(* *)
(**************************************************************************)
-include "delayed_updating/substitution/lift_prototerm_eq.ma".
-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_sn (f) (t:prototerm) (n:pnat):
- (ð\9d\9b\97f@â\9d¨nâ\9d©.â\86\91[â\87\82*[n]f]t) â\8a\86 â\86\91[f](ð\9d\9b\97n.t).
-#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
-@(ex2_intro … (𝗱n◗𝗺◗r))
+lemma lift_term_iref_pap_sn (f) (t:prototerm) (k:pnat):
+ (ð\9d\9b\95fï¼ â§£â\9d¨kâ\9d©.ð\9f ¡[â\87\82*[k]f]t) â\8a\86 ð\9f ¡[f](ð\9d\9b\95k.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_dx (f) (t) (n:pnat):
- ↑[f](𝛗n.t) ⊆ 𝛗f@❨n❩.↑[⇂*[n]f]t.
-#f #t #n #p * #q #Hq #H0 destruct
+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
-/3 width=1 by in_comp_iref, in_comp_lift_bi/
+<lift_path_d_sn <lift_path_m_sn
+/3 width=1 by in_comp_iref, in_comp_lift_path_term/
qed-.
-lemma lift_iref (f) (t) (n:pnat):
- (𝛗f@❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛗n.t).
-/3 width=1 by conj, lift_iref_sn, lift_iref_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_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.