(**************************************************************************)
(*       ___                                                              *)
(*      ||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                  *)
(*                                                                        *)
(**************************************************************************)

include "basic_2/reduction/fpbc_fleq.ma".
include "basic_2/computation/fpbg.ma".

(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)

(* Properties on lazy equivalence for closures ******************************)

lemma fpbg_fleq_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ →
                       ∀G2,L2,T2. ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G #L1 #L #T1 #T #H @(fpbg_ind … H) -G -L -T
[ /3 width=5 by fpbc_fpbg, fpbc_fleq_trans/
| /4 width=9 by fpbg_strap1, fpbc_fleq_trans/
]
qed-.

lemma fleq_fpbg_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
                       ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
#h #g #G #G2 #L #L2 #T #T2 #H @(fpbg_ind_dx … H) -G -L -T
[ /3 width=5 by fpbc_fpbg, fleq_fpbc_trans/
| /4 width=9 by fpbg_strap2, fleq_fpbc_trans/
]
qed-.