(* BASIC TERM VECTOR RELOCATION *********************************************)
inductive liftv (d,e:nat) : relation (list term) ≝
-| liftv_nil : liftv d e ◊ ◊
+| liftv_nil : liftv d e (◊) (◊)
| liftv_cons: ∀T1s,T2s,T1,T2.
⇧[d, e] T1 ≡ T2 → liftv d e T1s T2s →
liftv d e (T1 @ T1s) (T2 @ T2s)
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.
+qed-.
lemma liftv_inv_nil1: ∀T2s,d,e. ⇧[d, e] ◊ ≡ T2s → T2s = ◊.
-/2 width=5/ qed-.
+/2 width=5 by liftv_inv_nil1_aux/ qed-.
fact liftv_inv_cons1_aux: ∀T1s,T2s,d,e. ⇧[d, e] T1s ≡ T2s →
∀U1,U1s. T1s = U1 @ U1s →
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/
+| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
]
-qed.
+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-.
+/2 width=3 by liftv_inv_cons1_aux/ 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/
+[ /2 width=2 by liftv_nil, ex_intro/
| #T1 #T1s * #T2s #HT12s
- elim (lift_total T1 d e) /3 width=2/
+ elim (lift_total T1 d e) /3 width=2 by liftv_cons, ex_intro/
]
qed-.