(* Main properties **********************************************************)
-theorem lsuba_trans: ∀L1,L. L1 ⁝⊑ L → ∀L2. L ⁝⊑ L2 → L1 ⁝⊑ L2.
-#L1 #L #H elim H -L1 -L
+theorem lsuba_trans: ∀G,L1,L. G ⊢ L1 ⫃⁝ L → ∀L2. G ⊢ L ⫃⁝ L2 → G ⊢ L1 ⫃⁝ L2.
+#G #L1 #L #H elim H -L1 -L
[ #X #H >(lsuba_inv_atom1 … H) -H //
| #I #L1 #L #Y #HL1 #IHL1 #X #H
elim (lsuba_inv_pair1 … H) -H * #L2
[ #HL2 #H destruct /3 width=1/
| #W #V #A #HV #HW #HL2 #H1 #H2 #H3 destruct
- /3 width=3 by lsuba_abbr, lsuba_aaa_trans/
+ /3 width=3 by lsuba_beta, lsuba_aaa_trans/
]
| #L1 #L #W #V #A #HV #HW #HL1 #IHL1 #X #H
elim (lsuba_inv_pair1 … H) -H * #L2
- [ #HL2 #H destruct /3 width=5 by lsuba_abbr, lsuba_aaa_conf/
+ [ #HL2 #H destruct /3 width=5 by lsuba_beta, lsuba_aaa_conf/
| #W0 #V0 #A0 #_ #_ #_ #H destruct
]
]