+(* Basic inversion lemmas ***************************************************)
+
+fact liftv_inv_nil1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s → T1s = ◊ → T2s = ◊.
+#T1s #T2s #d #e * -T1s -T2s //
+#T1s #T2s #T1 #T2 #_ #_ #H destruct
+qed.
+
+lemma liftv_inv_nil1: ∀T2s,d,e. ⇧[d, e] ◊ ≡ T2s → T2s = ◊.
+/2 width=5/ qed-.
+
+fact liftv_inv_cons1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s →
+ ∀U1,U1s. T1s = U1 :: U1s →
+ ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
+ T2s = U2 :: U2s.
+#T1s #T2s #d #e * -T1s -T2s
+[ #U1 #U1s #H destruct
+| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5/
+]
+qed.
+
+lemma liftv_inv_cons1: ∀U1,U1s,T2s,d,e. ⇧[d, e] U1 :: U1s ≡ T2s →
+ ∃∃U2,U2s. ⇧[d, e] U1 ≡ U2 & ⇧[d, e] U1s ≡ U2s &
+ T2s = U2 :: U2s.
+/2 width=3/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma liftv_total: ∀d,e. ∀T1s:list term. ∃T2s. ⇧[d, e] T1s ≡ T2s.
+#d #e #T1s elim T1s -T1s
+[ /2 width=2/
+| #T1 #T1s * #T2s #HT12s
+ elim (lift_total T1 d e) /3 width=2/
+]
+qed-.