(**************************************************************************)
(*       ___                                                              *)
(*      ||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/unwind/unwind2_rmap_closed.ma".
include "delayed_updating/syntax/path_closed_height.ma".

(* TAILED UNWIND FOR RELOCATION MAP *****************************************)

(* Destructions with cpp and height *****************************************)

lemma nap_depth_unwind2_rmap_closed (o) (f) (q) (n):
      q ϵ 𝐂❨o,n❩ →
      f＠§❨♯q❩+♭q = ▶[f]q＠§❨♭q❩.
#o #f #q #n #Hq
<(path_closed_des_depth … Hq) in ⊢ (???%);
/2 width=2 by nap_plus_unwind2_rmap_closed_depth/
qed-.

lemma tls_depth_unwind2_rmap_closed (o) (f) (q) (n):
      q ϵ 𝐂❨o,n❩ →
      ⇂*[♯q]f ≗ ⇂*[♭q]▶[f]q.
#o #f #q #n #Hq elim Hq -q -n //
#q #n #k #_
>(npsucc_pred k) in ⊢ (%→?); <nplus_succ_dx #Hq #IH
<height_d_dx <depth_d_dx
>(path_closed_des_succ_depth … Hq)
@(stream_eq_trans … (tls_unwind2_rmap_d_dx …))
>nrplus_inj_dx >nrplus_inj_sn >nsucc_unfold
<(path_closed_des_succ_depth … Hq) -Hq
<nplus_comm <stream_tls_plus
<nplus_comm <stream_tls_plus
/2 width=1 by stream_tls_eq_repl/
qed-.