(* Forward lemmas on lenv refinement for atomic arity assignment ************)
lemma lsubsv_fwd_lsuba: ∀h,g,G,L1,L2. G ⊢ L1 ⫃¡[h, g] L2 → G ⊢ L1 ⫃⁝ L2.
-#h #g #G #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsuba_pair/
#L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #_ #_ #IHL12
lapply (snv_scast … HV H1W HVW H1l) -HV -H1W -HVW -H1l #HV
elim (snv_fwd_aaa … HV) -HV #A #HV
elim (snv_fwd_aaa … H2W) -H2W #B #HW
elim (aaa_inv_cast … HV) #HWA #_
lapply (lsuba_aaa_trans … HW … IHL12) #HWB
-lapply (aaa_mono … HWB … HWA) -HWB -HWA #H destruct /2 width=3/
+lapply (aaa_mono … HWB … HWA) -HWB -HWA #H destruct /2 width=3 by lsuba_beta/
qed-.