(* BASIC TERM VECTOR RELOCATION *********************************************)
-inductive liftv (d,e:nat) : relation (list term) ≝
-| liftv_nil : liftv d e (◊) (◊)
+inductive liftv (l,m:nat) : relation (list term) ≝
+| liftv_nil : liftv l m (◊) (◊)
| liftv_cons: ∀T1s,T2s,T1,T2.
- ⬆[d, e] T1 ≡ T2 → liftv d e T1s T2s →
- liftv d e (T1 @ T1s) (T2 @ T2s)
+ ⬆[l, m] T1 ≡ T2 → liftv l m T1s T2s →
+ liftv l m (T1 @ T1s) (T2 @ T2s)
.
-interpretation "relocation (vector)" 'RLift d e T1s T2s = (liftv d e T1s T2s).
+interpretation "relocation (vector)" 'RLift l m T1s T2s = (liftv l m T1s T2s).
(* Basic inversion lemmas ***************************************************)
-fact liftv_inv_nil1_aux: ∀T1s,T2s,d,e. ⬆[d, e] T1s ≡ T2s → T1s = ◊ → T2s = ◊.
-#T1s #T2s #d #e * -T1s -T2s //
+fact liftv_inv_nil1_aux: ∀T1s,T2s,l,m. ⬆[l, m] T1s ≡ T2s → T1s = ◊ → T2s = ◊.
+#T1s #T2s #l #m * -T1s -T2s //
#T1s #T2s #T1 #T2 #_ #_ #H destruct
qed-.
-lemma liftv_inv_nil1: ∀T2s,d,e. ⬆[d, e] ◊ ≡ T2s → T2s = ◊.
+lemma liftv_inv_nil1: ∀T2s,l,m. ⬆[l, m] ◊ ≡ T2s → T2s = ◊.
/2 width=5 by liftv_inv_nil1_aux/ qed-.
-fact liftv_inv_cons1_aux: ∀T1s,T2s,d,e. ⬆[d, e] T1s ≡ T2s →
+fact liftv_inv_cons1_aux: ∀T1s,T2s,l,m. ⬆[l, m] T1s ≡ T2s →
∀U1,U1s. T1s = U1 @ U1s →
- ∃∃U2,U2s. ⬆[d, e] U1 ≡ U2 & ⬆[d, e] U1s ≡ U2s &
+ ∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
T2s = U2 @ U2s.
-#T1s #T2s #d #e * -T1s -T2s
+#T1s #T2s #l #m * -T1s -T2s
[ #U1 #U1s #H destruct
| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
]
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 &
+lemma liftv_inv_cons1: ∀U1,U1s,T2s,l,m. ⬆[l, m] U1 @ U1s ≡ T2s →
+ ∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
T2s = U2 @ U2s.
/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
+lemma liftv_total: ∀l,m. ∀T1s:list term. ∃T2s. ⬆[l, m] T1s ≡ T2s.
+#l #m #T1s elim T1s -T1s
[ /2 width=2 by liftv_nil, ex_intro/
| #T1 #T1s * #T2s #HT12s
- elim (lift_total T1 d e) /3 width=2 by liftv_cons, ex_intro/
+ elim (lift_total T1 l m) /3 width=2 by liftv_cons, ex_intro/
]
qed-.