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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/substitution/fsubst.ma".
include "delayed_updating/syntax/prototerm_constructors.ma".
include "delayed_updating/syntax/prototerm_eq.ma".
include "delayed_updating/syntax/path_closed.ma".
include "delayed_updating/syntax/path_balanced.ma".
include "delayed_updating/syntax/path_structure.ma".
include "delayed_updating/notation/relations/black_rightarrow_dbf_3.ma".
include "ground/arith/nat_rplus.ma".
include "ground/xoa/ex_6_5.ma".

(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)

definition dbfr (r): relation2 prototerm prototerm ≝
           λt1,t2.
           ∃∃p,b,q,m,n. p●𝗔◗b●𝗟◗q = r &
           ⊗b ϵ 𝐁 & b ϵ 𝐂❨m❩ & q ϵ 𝐂❨n❩ & r◖𝗱↑n ϵ t1 &
           t1[⋔r←𝛕↑(m+n).(t1⋔(p◖𝗦))] ⇔ t2
.

interpretation
  "balanced focused reduction with delayed updating (prototerm)"
  'BlackRightArrowDBF t1 r t2 = (dbfr r t1 t2).