(* *)
(**************************************************************************)
+(**) (* reverse include *)
include "ground/lib/subset_ext_equivalence.ma".
-include "delayed_updating/substitution/lift_eq.ma".
+include "delayed_updating/substitution/lift_path_eq.ma".
include "delayed_updating/substitution/lift_prototerm.ma".
(* LIFT FOR PROTOTERM *******************************************************)
(* Constructions with subset_equivalence ************************************)
lemma lift_term_eq_repl_sn (f1) (f2) (t):
- f1 ≗ f2 → ↑[f1]t ⇔ ↑[f2]t.
+ f1 ≗ f2 → 🠡[f1]t ⇔ 🠡[f2]t.
/3 width=1 by subset_equivalence_ext_f1_exteq, lift_path_eq_repl/
qed.
lemma lift_term_eq_repl_dx (f) (t1) (t2):
- t1 ⇔ t2 → ↑[f]t1 ⇔ ↑[f]t2.
+ t1 ⇔ t2 → 🠡[f]t1 ⇔ 🠡[f]t2.
/2 width=1 by subset_equivalence_ext_f1_bi/
qed.
-lemma lift_term_after (f1) (f2) (t):
- ↑[f2]↑[f1]t ⇔ ↑[f2∘f1]t.
-#f1 #f2 #t @subset_eq_trans
-[
-| @subset_inclusion_ext_f1_compose
-| @subset_equivalence_ext_f1_exteq /2 width=5/
-]
-qed.
+lemma lift_term_grafted_sn (f) (t) (p):
+ 🠡[🠢[f]p](t⋔p) ⊆ (🠡[f]t)⋔(🠡[f]p).
+#f #t #p #q * #r #Hr #H0 destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lift_term_grafted_dx (f) (t) (p):
+ (🠡[f]t)⋔(🠡[f]p) ⊆ 🠡[🠢[f]p](t⋔p).
+#f #t #p #q * #r #Hr #H0
+elim (lift_path_inv_append_sn … (sym_eq … H0)) -H0
+#p0 #q0 #Hp0 #Hq0 #H0 destruct
+lapply (lift_path_inj … Hp0) -Hp0 #Hp0 destruct
+/2 width=1 by in_comp_lift_path_term/
+qed-.
+
+lemma lift_term_grafted (f) (t) (p):
+ 🠡[🠢[f]p](t⋔p) ⇔ (🠡[f]t)⋔(🠡[f]p).
+/3 width=1 by lift_term_grafted_sn, lift_term_grafted_dx, conj/ qed.
+
+lemma lift_term_grafted_S (f) (t) (p):
+ 🠡[🠢[f]p](t⋔(p◖𝗦)) ⇔ (🠡[f]t)⋔((🠡[f]p)◖𝗦).
+// qed.