(* *)
(**************************************************************************)
-include "ground/relocation/tr_compose.ma".
-include "ground/relocation/tr_uni.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 ***********************************************************)
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_node_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_edge_L ⇒ lift_gen (A) (λg,p. k g (𝗟◗p)) (⫯f) q
- | label_edge_A ⇒ lift_gen (A) (λg,p. k g (𝗔◗p)) f q
- | label_edge_S ⇒ lift_gen (A) (λg,p. k g (𝗦◗p)) 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❩.
+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_d_lcons_sn (A) (k) (p) (l) (n) (f):
- ↑❨k, f∘𝐮❨ninj n❩, l◗p❩ = ↑{A}❨k, f, 𝗱n◗l◗p❩.
+lemma lift_m_sn (A) (k) (p) (f):
+ ↑❨(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.
+(* Basic constructions with proj_rmap ***************************************)
-lemma lift_path_d_lcons_sn (f) (p) (l) (n):
- ↑[f∘𝐮❨ninj n❩](l◗p) = ↑[f](𝗱n◗l◗p).
+lemma lift_rmap_empty (f):
+ f = ↑[𝐞]f.
// qed.
-(* Basic constructions with proj_rmap ***************************************)
-
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.
+// qed.
lemma lift_rmap_L_sn (f) (p):
↑[p](⫯f) = ↑[𝗟◗p]f.
lemma lift_rmap_append (p2) (p1) (f):
↑[p2]↑[p1]f = ↑[p1●p2]f.
#p2 #p1 elim p1 -p1 // * [ #n ] #p1 #IH #f //
-[ <lift_rmap_A_sn <lift_rmap_A_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 //
]
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 (𝗟◗p)) →
- (∀p. Q p → Q (𝗔◗p)) →
- (∀p. Q p → Q (𝗦◗p)) →
- ∀p. Q p.
-#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #p
-elim p -p [| * [ #n * ] ]
-/2 width=1 by/
-qed-.
+(* Advanced constructions with proj_rmap and path_rcons *********************)
+
+lemma lift_rmap_d_dx (f) (p) (n):
+ (𝐢) = ↑[p◖𝗱n]f.
+// qed.
+
+lemma lift_rmap_m_dx (f) (p):
+ ↑[p]f = ↑[p◖𝗺]f.
+// qed.
+
+lemma lift_rmap_L_dx (f) (p):
+ (⫯↑[p]f) = ↑[p◖𝗟]f.
+// qed.
+
+lemma lift_rmap_A_dx (f) (p):
+ ↑[p]f = ↑[p◖𝗔]f.
+// qed.
+
+lemma lift_rmap_S_dx (f) (p):
+ ↑[p]f = ↑[p◖𝗦]f.
+// qed.
+
+lemma lift_rmap_pap_d_dx (f) (p) (n) (m):
+ m = ↑[p◖𝗱n]f@❨m❩.
+// qed.