-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_empty (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):
+ f = ↑[𝐞]f.
+// qed.
+
+lemma lift_rmap_d_sn (f) (p) (n):
+ ↑[p](f∘𝐮❨ninj n❩) = ↑[𝗱n◗p]f.
+#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.
+// 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_m_sn <lift_rmap_m_sn //
+| <lift_rmap_A_sn <lift_rmap_A_sn //
+| <lift_rmap_S_sn <lift_rmap_S_sn //
+]
+qed.
+
+(* Advanced constructions with proj_rmap and path_rcons *********************)
+
+lemma lift_rmap_d_dx (f) (p) (n):
+ (↑[p]f)∘𝐮❨ninj n❩ = ↑[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.
+
+(* 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-.