+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+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_uni.ma".
+include "ground/relocation/tr_pap_tls.ma".
+
+(* LIFT FOR PATH ***********************************************************)
+
+definition lift_continuation (A:Type[0]) ≝
+ tr_map → path → A.
+
+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 ⇒ lift_gen (A) (λg,p. k g (𝗱(f@❨n❩)◗p)) (⇂*[n]f) 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
+ ]
+].
+
+interpretation
+ "lift (gneric)"
+ 'UpArrow A k f p = (lift_gen A k f p).
+
+definition proj_path: lift_continuation … ≝
+ λf,p.p.
+
+definition proj_rmap: lift_continuation … ≝
+ λf,p.f.
+
+interpretation
+ "lift (path)"
+ 'UpArrow f p = (lift_gen ? proj_path f p).
+
+interpretation
+ "lift (relocation map)"
+ 'UpArrow p f = (lift_gen ? proj_rmap f p).
+
+(* Basic constructions ******************************************************)
+
+lemma lift_empty (A) (k) (f):
+ k f (𝐞) = ↑{A}❨k, f, 𝐞❩.
+// qed.
+
+lemma lift_d_sn (A) (k) (p) (n) (f):
+ ↑❨(λg,p. k g (𝗱(f@❨n❩)◗p)), ⇂*[n]f, p❩ = ↑{A}❨k, f, 𝗱n◗p❩.
+// qed.
+
+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):
+ ↑❨(λg,p. k g (𝗟◗p)), ⫯f, p❩ = ↑{A}❨k, f, 𝗟◗p❩.
+// qed.
+
+lemma lift_A_sn (A) (k) (p) (f):
+ ↑❨(λg,p. k g (𝗔◗p)), f, p❩ = ↑{A}❨k, f, 𝗔◗p❩.
+// qed.
+
+lemma lift_S_sn (A) (k) (p) (f):
+ ↑❨(λg,p. k g (𝗦◗p)), f, p❩ = ↑{A}❨k, f, 𝗦◗p❩.
+// qed.
+
+(* Basic constructions with proj_path ***************************************)
+
+lemma lift_path_empty (f):
+ (𝐞) = ↑[f]𝐞.
+// qed.
+
+(* Basic constructions with proj_rmap ***************************************)
+
+lemma lift_rmap_empty (f):
+ f = ↑[𝐞]f.
+// qed.
+
+lemma lift_rmap_d_sn (f) (p) (n):
+ ↑[p](⇂*[ninj n]f) = ↑[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.
+// 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):
+ ⇂*[ninj n](↑[p]f) = ↑[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):
+ ↑[p]f@❨m+n❩ = ↑[p◖𝗱n]f@❨m❩+↑[p]f@❨n❩.
+// qed.