(* Properties concerning basic local environment slicing ********************)
(* Note: the constant 0 cannot be generalized *)
-lemma lsuba_ldrop_O1_conf: ∀L1,L2. L1 ⁝⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 →
- ∃∃K2. K1 ⁝⊑ K2 & ⇩[0, e] L2 ≡ K2.
-#L1 #L2 #H elim H -L1 -L2
+lemma lsuba_ldrop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K1,e. ⇩[0, e] L1 ≡ K1 →
+ ∃∃K2. G ⊢ K1 ⁝⊑ K2 & ⇩[0, e] L2 ≡ K2.
+#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3/
| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
- elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
+ elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
<(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
| elim (IHL12 … HLK1) -L1 /3 width=3/
]
-| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K1 #e #H
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
qed-.
(* Note: the constant 0 cannot be generalized *)
-lemma lsuba_ldrop_O1_trans: ∀L1,L2. L1 ⁝⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. K1 ⁝⊑ K2 & ⇩[0, e] L1 ≡ K1.
-#L1 #L2 #H elim H -L1 -L2
+lemma lsuba_ldrop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⁝⊑ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. G ⊢ K1 ⁝⊑ K2 & ⇩[0, e] L1 ≡ K1.
+#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3/
| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
<(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
| elim (IHL12 … HLK2) -L2 /3 width=3/
]
-| #L1 #L2 #V #W #A #HV #HW #_ #IHL12 #K2 #e #H
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
elim (IHL12 L2 0) -IHL12 // #X #HL12 #H