arithmetics for λδ

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / pnat_iter.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/pnat_iter.ma b/matita/matita/contribs/lambdadelta/ground/arith/pnat_iter.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground/notation/functions/exp_3.ma".
+include "ground/arith/pnat.ma".
+
+(* POSITIVE INTEGERS ********************************************************)
+
+rec definition piter (p:pnat) (A:Type[0]) (f:A→A) (a:A) ≝
+match p with
+[ punit   ⇒ f a
+| psucc q ⇒ f (piter q A f a)
+].
+
+interpretation
+  "iterated function (positive integers)"
+  'Exp A f p = (piter p A f).
+
+(* Basic rewrites ***********************************************************)
+
+lemma piter_unit (A) (f) (a): f a = (f^{A}𝟏) a.
+// qed.
+
+lemma piter_succ (A) (f) (p) (a): f (f^p a) = f^{A}(↑p) a.
+// qed.
+
+(* Advanced rewrites ********************************************************)
+
+lemma piter_appl (A) (f) (p) (a): f (f^p a) = f^{A}p (f a).
+#A #f #p elim p -p //
+#p #IH #a <piter_succ <piter_succ //
+qed.