--- /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/multiple/fleq.ma".
+include "basic_2/reduction/fpb_lleq.ma".
+
+(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
+
+(* Properties on lazy equivalence for closures ******************************)
+
+lemma fleq_fpb_trans: ∀h,o,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≡[0] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≡[0] ⦃G2, L2, U2⦄.
+#h #o #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
+#K2 #HK12 #G2 #L2 #U2 #H12 elim (lleq_fpb_trans … HK12 … H12) -K2
+/3 width=5 by fleq_intro, ex2_3_intro/
+qed-.
+
+(* Inversion lemmas on lazy equivalence for closures ************************)
+
+lemma fpb_inv_fleq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #H elim (fleq_inv_gen … H) -H
+ /3 width=8 by lleq_fwd_length, fqu_inv_eq/
+| #T2 #_ #HnT #H elim (fleq_inv_gen … H) -H /2 width=1 by/
+| #L2 #_ #HnL #H elim (fleq_inv_gen … H) -H /2 width=1 by/
+]
+qed-.
+