+(**************************************************************************)
+(* ___ *)
+(* ||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/btpredstarrestricted_8.ma".
+include "basic_2/computation/fpbg.ma".
+
+(* RESTRICTED "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES ***********)
+
+inductive fpbr (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpbr_inj : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → fpbr h g G1 L1 T1 G2 L2 T2
+| fpbr_step: ∀G,G2,L,L2,T,T2. fpbr h g G1 L1 T1 G L T → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ fpbr h g G1 L1 T1 G2 L2 T2
+.
+
+interpretation "restricted 'big tree' proper parallel computation (closure)"
+ 'BTPRedStarRestricted h g G1 L1 T1 G2 L2 T2 = (fpbr h g G1 L1 T1 G2 L2 T2).
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fpbr_fwd_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
+/3 width=5 by fpbg_strap1, fpbc_fpbg, fpbc_fqu/
+qed-.
+
+lemma fpbr_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
+/3 width=5 by fpbs_strap1, fqup_fpbs, fqu_fqup/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fqu_fpbs_fpbr: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fpbs_ind … H) -G2 -L2 -T2
+/2 width=5 by fpbr_inj, fpbr_step/
+qed.
+
+lemma fpbr_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fqu_fpbs_fpbr, fpbr_fwd_fpbs/ qed-.
+
+(* Note: this is used in the closure proof *)
+lemma fpbr_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(fpbs_ind … HT2) -G2 -L2 -T2
+/2 width=5 by fpbr_step/
+qed-.
+
+lemma fqup_fpbr_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #HT1 @(fqup_ind … HT1) -G -L -T
+/3 width=5 by fpbr_strap2/
+qed-.
+
+(* Note: this is used in the closure proof *)
+lemma fqup_fpbr: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
+/3 width=5 by fqu_fpbs_fpbr, fqus_fpbs/
+qed.