--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/notation/relations/lazybtpredstarproper_8.ma".
+include "basic_2/reduction/fpb.ma".
+include "basic_2/computation/fpbs.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
+ λh,o,G1,L1,T1,G2,L2,T2.
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+
+interpretation "'qrst' proper parallel computation (closure)"
+ 'LazyBTPRedStarProper h o G1 L1 T1 G2 L2 T2 = (fpbg h o G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fpb_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
+/2 width=5 by ex2_3_intro/ qed.
+
+lemma fpbg_fpbq_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ >≡[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
+/3 width=9 by fpbs_strap1, ex2_3_intro/
+qed-.