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[helm.git] / matita / contribs / library_auto / auto / nat / sigma_and_pi.ma

diff --git a/matita/contribs/library_auto/auto/nat/sigma_and_pi.ma b/matita/contribs/library_auto/auto/nat/sigma_and_pi.ma
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+(**************************************************************************)
+(*       ___                                                               *)
+(*      ||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.