+(**************************************************************************)
+(* ___ *)
+(* ||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 "ground/lib/subset_equivalence.ma".
+include "delayed_updating/syntax/preterm.ma".
+include "delayed_updating/substitution/lift_structure.ma".
+include "delayed_updating/substitution/lift_prototerm.ma".
+
+(* LIFT FOR PRETERM *********************************************************)
+
+(* Constructions with subset_equivalence ************************************)
+
+lemma lift_grafted_S_sn (f) (t) (p):
+ ↑[↑[p]f](t⋔(p◖𝗦)) ⊆ (↑[f]t)⋔((⊗p)◖𝗦).
+#f #t #p #q * #r #Hr #H0 destruct
+@(ex2_intro … Hr) -Hr
+<list_append_rcons_sn <list_append_rcons_sn
+<lift_append_proper_dx //
+qed-.
+
+lemma lift_grafted_S_dx (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+ (↑[f]t)⋔((⊗p)◖𝗦) ⊆ ↑[↑[p]f](t⋔(p◖𝗦)).
+#f #t #p #Hp #Ht #q * #r #Hr
+<list_append_rcons_sn #H0
+elim (lift_inv_append_proper_dx … (sym_eq … H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+elim (lift_path_inv_S_sn … (sym_eq … Hq0)) -Hq0
+#r1 #r2 #Hr1 #Hr2 #H0 destruct
+lapply (preterm_in_root_append_inv_structure_empty_dx … p0 … Ht Hr1)
+[ /2 width=2 by ex_intro/ ] -Hr1 #Hr1 destruct
+/2 width=1 by in_comp_lift_bi/
+qed-.
+
+lemma lift_grafted_S (f) (t) (p): p ϵ ▵t → t ϵ 𝐓 →
+ ↑[↑[p]f](t⋔(p◖𝗦)) ⇔ (↑[f]t)⋔((⊗p)◖𝗦).
+/3 width=1 by lift_grafted_S_sn, conj, lift_grafted_S_dx/ qed.