+
+lemma lleq_fpbs_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ L1 ≡[T1, 0] L → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_strap2, fpbq_lleq/ qed-.
+
+lemma cpxs_fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+ ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
+
+lemma cpxs_fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+ ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
+
+lemma fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
+ ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
+
+lemma cpxs_fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+ ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
+
+lemma lpxs_lleq_fpbs: ∀h,g,G,L1,L,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L →
+ L ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+/3 width=3 by lpxs_fpbs_trans, lleq_fpbs/ qed.
+
+(* Note: this is used in the closure proof *)
+lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
+/4 width=5 by fpbs_strap1, fpbq_fpbs, lpr_fpbq, cpr_fpbq/
+qed.