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/lt_arith.ma".
  18
  19 let rec sigma n f \def
  20   match n with
  21   [ O \Rightarrow O
  22   | (S p) \Rightarrow (f p)+(sigma p f)].
  23
  24 let rec pi n f \def
  25   match n with
  26   [ O \Rightarrow (S O)
  27   | (S p) \Rightarrow (f p)*(pi p f)].
  28
  29 theorem eq_sigma: \forall f,g:nat \to nat.
  30 \forall n:nat. (\forall m:nat. m < n \to f m = g m) \to
  31 (sigma n f) = (sigma n g).
  32 intros 3.elim n.
  33 simplify.reflexivity.
  34 simplify.
  35 apply eq_f2.apply H1.simplify. apply le_n.
  36 apply H.intros.apply H1.
  37 apply trans_lt ? n1.assumption.simplify.apply le_n.
  38 qed.
  39
  40 theorem eq_pi: \forall f,g:nat \to nat.
  41 \forall n:nat. (\forall m:nat. m < n \to f m = g m) \to
  42 (pi n f) = (pi n g).
  43 intros 3.elim n.
  44 simplify.reflexivity.
  45 simplify.
  46 apply eq_f2.apply H1.simplify. apply le_n.
  47 apply H.intros.apply H1.
  48 apply trans_lt ? n1.assumption.simplify.apply le_n.
  49 qed.