helm/matita/library/nat/sigma_and_pi.ma

   1 (**************************************************************************)
   2 (*       ___                                                                *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
   8 (*      ||A||       E.Tassi, S.Zacchiroli                                 *)
   9 (*      \   /                                                             *)
  10 (*       \ /        Matita is distributed under the terms of the          *)
  11 (*        v         GNU Lesser General Public License Version 2.1         *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/nat/sigma_and_pi".
  16
  17 include "nat/factorial.ma".
  18 include "nat/lt_arith.ma".
  19
  20 let rec sigma n f \def
  21   match n with
  22   [ O \Rightarrow O
  23   | (S p) \Rightarrow (f p)+(sigma p f)].
  24
  25 let rec pi n f \def
  26   match n with
  27   [ O \Rightarrow (S O)
  28   | (S p) \Rightarrow (f p)*(pi p f)].
  29
  30 theorem eq_sigma: \forall f,g:nat \to nat.
  31 \forall n:nat. (\forall m:nat. m < n \to f m = g m) \to
  32 (sigma n f) = (sigma n g).
  33 intros 3.elim n.
  34 simplify.reflexivity.
  35 simplify.
  36 apply eq_f2.apply H1.simplify. apply le_n.
  37 apply H.intros.apply H1.
  38 apply trans_lt ? n1.assumption.simplify.apply le_n.
  39 qed.
  40
  41 theorem eq_pi: \forall f,g:nat \to nat.
  42 \forall n:nat. (\forall m:nat. m < n \to f m = g m) \to
  43 (pi n f) = (pi n g).
  44 intros 3.elim n.
  45 simplify.reflexivity.
  46 simplify.
  47 apply eq_f2.apply H1.simplify. apply le_n.
  48 apply H.intros.apply H1.
  49 apply trans_lt ? n1.assumption.simplify.apply le_n.
  50 qed.
  51
  52 theorem eq_fact_pi: \forall n. n! = pi n S.
  53 intro.elim n.
  54 simplify.reflexivity.
  55 change with (S n1)*n1! = (S n1)*(pi n1 S).
  56 apply eq_f.assumption.
  57 qed.