(* Properties concerning basic local environment slicing ********************)
(* Note: the constant 0 cannot be generalized *)
-lemma lsubsv_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
+lemma lsubsv_ldrop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
∀K1,e. ⇩[0, e] L1 ≡ K1 →
- ∃∃K2. h ⊢ K1 ¡⊑[g] K2 & ⇩[0, e] L2 ≡ K2.
-#h #g #L1 #L2 #H elim H -L1 -L2
+ ∃∃K2. G ⊢ K1 ¡⊑[h, g] K2 & ⇩[0, e] L2 ≡ K2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3/
| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK1
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
- elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ 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 #V1 #V2 #W1 #W2 #l #HV1 #HVW1 #HW12 #HW2 #HWV2 #_ #IHL12 #K1 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK1
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K1 #e #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
[ destruct
- elim (IHL12 L1 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=6/
+ elim (IHL12 L1 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=4/
| elim (IHL12 … HLK1) -L1 /3 width=3/
]
]
-qed.
+qed-.
(* Note: the constant 0 cannot be generalized *)
-lemma lsubsv_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
+lemma lsubsv_ldrop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. h ⊢ K1 ¡⊑[g] K2 & ⇩[0, e] L1 ≡ K1.
-#h #g #L1 #L2 #H elim H -L1 -L2
+ ∃∃K1. G ⊢ K1 ¡⊑[h, g] K2 & ⇩[0, e] L1 ≡ K1.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3/
| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK2
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
- elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=3/
+ elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=3/
| elim (IHL12 … HLK2) -L2 /3 width=3/
]
-| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HVW1 #HW12 #HW2 #HWV2 #_ #IHL12 #K2 #e #H
- elim (ldrop_inv_O1 … H) -H * #He #HLK2
+| #L1 #L2 #W #V #l #H1W #HV #HVW #H2W #H1l #H2l #_ #IHL12 #K2 #e #H
+ elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
[ destruct
- elim (IHL12 L2 0 ?) -IHL12 // #X #HL12 #H
- <(ldrop_inv_refl … H) in HL12; -H /3 width=6/
+ elim (IHL12 L2 0) -IHL12 // #X #HL12 #H
+ <(ldrop_inv_O2 … H) in HL12; -H /3 width=4/
| elim (IHL12 … HLK2) -L2 /3 width=3/
]
]
-qed.
+qed-.