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

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  14
  15 include "ground/notation/functions/exp_3.ma".
  16 include "ground/arith/pnat.ma".
  17
  18 (* ITERATED FUNCTION FOR POSITIVE INTEGERS **********************************)
  19
  20 (* Note: see also: lib/arithemetics/bigops.ma *)
  21 rec definition piter (p:pnat) (A:Type[0]) (f:A→A) (a:A) ≝
  22 match p with
  23 [ punit   ⇒ f a
  24 | psucc q ⇒ f (piter q A f a)
  25 ].
  26
  27 interpretation
  28   "iterated function (positive integers)"
  29   'Exp A f p = (piter p A f).
  30
  31 (* Basic constructions ******************************************************)
  32
  33 lemma piter_unit (A) (f) (a): f a = (f^{A}𝟏) a.
  34 // qed.
  35
  36 lemma piter_succ (A) (f) (p) (a): f (f^p a) = f^{A}(↑p) a.
  37 // qed.
  38
  39 (* Advanced constructions ***************************************************)
  40
  41 lemma piter_appl (A) (f) (p) (a): f (f^p a) = f^{A}p (f a).
  42 #A #f #p elim p -p //
  43 #p #IH #a <piter_succ <piter_succ //
  44 qed.