(* LIFT FOR PROTOTERM *******************************************************)
-lemma lift_term_iref_sn (f) (t:prototerm) (n:pnat):
- (𝛕f@⧣❨n❩.↑[⇂*[n]f]t) ⊆ ↑[f](𝛕n.t).
-#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
-@(ex2_intro … (𝗱n◗𝗺◗r))
+lemma lift_term_iref_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_term_iref_dx (f) (t) (n:pnat):
- ↑[f](𝛕n.t) ⊆ 𝛕f@⧣❨n❩.↑[⇂*[n]f]t.
-#f #t #n #p * #q #Hq #H0 destruct
+lemma lift_term_iref_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_term_iref (f) (t) (n:pnat):
- (𝛕f@⧣❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛕n.t).
+lemma lift_term_iref (f) (t) (k:pnat):
+ (𝛕f@⧣❨k❩.↑[⇂*[k]f]t) ⇔ ↑[f](𝛕k.t).
/3 width=1 by conj, lift_term_iref_sn, lift_term_iref_dx/
qed.
-lemma lift_term_iref_uni (t) (m) (n):
- (𝛕(n+m).t) ⇔ ↑[𝐮❨m❩](𝛕n.t).
-#t #m #n
+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 …))
<tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
/3 width=1 by iref_eq_repl, lift_term_id/