(* *)
(**************************************************************************)
-include "ground/relocation/tr_compose_pap.ma".
-include "ground/relocation/tr_uni_pap.ma".
-include "delayed_updating/syntax/path.ma".
include "delayed_updating/notation/functions/uparrow_4.ma".
include "delayed_updating/notation/functions/uparrow_2.ma".
+include "delayed_updating/syntax/path.ma".
+include "ground/relocation/tr_id_pap.ma".
-(* LIFT FOR PATH ***********************************************************)
+(* LIFT FOR PATH ************************************************************)
definition lift_continuation (A:Type[0]) ≝
tr_map → path → A.
-(* Note: inner numeric labels are not liftable, so they are removed *)
rec definition lift_gen (A:Type[0]) (k:lift_continuation A) (f) (p) on p ≝
match p with
[ list_empty ⇒ k f (𝐞)
| list_lcons l q ⇒
match l with
- [ label_d n ⇒
- match q with
- [ list_empty ⇒ lift_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (f∘𝐮❨n❩) q
- | list_lcons _ _ ⇒ lift_gen (A) k (f∘𝐮❨n❩) q
- ]
- | label_m ⇒ lift_gen (A) k f q
+ [ label_d n ⇒ lift_gen (A) (λg,p. k g (𝗱(f@⧣❨n❩)◗p)) (𝐢) q
+ | label_m ⇒ lift_gen (A) (λg,p. k g (𝗺◗p)) f q
| label_L ⇒ lift_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
| label_A ⇒ lift_gen (A) (λg,p. k g (𝗔◗p)) f q
| label_S ⇒ lift_gen (A) (λg,p. k g (𝗦◗p)) f q
k f (𝐞) = ↑{A}❨k, f, 𝐞❩.
// qed.
-lemma lift_d_empty_sn (A) (k) (n) (f):
- ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), f∘𝐮❨ninj n❩, 𝐞❩ = ↑{A}❨k, f, 𝗱n◗𝐞❩.
-// qed.
-
-lemma lift_d_lcons_sn (A) (k) (p) (l) (n) (f):
- ↑❨k, f∘𝐮❨ninj n❩, l◗p❩ = ↑{A}❨k, f, 𝗱n◗l◗p❩.
+lemma lift_d_sn (A) (k) (p) (n) (f):
+ ↑❨(λg,p. k g (𝗱(f@⧣❨n❩)◗p)), 𝐢, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
// qed.
lemma lift_m_sn (A) (k) (p) (f):
- ↑❨k, f, p❩ = ↑{A}❨k, f, 𝗺◗p❩.
+ ↑❨(λg,p. k g (𝗺◗p)), f, p❩ = ↑{A}❨k, f, 𝗺◗p❩.
// qed.
lemma lift_L_sn (A) (k) (p) (f):
(𝐞) = ↑[f]𝐞.
// qed.
-lemma lift_path_d_empty_sn (f) (n):
- 𝗱(f@❨n❩)◗𝐞 = ↑[f](𝗱n◗𝐞).
-// qed.
-
-lemma lift_path_d_lcons_sn (f) (p) (l) (n):
- ↑[f∘𝐮❨ninj n❩](l◗p) = ↑[f](𝗱n◗l◗p).
-// qed.
-
-lemma lift_path_m_sn (f) (p):
- ↑[f]p = ↑[f](𝗺◗p).
-// qed.
-
(* Basic constructions with proj_rmap ***************************************)
lemma lift_rmap_empty (f):
// qed.
lemma lift_rmap_d_sn (f) (p) (n):
- ↑[p](f∘𝐮❨ninj n❩) = ↑[𝗱n◗p]f.
-#f * // qed.
+ ↑[p]𝐢 = ↑[𝗱n◗p]f.
+// qed.
lemma lift_rmap_m_sn (f) (p):
↑[p]f = ↑[𝗺◗p]f.
↑[p]f = ↑[𝗦◗p]f.
// qed.
+(* Advanced cinstructionswith proj_rmap and tr_id ***************************)
+
+lemma lift_rmap_id (p):
+ (𝐢) = ↑[p]𝐢.
+#p elim p -p //
+* [ #n ] #p #IH //
+qed.
+
(* Advanced constructions with proj_rmap and path_append ********************)
lemma lift_rmap_append (p2) (p1) (f):
↑[p2]↑[p1]f = ↑[p1●p2]f.
#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
-[ <lift_rmap_m_sn <lift_rmap_m_sn //
+[ <lift_rmap_d_sn <lift_rmap_d_sn //
+| <lift_rmap_m_sn <lift_rmap_m_sn //
| <lift_rmap_A_sn <lift_rmap_A_sn //
| <lift_rmap_S_sn <lift_rmap_S_sn //
]
(* Advanced constructions with proj_rmap and path_rcons *********************)
lemma lift_rmap_d_dx (f) (p) (n):
- (↑[p]f)∘𝐮❨ninj n❩ = ↑[p◖𝗱n]f.
+ (𝐢) = ↑[p◖𝗱n]f.
// qed.
lemma lift_rmap_m_dx (f) (p):
// qed.
lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
- ↑[p]f@❨m+n❩ = ↑[p◖𝗱n]f@❨m❩.
-#f #p #n #m
-<lift_rmap_d_dx <tr_compose_pap <tr_uni_pap //
-qed.
-
-(* Advanced eliminations with path ******************************************)
-
-lemma path_ind_lift (Q:predicate …):
- Q (𝐞) →
- (∀n. Q (𝐞) → Q (𝗱n◗𝐞)) →
- (∀n,l,p. Q (l◗p) → Q (𝗱n◗l◗p)) →
- (∀p. Q p → Q (𝗺◗p)) →
- (∀p. Q p → Q (𝗟◗p)) →
- (∀p. Q p → Q (𝗔◗p)) →
- (∀p. Q p → Q (𝗦◗p)) →
- ∀p. Q p.
-#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #p
-elim p -p [| * [ #n * ] ]
-/2 width=1 by/
-qed-.
+ m = ↑[p◖𝗱n]f@⧣❨m❩.
+// qed.