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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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notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
   non associative with precedence 45
   for @{ 'FocalizedPConv $L1 $T1 $L2 $T2 }.

notation "hvbox( ⦃ term 46 L1 , break term 46 T1 ⦄ ⬌ ⬌ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
   non associative with precedence 45
   for @{ 'FocalizedPConvAlt $L1 $T1 $L2 $T2 }.

include "basic_2/reducibility/fpr.ma".

(* CONTEXT-FREE PARALLEL CONVERSION ON CLOSURES *****************************)

definition fpc: bi_relation lenv term ≝
   λL1,T1,L2,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ ∨ ⦃L2, T2⦄ ➡ ⦃L1, T1⦄.

interpretation
   "context-free parallel conversion (closure)"
   'FocalizedPConv L1 T1 L2 T2 = (fpc L1 T1 L2 T2).

(* Basic properties *********************************************************)

lemma fpc_refl: bi_reflexive … fpc.
/2 width=1/ qed.

lemma fpc_sym: bi_symmetric … fpc.
#L1 #L2 #T1 #T2 * /2 width=1/
qed.

(* Basic forward lemmas *****************************************************)

lemma fpc_fwd_fpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⬌ ⦃L2, T2⦄ →
                   ∃∃L,T. ⦃L1, T1⦄ ➡ ⦃L, T⦄ & ⦃L2, T2⦄ ➡ ⦃L, T⦄.
#L1 #L2 #T1 #T2 * /2 width=4/
qed.