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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "ground/notation/functions/exp_3.ma".
16 include "ground/arith/pnat.ma".
18 (* POSITIVE INTEGERS ********************************************************)
20 rec definition piter (p:pnat) (A:Type[0]) (f:A→A) (a:A) ≝
23 | psucc q ⇒ f (piter q A f a)
27 "iterated function (positive integers)"
28 'Exp A f p = (piter p A f).
30 (* Basic rewrites ***********************************************************)
32 lemma piter_unit (A) (f) (a): f a = (f^{A}𝟏) a.
35 lemma piter_succ (A) (f) (p) (a): f (f^p a) = f^{A}(↑p) a.
38 (* Advanced rewrites ********************************************************)
40 lemma piter_appl (A) (f) (p) (a): f (f^p a) = f^{A}p (f a).
42 #p #IH #a <piter_succ <piter_succ //