-lemma csx_lpxs_conf: ∀h,g,G,L1. ∀T:term. ⦃G, L1⦄ ⊢ ⬊*[h, g] T →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L1 #T #HT #L2 #H @(lpxs_ind … H) -L2 /2 width=3 by csx_lpx_conf/
+lemma csx_fpb_conf: ∀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.
+#h #g #G1 #L1 #T1 #HT1 #G2 #L2 #T2 *
+/2 width=5 by csx_cpx_trans, csx_fquq_conf, csx_llpx_conf/
+qed-.
+
+lemma csx_fpbs_conf: ∀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.
+#h #g #G1 #L1 #T1 #HT1 #G2 #L2 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+/2 width=5 by csx_fpb_conf/