(* *)
(**************************************************************************)
-include "delayed_updating/substitution/lift_prototerm_eq.ma".
-include "delayed_updating/substitution/lift_uni.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".
(* LIFT FOR PROTOTERM *******************************************************)
-lemma lift_iref_bi (t1) (t2) (n):
- t1 ⇔ t2 → 𝛗n.t1 ⇔ 𝛗n.t2.
-/2 width=1 by subset_equivalence_ext_f1_bi/
-qed.
-
lemma lift_iref_sn (f) (t:prototerm) (n:pnat):
- (𝛗f@❨n❩.t) ⊆ ↑[f](𝛗n.t).
-#f #t #n #p * #q #Hq #H0 destruct
-@(ex2_intro … (𝗱n◗𝗺◗q))
+ (𝛗f@⧣❨n❩.↑[⇂*[n]f]t) ⊆ ↑[f](𝛗n.t).
+#f #t #n #p * #q * #r #Hr #H1 #H2 destruct
+@(ex2_intro … (𝗱n◗𝗺◗r))
/2 width=1 by in_comp_iref/
qed-.
lemma lift_iref_dx (f) (t) (n:pnat):
- ↑[f](𝛗n.t) ⊆ 𝛗f@❨n❩.t.
+ ↑[f](𝛗n.t) ⊆ 𝛗f@⧣❨n❩.↑[⇂*[n]f]t.
#f #t #n #p * #q #Hq #H0 destruct
elim (in_comp_inv_iref … Hq) -Hq #p #H0 #Hp destruct
-/2 width=1 by in_comp_iref/
+/3 width=1 by in_comp_iref, in_comp_lift_path_term/
qed-.
lemma lift_iref (f) (t) (n:pnat):
- (𝛗f@❨n❩.t) ⇔ ↑[f](𝛗n.t).
+ (𝛗f@⧣❨n❩.↑[⇂*[n]f]t) ⇔ ↑[f](𝛗n.t).
/3 width=1 by conj, lift_iref_sn, lift_iref_dx/
qed.
lemma lift_iref_uni (t) (m) (n):
(𝛗(n+m).t) ⇔ ↑[𝐮❨m❩](𝛗n.t).
-// qed.
+#t #m #n
+@(subset_eq_trans … (lift_iref …))
+<tr_uni_pap >nsucc_pnpred <tr_tls_succ_uni
+/3 width=1 by iref_eq_repl, lift_term_id/
+qed.