--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/lfxs_fqup.ma".
+include "basic_2/i_static/tc_lfxs.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+(* Advanced properties ******************************************************)
+
+lemma tc_lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀T. reflexive … (tc_lfxs R T).
+/3 width=1 by lfxs_refl, inj/ qed.
+
+(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
+lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
+ ∀L,V1,V2. LTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⦻**[R, T] L.ⓑ{I}V2.
+#R #HR #L #V1 #V2 #H elim H -V2
+/3 width=3 by tc_lfxs_step_dx, lfxs_pair, inj/
+qed.
+
+(* Advanced eliminators *****************************************************)
+
+lemma tc_lfxs_ind_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,T. ∀R0:predicate …. R0 L1 →
+ (∀L,L2. L1 ⦻**[R, T] L → L ⦻*[R, T] L2 → R0 L → R0 L2) →
+ ∀L2. L1 ⦻**[R, T] L2 → R0 L2.
+#R #HR #L1 #T #R0 #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by lfxs_refl/
+qed-.
+
+lemma tc_lfxs_ind_dx: ∀R. (∀L. reflexive … (R L)) →
+ ∀L2,T. ∀R0:predicate …. R0 L2 →
+ (∀L1,L. L1 ⦻*[R, T] L → L ⦻**[R, T] L2 → R0 L → R0 L1) →
+ ∀L1. L1 ⦻**[R, T] L2 → R0 L1.
+#R #HR #L2 #R0 #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by lfxs_refl/
+qed-.