(* LIFT FOR PATH ***********************************************************)
+definition lift_continuation (A:Type[0]) ≝
+ path → tr_map → A.
+
(* Note: inner numeric labels are not liftable, so they are removed *)
-rec definition lift_gen (A:Type[0]) (k:?→?→A) (p) (f) on p ≝
+rec definition lift_gen (A:Type[0]) (k:lift_continuation A) (p) (f) on p ≝
match p with
-[ list_empty ⇒ k 𝐞 f
+[ list_empty ⇒ k (𝐞) f
| list_lcons l q ⇒
match l with
[ label_node_d n ⇒
match q with
- [ list_empty ⇒ lift_gen (A) (λp. k (𝗱❨f@❨n❩❩◗p)) q f
+ [ list_empty ⇒ lift_gen (A) (λp. k (𝗱(f@❨n❩)◗p)) q (f∘𝐮❨n❩)
| list_lcons _ _ ⇒ lift_gen (A) k q (f∘𝐮❨n❩)
]
| label_edge_L ⇒ lift_gen (A) (λp. k (𝗟◗p)) q (⫯f)
(* Basic constructions ******************************************************)
-lemma lift_L (A) (k) (p) (f):
+lemma lift_empty (A) (k) (f):
+ k (𝐞) f = ↑{A}❨k, 𝐞, f❩.
+// qed.
+
+lemma lift_d_empty_sn (A) (k) (n) (f):
+ ↑❨(λp. k (𝗱(f@❨n❩)◗p)), 𝐞, f∘𝐮❨ninj n❩❩ = ↑{A}❨k, 𝗱n◗𝐞, f❩.
+// qed.
+
+lemma lift_d_lcons_sn (A) (k) (p) (l) (n) (f):
+ ↑❨k, l◗p, f∘𝐮❨ninj n❩❩ = ↑{A}❨k, 𝗱n◗l◗p, f❩.
+// qed.
+
+lemma lift_L_sn (A) (k) (p) (f):
↑❨(λp. k (𝗟◗p)), p, ⫯f❩ = ↑{A}❨k, 𝗟◗p, f❩.
// qed.
+lemma lift_A_sn (A) (k) (p) (f):
+ ↑❨(λp. k (𝗔◗p)), p, f❩ = ↑{A}❨k, 𝗔◗p, f❩.
+// qed.
+
+lemma lift_S_sn (A) (k) (p) (f):
+ ↑❨(λp. k (𝗦◗p)), p, f❩ = ↑{A}❨k, 𝗦◗p, f❩.
+// qed.
+
(* Basic constructions with proj_path ***************************************)
-lemma lift_append (p) (f) (q):
- q●↑[f]p = ↑❨(λp. proj_path (q●p)), p, f❩.
-#p elim p -p
-[ //
-| #l #p #IH #f #q cases l
- [
- | <lift_L in ⊢ (???%);
- >(list_append_rcons_sn ? q) in ⊢ (???(??(λ_.%)??));
-
- <IH
- normalize >IH
- | //
-
-(* Constructions with append ************************************************)
-
-theorem lift_append_A (p2) (p1) (f):
- (↑[f]p1)●𝗔◗↑[↑[p1]f]p2 = ↑[f](p1●𝗔◗p2).
-#p2 #p1 elim p1 -p1
-[ #f normalize
+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.
+
+(* Basic constructions with proj_rmap ***************************************)
+
+lemma lift_rmap_d_sn (f) (p) (n):
+ ↑[p](f∘𝐮❨ninj n❩) = ↑[𝗱n◗p]f.
+#f * // qed.
+
+lemma lift_rmap_L_sn (f) (p):
+ ↑[p](⫯f) = ↑[𝗟◗p]f.
+// qed.
+
+lemma lift_rmap_A_sn (f) (p):
+ ↑[p]f = ↑[𝗔◗p]f.
+// qed.
+
+lemma lift_rmap_S_sn (f) (p):
+ ↑[p]f = ↑[𝗦◗p]f.
+// 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_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-.