matita/matita/contribs/lambdadelta/ground/arith/pnat_iter.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground/notation/functions/exp_3.ma".
  16 include "ground/arith/pnat.ma".
  17
  18 (* POSITIVE INTEGERS ********************************************************)
  19
  20 rec definition piter (p:pnat) (A:Type[0]) (f:A→A) (a:A) ≝
  21 match p with
  22 [ punit   ⇒ f a
  23 | psucc q ⇒ f (piter q A f a)
  24 ].
  25
  26 interpretation
  27   "iterated function (positive integers)"
  28   'Exp A f p = (piter p A f).
  29
  30 (* Basic rewrites ***********************************************************)
  31
  32 lemma piter_unit (A) (f) (a): f a = (f^{A}𝟏) a.
  33 // qed.
  34
  35 lemma piter_succ (A) (f) (p) (a): f (f^p a) = f^{A}(↑p) a.
  36 // qed.
  37
  38 (* Advanced rewrites ********************************************************)
  39
  40 lemma piter_appl (A) (f) (p) (a): f (f^p a) = f^{A}p (f a).
  41 #A #f #p elim p -p //
  42 #p #IH #a <piter_succ <piter_succ //
  43 qed.