(* *)
(**************************************************************************)
-include "ground_2/notation/functions/uparrowstar_2.ma".
+include "ground/notation/functions/uparrowstar_2.ma".
include "apps_2/notation/functional/uparrow_2.ma".
include "static_2/relocation/lifts.ma".
rec definition flifts f U on U ≝ match U with
[ TAtom I ⇒ match I with
[ Sort _ ⇒ U
- | LRef i ⇒ #(f@❴i❵)
+ | LRef i ⇒ #(f@⧣❨i❩)
| GRef _ ⇒ U
]
| TPair I V T ⇒ match I with
- [ Bind2 p I ⇒ ⓑ{p,I}(flifts f V).(flifts (⫯f) T)
- | Flat2 I ⇒ ⓕ{I}(flifts f V).(flifts f T)
+ [ Bind2 p I ⇒ ⓑ[p,I](flifts f V).(flifts (⫯f) T)
+ | Flat2 I ⇒ ⓕ[I](flifts f V).(flifts f T)
]
].
interpretation "uniform functional relocation (term)"
'UpArrow i T = (flifts (uni i) T).
+(* Basic properties *********************************************************)
+
+lemma flifts_lref (f) (i): ↑*[f](#i) = #(f@⧣❨i❩).
+// qed.
+
+lemma flifts_bind (f) (p) (I) (V) (T): ↑*[f](ⓑ[p,I]V.T) = ⓑ[p,I]↑*[f]V.↑*[⫯f]T.
+// qed.
+
+lemma flifts_flat (f) (I) (V) (T): ↑*[f](ⓕ[I]V.T) = ⓕ[I]↑*[f]V.↑*[f]T.
+// qed.
+
(* Main properties **********************************************************)
-theorem flifts_lifts: â\88\80T,f. â¬\86*[f]T ≘ ↑*[f]T.
+theorem flifts_lifts: â\88\80T,f. â\87§*[f]T ≘ ↑*[f]T.
#T elim T -T *
/2 width=1 by lifts_sort, lifts_lref, lifts_gref, lifts_bind, lifts_flat/
qed.
(* Main inversion properties ************************************************)
-theorem flifts_inv_lifts: â\88\80f,T1,T2. â¬\86*[f]T1 ≘ T2 → ↑*[f]T1 = T2.
+theorem flifts_inv_lifts: â\88\80f,T1,T2. â\87§*[f]T1 ≘ T2 → ↑*[f]T1 = T2.
#f #T1 #T2 #H elim H -f -T1 -T2 //
[ #f #i1 #i2 #H <(at_inv_total … H) //
| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT <IHV <IHT -V2 -T2 //
(* Derived properties *******************************************************)
+lemma flifts_comp: ∀f1,f2. f1 ≡ f2 → ∀T. ↑*[f1]T = ↑*[f2]T.
+/3 width=3 by flifts_inv_lifts, lifts_eq_repl_fwd/ qed.
+
+(* Derived properties with uniform relocation *******************************)
+
lemma flifts_lref_uni: ∀l,i. ↑[l](#i) = #(l+i).
/3 width=1 by flifts_inv_lifts, lifts_lref_uni/ qed.
-(*
-lemma flift_join: ∀e1,e2,T. ⬆[e1, e2] ↑[0, e1] T ≡ ↑[0, e1 + e2] T.
-#e1 #e2 #T
-lapply (flift_lift T 0 (e1+e2)) #H
-elim (lift_split … H e1 e1) -H // #U #H
->(flift_inv_lift … H) -H //
-qed.
-*)