatch
+ (*change with (m \le (S n1)+m).
+ apply le_plus_n*)
+ | autobatch
+ (*rewrite > (sym_plus m).
+ apply le_n*)
+ ]
+ | apply H.
+ intros.
+ apply H1
+ [ assumption
+ | autobatch
+ (*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
+[ autobatch
+ (*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.
+ autobatch
+ (*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
+[ autobatch
+ (*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.
+ autobatch
+ (*apply eq_f2
+ [ apply sym_times
+ | reflexivity
+ ]*)
+]
+qed.