/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.
+lemma tc_lfxs_pair_refl: ∀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/
+/3 width=3 by tc_lfxs_step_dx, lfxs_pair_refl, inj/
qed.
(* Advanced eliminators *****************************************************)
#R #HR #L2 #R0 #HL2 #IHL2 #L1 #HL12
@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by lfxs_refl/
qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma tc_lfxs_inv_bind_void: ∀R. (∀L. reflexive … (R L)) →
+ ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
+ L1 ⪤**[R, V] L2 ∧ L1.ⓧ ⪤**[R, T] L2.ⓧ.
+#R #HR #p #I #L1 #L2 #V #T #H @(tc_lfxs_ind_sn … HR … H) -L2
+[ /3 width=1 by tc_lfxs_refl, conj/
+| #L #L2 #_ #H * elim (lfxs_inv_bind_void … H) -H /3 width=3 by tc_lfxs_step_dx, conj/
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma tc_lfxs_fwd_bind_dx_void: ∀R. (∀L. reflexive … (R L)) →
+ ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
+ L1.ⓧ ⪤**[R, T] L2.ⓧ.
+#R #HR #p #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_bind_void … H) -H //
+qed-.