+lemma fpbs_lpxs_trans:
+ ∀G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L,T2❫ →
+ ∀L2. ❪G2,L❫ ⊢ ⬈* L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+#G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lpxs_ind_dx … H) -L2
+/3 width=5 by fpbs_strap1, fpbq_lpx/
+qed-.
+
+lemma lpxs_fpbs_trans:
+ ∀G1,G2,L,L2,T1,T2. ❪G1,L,T1❫ ≥ ❪G2,L2,T2❫ →
+ ∀L1. ❪G1,L1❫ ⊢ ⬈* L → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+#G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lpxs_ind_sn … H) -L1
+/3 width=5 by fpbs_strap2, fpbq_lpx/
+qed-.
+
+(* Basic_2A1: uses: lpxs_lleq_fpbs *)
+lemma lpxs_feqx_fpbs:
+ ∀G1,L1,L,T1. ❪G1,L1❫ ⊢ ⬈* L →
+ ∀G2,L2,T2. ❪G1,L,T1❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+/3 width=3 by lpxs_fpbs_trans, feqx_fpbs/ qed.
+
+lemma fpbs_lpx_trans:
+ ∀G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ≥ ❪G2,L,T2❫ →
+ ∀L2. ❪G2,L❫ ⊢ ⬈ L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+/3 width=3 by fpbs_lpxs_trans, lpx_lpxs/ qed-.
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma fqus_lpxs_fpbs:
+ ∀G1,G2,L1,L,T1,T2. ❪G1,L1,T1❫ ⬂* ❪G2,L,T2❫ →
+ ∀L2. ❪G2,L❫ ⊢ ⬈* L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
+
+(* Properties with extended context-sensitive parallel rt-computation *******)
+
+lemma cpxs_fqus_lpxs_fpbs:
+ ∀G1,L1,T1,T. ❪G1,L1❫ ⊢ T1 ⬈* T →
+ ∀G2,L,T2. ❪G1,L1,T❫ ⬂* ❪G2,L,T2❫ →
+ ∀L2.❪G2,L❫ ⊢ ⬈* L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
+
+lemma fpbs_cpxs_teqx_fqup_lpx_trans:
+ ∀G1,G3,L1,L3,T1,T3. ❪G1,L1,T1❫ ≥ ❪G3,L3,T3❫ →
+ ∀T4. ❪G3,L3❫ ⊢ T3 ⬈* T4 → ∀T5. T4 ≛ T5 →
+ ∀G2,L4,T2. ❪G3,L3,T5❫ ⬂+ ❪G2,L4,T2❫ →
+ ∀L2. ❪G2,L4❫ ⊢ ⬈ L2 → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫.
+#G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
+@(fpbs_lpx_trans … HL42) -L2 (**) (* full auto too slow *)
+@(fpbs_fqup_trans … H34) -G2 -L4 -T2
+/3 width=3 by fpbs_cpxs_trans, fpbs_teqx_trans/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: fpbs_intro_alt *)
+lemma fpbs_intro_star:
+ ∀G1,L1,T1,T. ❪G1,L1❫ ⊢ T1 ⬈* T →
+ ∀G,L,T0. ❪G1,L1,T❫ ⬂* ❪G,L,T0❫ →
+ ∀L0. ❪G,L❫ ⊢ ⬈* L0 →
+ ∀G2,L2,T2. ❪G,L0,T0❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ .
+/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_feqx/ qed.
+
+(* Advanced inversion lemmas *************************************************)