+
+(* Advanced properties ******************************************************)
+
+lemma fpbs_intro_alt: ∀h,g,G1,G2,L1,L0,L,L2,T1,T,T2.
+ ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ →
+ ⦃G2, L0⦄ ⊢ ➡*[h, g] L → L ≡[T2, 0] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ .
+/3 width=7 by fpbsa_inv_fpbs, ex4_3_intro/ qed.
+
+(* Advanced inversion lemmas *************************************************)
+
+lemma fpbs_inv_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T &
+ ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ &
+ ⦃G2, L0⦄ ⊢ ➡*[h, g] L & L ≡[T2, 0] L2.
+/2 width=1 by fpbs_fpbsa/ qed-.