include "basic_2/static/lsuba_aaa.ma".
-(* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
+(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
(* 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. Transitive … (lsuba G).
+#G #L1 #L #H elim H -L1 -L
[ #X #H >(lsuba_inv_atom1 … H) -H //
-| #I #L1 #L #V #HL1 #IHL1 #X #H
+| #I #L1 #L #Y #HL1 #IHL1 #X #H
elim (lsuba_inv_pair1 … H) -H * #L2
- [ #HL2 #H destruct /3 width=1/
- | #V #A #HLV #HL2V #HL2 #H1 #H2 destruct /3 width=3/
+ [ #HL2 #H destruct /3 width=1 by lsuba_pair/
+ | #W #V #A #HV #HW #HL2 #H1 #H2 #H3 destruct
+ /3 width=3 by lsuba_beta, lsuba_aaa_trans/
]
-| #L1 #L #V1 #W #A1 #HV1 #HW #HL1 #IHL1 #X #H
+| #L1 #L #W #V #A #HV #HW #HL1 #IHL1 #X #H
elim (lsuba_inv_pair1 … H) -H * #L2
- [ #HL2 #H destruct /3 width=5/
- | #V #A2 #_ #_ #_ #_ #H destruct
+ [ #HL2 #H destruct /3 width=5 by lsuba_beta, lsuba_aaa_conf/
+ | #W0 #V0 #A0 #_ #_ #_ #H destruct
]
]
-qed.
+qed-.