(* Main properties **********************************************************)
(* Basic_1: includes: lifts_inj *)
(* Main properties **********************************************************)
(* Basic_1: includes: lifts_inj *)
[ #T2s #H >(liftsv_inv_nil2 … H) -H //
| #T1s #Us #T1 #U #HT1U #_ #IHT1Us #X #H destruct
elim (liftsv_inv_cons2 … H) -H #T2 #T2s #HT2U #HT2Us #H destruct
[ #T2s #H >(liftsv_inv_nil2 … H) -H //
| #T1s #Us #T1 #U #HT1U #_ #IHT1Us #X #H destruct
elim (liftsv_inv_cons2 … H) -H #T2 #T2s #HT2U #HT2Us #H destruct
[ #U2s #H >(liftsv_inv_nil1 … H) -H //
| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
elim (liftsv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
[ #U2s #H >(liftsv_inv_nil1 … H) -H //
| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
elim (liftsv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
-theorem liftsv_trans: ∀T1s,Ts,f1. ⬆*[f1] T1s ≡ Ts → ∀T2s,f2. ⬆*[f2] Ts ≡ T2s →
+theorem liftsv_trans: ∀f1,T1s,Ts. ⬆*[f1] T1s ≡ Ts → ∀T2s,f2. ⬆*[f2] Ts ≡ T2s →
[ #T2s #f2 #H >(liftsv_inv_nil1 … H) -T2s /2 width=3 by liftsv_nil/
| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #f2 #H elim (liftsv_inv_cons1 … H) -H
#T2 #T2s #HT2 #HT2s #H destruct /3 width=6 by lifts_trans, liftsv_cons/
[ #T2s #f2 #H >(liftsv_inv_nil1 … H) -T2s /2 width=3 by liftsv_nil/
| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #f2 #H elim (liftsv_inv_cons1 … H) -H
#T2 #T2s #HT2 #HT2s #H destruct /3 width=6 by lifts_trans, liftsv_cons/