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_term_iref_sn (f) (t:prototerm) (n:pnat):
- (๐f๏ผ โงฃโจnโฉ.โ[โ*[n]f]t) โ โ[f](๐n.t).
-#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
-@(ex2_intro โฆ (๐ฑnโ๐บโr))
+lemma lift_term_iref_pap_sn (f) (t:prototerm) (k:pnat):
+ (๐f๏ผ โงฃโจkโฉ.โ[โ*[k]f]t) โ โ[f](๐k.t).
+#f #t #k #p * #q * #r #Hr #H1 #H2 destruct
+@(ex2_intro โฆ (๐ฑkโ๐บโr))
/2 width=1 by in_comp_iref/
qed-.
-lemma lift_term_iref_dx (f) (t) (n:pnat):
- โ[f](๐n.t) โ ๐f๏ผ โงฃโจnโฉ.โ[โ*[n]f]t.
-#f #t #n #p * #q #Hq #H0 destruct
+lemma lift_term_iref_pap_dx (f) (t) (k:pnat):
+ โ[f](๐k.t) โ ๐f๏ผ โงฃโจkโฉ.โ[โ*[k]f]t.
+#f #t #k #p * #q #Hq #H0 destruct
elim (in_comp_inv_iref โฆ 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):
- (๐f๏ผ โงฃโจnโฉ.โ[โ*[n]f]t) โ โ[f](๐n.t).
-/3 width=1 by conj, lift_term_iref_sn, lift_term_iref_dx/
+lemma lift_term_iref_pap (f) (t) (k:pnat):
+ (๐f๏ผ โงฃโจkโฉ.โ[โ*[k]f]t) โ โ[f](๐k.t).
+/3 width=1 by conj, lift_term_iref_pap_sn, lift_term_iref_pap_dx/
qed.
-lemma lift_term_iref_uni (t) (m) (n):
- (๐(n+m).t) โ โ[๐ฎโจmโฉ](๐n.t).
-#t #m #n
-@(subset_eq_trans โฆ (lift_term_iref โฆ))
+lemma lift_term_iref_nap (f) (t) (n):
+ (๐โ(f๏ผ ยงโจnโฉ).โ[โ*[โn]f]t) โ โ[f](๐โn.t).
+#f #t #n
+>tr_pap_succ_nap //
+qed.
+
+lemma lift_term_iref_uni (t) (n) (k):
+ (๐(k+n).t) โ โ[๐ฎโจnโฉ](๐k.t).
+#t #n #k
+@(subset_eq_trans โฆ (lift_term_iref_pap โฆ))
<tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
/3 width=1 by iref_eq_repl, lift_term_id/
qed.