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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/unwind1/unwind.ma".
include "delayed_updating/substitution/fsubst.ma".
include "delayed_updating/syntax/prototerm_constructors.ma".
include "delayed_updating/syntax/prototerm_equivalence.ma".
include "delayed_updating/syntax/path_structure.ma".
include "delayed_updating/syntax/path_balanced.ma".
include "delayed_updating/syntax/path_depth.ma".
include "delayed_updating/notation/relations/black_rightarrow_df_4.ma".
include "ground/xoa/ex_1_2.ma".
include "ground/xoa/and_4.ma".

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

definition dfr (p) (q): relation2 prototerm prototerm ≝
           λt1,t2. ∃∃b,n.
           let r ≝ p●𝗔◗b●𝗟◗q in
           ∧∧ (⊗b ϵ 𝐁 ∧ 𝟎 = ❘b❘) & ↑❘q❘ = (▼[r]𝐢)@❨n❩ & r◖𝗱n ϵ t1 &
              t1[⋔r←𝛗(n+❘b❘).(t1⋔(p◖𝗦))] ⇔ t2
.

interpretation
  "focused balanced reduction with delayed updating (prototerm)"
  'BlackRightArrowDF t1 p q t2 = (dfr p q t1 t2).