lemma lsx_lreq_conf: ∀h,o,G,L1,T,l. G ⊢ ⬊*[h, o, T, l] L1 →
                    ∀L2. L1 ⩬[l, ∞] L2 → G ⊢ ⬊*[h, o, T, l] L2.
#h #o #G #L1 #T #l #H @(lsx_ind … H) -L1
#L1 #_ #IHL1 #L2 #HL12 @lsx_intro
#L3 #HL23 #HnL23 elim (lreq_lpx_trans_lleq … HL12 … HL23) -HL12 -HL23
#L0 #HL03 #HL10 #H elim (H T) -H /4 width=4 by/
qed-.

(* this is better expressed with the binder \chi *)

lemma lsx_fwd_bind_dx: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬈*[h, o, ⓑ{a,I}V.T, l] L →
                       G ⊢ ⬈*[h, o, T, ⫯l] L.ⓑ{I}V.
#h #o #a #I #G #L #V1 #T #l #H @(lsx_ind … H) -L
#L1 #_ #IHL1 @lsx_intro
#Y #H #HT elim (lpx_inv_pair1 … H) -H
#L2 #V2 #HL12 #_ #H destruct
@(lsx_lreq_conf … (L2.ⓑ{I}V1)) /2 width=1 by lreq_succ/
@IHL1 // #H @HT -IHL1 -HL12 -HT
@(lleq_lreq_trans … (L2.ⓑ{I}V1))
/2 width=2 by lleq_fwd_bind_dx, lreq_succ/
qed-.