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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                  *)
(*                                                                        *)
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include "ground/arith/pnat_plus.ma".
include "ground/relocation/tr_map.ma".
include "delayed_updating/notation/functions/verticalbarat_2.ma".

(* POSITIVE APPLICATION FOR STACKS OF BASIC RELOCATION MAPS *****************)

rec definition sbr_pap (i: pnat) on i: tr_map → pnat.
* #p #f cases i -i
[ @(p)
| #i cases p -p
  [ lapply (sbr_pap i f) -sbr_pap -i -f
    #i @(↑i)
  | #p @(i+↑p)
  ]
]
defined.

interpretation
  "functional positive application (stack of relocation maps)"
  'VerticalBarAt f i = (sbr_pap i f).

(* Basic constructions ******************************************************)

lemma sbr_pap_unfold_unit (f):
      ∀p. p = (p⨮f)❘@❨𝟏❩.
// qed.

lemma sbr_pap_unfold_unit_succ (f):
      ∀i. ↑(f❘@❨i❩) = (𝟏⨮f)❘@❨↑i❩.
// qed.

lemma sbr_pap_unfold_succ_bi (f):
      ∀p,i. i+↑p = (↑p⨮f)❘@❨↑i❩.
// qed.