(* *)
(**************************************************************************)
-include "ground/relocation/tr_compose_compose.ma".
-include "ground/relocation/tr_compose_pn.ma".
-include "delayed_updating/substitution/lift_eq.ma".
-
-lemma lift_path_after (p) (f1) (f2):
- ↑[f2]↑[f1]p = ↑[f2∘f1]p.
-#p @(path_ind_lift … p) -p // [ #n #l #p | #p ] #IH #f1 #f2
-[ <lift_path_d_lcons_sn <lift_path_d_lcons_sn
- >(lift_path_eq_repl … (tr_compose_assoc …)) //
-| <lift_path_L_sn <lift_path_L_sn <lift_path_L_sn
- >tr_compose_push_bi //
-]
-qed.
-
-include "delayed_updating/substitution/lift_prototerm.ma".
-
-axiom lift_term_after (t) (f1) (f2):
- ↑[f2]↑[f1]t ⇔ ↑[f2∘f1]t.
-
-include "delayed_updating/syntax/prototerm_constructors.ma".
+include "delayed_updating/substitution/lift_prototerm_id.ma".
+include "delayed_updating/substitution/lift_path_uni.ma".
+include "delayed_updating/syntax/prototerm_constructors_eq.ma".
+include "ground/relocation/nap.ma".
(* LIFT FOR PROTOTERM *******************************************************)
-lemma lift_iref_after_sn (f) (t) (n:pnat):
- (↑[f∘𝐮❨n❩]t) ⊆ ↑[f](𝛗n.t).
-#f #t #n #p * #q #Hq #H0 destruct
-@(ex2_intro … (𝗱n◗𝗺◗q))
+lemma lift_term_iref_pap_sn (f) (t:prototerm) (k:pnat):
+ (𝛕f@⧣❨k❩.↑[⇂*[k]f]t) ⊆ ↑[f](𝛕k.t).
+#f #t #k #p * #q * #r #Hr #H1 #H2 destruct
+@(ex2_intro … (𝗱k◗𝗺◗r))
/2 width=1 by in_comp_iref/
qed-.
-lemma lift_iref_after_dx (f) (t) (n:pnat):
- ↑[f](𝛗n.t) ⊆ ↑[f∘𝐮❨n❩]t.
-#f #t #n #p * #q #Hq #H0 destruct
-elim (in_comp_inv_iref … Hq) -Hq #p #Hp #Ht destruct
-/2 width=1 by in_comp_lift_bi/
+lemma lift_term_iref_pap_dx (f) (t) (k:pnat):
+ ↑[f](𝛕k.t) ⊆ 𝛕f@⧣❨k❩.↑[⇂*[k]f]t.
+#f #t #k #p * #q #Hq #H0 destruct
+elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
+<lift_path_d_sn <lift_path_m_sn
+/3 width=1 by in_comp_iref, in_comp_lift_path_term/
qed-.
-lemma lift_iref_after (f) (t) (n:pnat):
- ↑[f∘𝐮❨n❩]t ⇔ ↑[f](𝛗n.t).
-/3 width=1 by conj, lift_iref_after_sn, lift_iref_after_dx/
+lemma lift_term_iref_pap (f) (t) (k:pnat):
+ (𝛕f@⧣❨k❩.↑[⇂*[k]f]t) ⇔ ↑[f](𝛕k.t).
+/3 width=1 by conj, lift_term_iref_pap_sn, lift_term_iref_pap_dx/
+qed.
+
+lemma lift_term_iref_nap (f) (t) (n):
+ (𝛕↑(f@§❨n❩).↑[⇂*[↑n]f]t) ⇔ ↑[f](𝛕↑n.t).
+#f #t #n
+>tr_pap_succ_nap //
qed.
-lemma lift_iref (f) (t) (n:pnat):
- ↑[f]↑[𝐮❨n❩]t ⇔ ↑[f](𝛗n.t).
-/3 width=3 by lift_term_after, subset_eq_trans/
+lemma lift_term_iref_uni (t) (n) (k):
+ (𝛕(k+n).t) ⇔ ↑[𝐮❨n❩](𝛕k.t).
+#t #n #k
+@(subset_eq_trans … (lift_term_iref_pap …))
+<tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
+/3 width=1 by iref_eq_repl, lift_term_id/
qed.