include "delayed_updating/substitution/lift_structure.ma".
include "delayed_updating/substitution/lift_prototerm.ma".
+lemma pippo (p):
+ ⊗p ϵ 𝐈.
+#p @(list_ind_rcons … p) -p
+[ <structure_empty //
+| #p * [ #n ] #IH
+ [ <structure_d_dx //
+ | <structure_m_dx //
+ | <structure_L_dx //
+ | <structure_A_dx //
+ | <structure_S_dx //
+ ]
+]
+qed.
+
(* 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
+lemma lift_grafted_sn (f) (t) (p): p ϵ 𝐈 →
+ ↑[↑[p]f](t⋔p) ⊆ (↑[f]t)⋔(⊗p).
+#f #t #p #Hp #q * #r #Hr #H0 destruct
@(ex2_intro … Hr) -Hr
-<list_append_rcons_sn <list_append_rcons_sn
-<lift_append_proper_dx //
+<lift_append_inner_sn //
+qed-.
+
+lemma lift_grafted_dx (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+ (↑[f]t)⋔(⊗p) ⊆ ↑[↑[p]f](t⋔p).
+#f #t #p #H1p #H2p #Ht #q * #r #Hr #H0
+elim (lift_inv_append_inner_sn … (sym_eq … H0)) -H0 //
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+<(Ht … Hp0) [|*: /2 width=2 by ex_intro/ ] -p
+/2 width=1 by in_comp_lift_bi/
qed-.
+lemma lift_grafted (f) (t) (p): p ϵ 𝐈 → p ϵ ▵t → t ϵ 𝐓 →
+ ↑[↑[p]f](t⋔p) ⇔ (↑[f]t)⋔(⊗p).
+/3 width=1 by lift_grafted_sn, conj, lift_grafted_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
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.
+#f #t #p #Hp #Ht
+@conj
+[ >lift_rmap_S_dx >structure_S_dx
+ @lift_grafted_sn //
+| /2 width=1 by lift_grafted_S_dx/
+]
+qed.