(* Main properties **********************************************************)
(* Basic_1: includes: lifts_inj *)
-theorem liftsv_inj: ∀T1s,Us,t. ⬆*[t] T1s ≡ Us →
- ∀T2s. ⬆*[t] T2s ≡ Us → T1s = T2s.
-#T1s #Us #t #H elim H -T1s -Us
-[ #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
+theorem liftsv_inj: ∀T1c,Us,f. ⬆*[f] T1c ≡ Us →
+ ∀T2c. ⬆*[f] T2c ≡ Us → T1c = T2c.
+#T1c #Us #f #H elim H -T1c -Us
+[ #T2c #H >(liftsv_inv_nil2 … H) -H //
+| #T1c #Us #T1 #U #HT1U #_ #IHT1Us #X #H destruct
+ elim (liftsv_inv_cons2 … H) -H #T2 #T2c #HT2U #HT2Us #H destruct
>(lifts_inj … HT1U … HT2U) -U /3 width=1 by eq_f/
]
qed-.
(* Basic_2A1: includes: liftv_mono *)
-theorem liftsv_mono: ∀Ts,U1s,t. ⬆*[t] Ts ≡ U1s →
- ∀U2s. ⬆*[t] Ts ≡ U2s → U1s = U2s.
-#Ts #U1s #t #H elim H -Ts -U1s
-[ #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_mono: ∀Ts,U1c,f. ⬆*[f] Ts ≡ U1c →
+ ∀U2c. ⬆*[f] Ts ≡ U2c → U1c = U2c.
+#Ts #U1c #f #H elim H -Ts -U1c
+[ #U2c #H >(liftsv_inv_nil1 … H) -H //
+| #Ts #U1c #T #U1 #HTU1 #_ #IHTU1c #X #H destruct
+ elim (liftsv_inv_cons1 … H) -H #U2 #U2c #HTU2 #HTU2c #H destruct
>(lifts_mono … HTU1 … HTU2) -T /3 width=1 by eq_f/
]
qed-.
(* Basic_1: includes: lifts1_xhg (right to left) *)
(* Basic_2A1: includes: liftsv_liftv_trans_le *)
-theorem liftsv_trans: ∀T1s,Ts,t1. ⬆*[t1] T1s ≡ Ts → ∀T2s,t2. ⬆*[t2] Ts ≡ T2s →
- ∀t. t2 ⊚ t1 ≡ t → ⬆*[t] T1s ≡ T2s.
-#T1s #Ts #t1 #H elim H -T1s -Ts
-[ #T2s #t2 #H >(liftsv_inv_nil1 … H) -T2s /2 width=3 by liftsv_nil/
-| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #t2 #H elim (liftsv_inv_cons1 … H) -H
- #T2 #T2s #HT2 #HT2s #H destruct /3 width=6 by lifts_trans, liftsv_cons/
+theorem liftsv_trans: ∀T1c,Ts,f1. ⬆*[f1] T1c ≡ Ts → ∀T2c,f2. ⬆*[f2] Ts ≡ T2c →
+ ∀f. f2 ⊚ f1 ≡ f → ⬆*[f] T1c ≡ T2c.
+#T1c #Ts #f1 #H elim H -T1c -Ts
+[ #T2c #f2 #H >(liftsv_inv_nil1 … H) -T2c /2 width=3 by liftsv_nil/
+| #T1c #Ts #T1 #T #HT1 #_ #IHT1c #X #f2 #H elim (liftsv_inv_cons1 … H) -H
+ #T2 #T2c #HT2 #HT2c #H destruct /3 width=6 by lifts_trans, liftsv_cons/
]
qed-.