matita/matita/contribs/lambdadelta/ground/arith/nat_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/arith/pnat_iter.ma".
  16 include "ground/arith/nat.ma".
  17
  18 (* NON-NEGATIVE INTEGERS ****************************************************)
  19
  20 definition niter (n:nat) (A:Type[0]) (f:A→A) (a:A) ≝
  21 match n with
  22 [ nzero  ⇒ a
  23 | ninj p ⇒ f^{A}p a
  24 ]
  25 .
  26
  27 interpretation
  28   "iterated function (non-negative integers)"
  29   'Exp A f n = (niter n A f).
  30
  31 (* Basic rewrites ***********************************************************)
  32
  33 lemma niter_zero (A) (f) (a): a = (f^{A}𝟎) a.
  34 // qed.
  35
  36 lemma niter_inj (A) (f) (p) (a): f^p a = f^{A}(ninj p) a.
  37 // qed.
  38
  39 (* Advanced rewrites ********************************************************)
  40
  41 lemma niter_appl (A) (f) (n) (a): f (f^n a) = f^{A}n (f a).
  42 #A #f * //
  43 #p #a <niter_inj <niter_inj <piter_appl //
  44 qed.