#d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
qed-.
+lemma ltpss_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
+ (∀L1,L. L1 [d, e] ▶ L → L [d, e] ▶* L2 → R L → R L1) →
+ ∀L1. L1 [d, e] ▶* L2 → R L1.
+#d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
+qed-.
+
(* Basic properties *********************************************************)
lemma ltpss_strap: ∀L1,L,L2,d,e.