-(* Note: this is used in the closure proof *)
-lemma fqup_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
-/3 width=5 by fqus_fpbs, fpb_fqu, ex2_3_intro/
-qed.
-
-lemma cpxs_fpbg: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 →
- (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, o] ⦃G, L, T2⦄.
-#h #o #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
-/4 width=5 by cpxs_fpbs, fpb_cpx, ex2_3_intro/
-qed.
-
-lemma lstas_fpbg: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → (T1 = T2 → ⊥) →
- ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → ⦃G, L, T1⦄ >≡[h, o] ⦃G, L, T2⦄.
-/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
-
-lemma lpxs_fpbg: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
- (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, o] ⦃G, L2, T⦄.
-#h #o #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
-/4 width=5 by fpb_lpx, lpxs_lleq_fpbs, ex2_3_intro/
-qed.
+(* Advanced properties of parallel rst-computation on closures **************)
+
+lemma fpbs_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h] ⦃G2, L2, U2⦄.
+#h #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_inv_fpbg … H) -H
+[ #H12 #G2 #L2 #U2 #H2 elim (fdeq_fpb_trans … H12 … H2) -F2 -K2 -T2
+ /3 width=5 by fdeq_fpbs, ex2_3_intro/
+| * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
+ @(ex2_3_intro … H4) -H4 /3 width=5 by fpbs_strap1, fpb_fpbq/
+]
+qed-.