+/5 width=1 by fsb_intro, fpbc_fpbg, fpbu_fpbc/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fsb_fpbs_trans: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h, g] T1 →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⦥[h, g] T2.
+/4 width=5 by fsba_inv_fsb, fsb_fsba, fsba_fpbs_trans/ qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fsb_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
+ (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h, g] T1 →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ >⋕[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) →
+ ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⦥[h, g] T1 → R G1 L1 T1.
+#h #g #R #IH #G1 #L1 #T1 #H @(fsba_ind_alt h g … G1 L1 T1)
+/3 width=1 by fsba_inv_fsb, fsb_fsba/