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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 "delayed_updating/syntax/path_labels.ma".
include "delayed_updating/notation/functions/downarrowrightsharp_2.ma".
include "ground/arith/pnat_plus.ma".

(* HEAD BY DEPTH FOR PATH ***************************************************)

rec definition path_dhd (m) (p) on p: path ≝
match p with
[ list_empty     ⇒ 𝗟∗∗m
| list_lcons l q ⇒
  match l with
  [ label_d n ⇒ l◗(path_dhd (m+n) q)
  | label_m   ⇒ l◗(path_dhd m q)
  | label_L   ⇒
    match m with
    [ punit   ⇒ l◗𝐞
    | psucc o ⇒ l◗(path_dhd o q)
    ]
  | label_A   ⇒ l◗(path_dhd m q)
  | label_S   ⇒ l◗(path_dhd m q)
  ]
].

interpretation
  "head by depth (reversed path)"
  'DownArrowRightSharp n p = (path_dhd n p).

(* basic constructions ****************************************************)

lemma path_dhd_empty (n:pnat):
      (𝗟∗∗n) = ↳⧣[n]𝐞.
// qed.

lemma path_dhd_d_sn (p) (n) (m):
      (𝗱m◗↳⧣[n+m]p) = ↳⧣[n](𝗱m◗p).
// qed.

lemma path_dhd_m_sn (p) (n):
      (𝗺◗↳⧣[n]p) = ↳⧣[n](𝗺◗p).
// qed.

lemma path_dhd_L_sn_unit (p):
      (𝗟◗𝐞) = ↳⧣[𝟏](𝗟◗p).
// qed.

lemma path_dhd_L_sn_succ (p) (n):
      (𝗟◗↳⧣[n]p) = ↳⧣[↑n](𝗟◗p).
// qed.

lemma path_dhd_A_sn (p) (n):
      (𝗔◗↳⧣[n]p) = ↳⧣[n](𝗔◗p).
// qed.

lemma path_dhd_S_sn (p) (n):
      (𝗦◗↳⧣[n]p) = ↳⧣[n](𝗦◗p).
// qed.