+
+(* Eliminators with transitive closure **************************************)
+
+lemma lex_CTC_ind_sn (R) (L2): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
+ ∀Q:predicate lenv. Q L2 →
+ (∀L1,L. L1 ⪤[R] L → L ⪤[CTC … R] L2 → Q L → Q L1) →
+ ∀L1. L1 ⪤[CTC … R] L2 → Q L1.
+#R #L2 #H1R #H2R #Q #IH1 #IH2 #L1 #H
+lapply (lex_inv_CTC … H1R … H) -H #H
+@(TC_star_ind_dx ???????? H) -H
+/3 width=4 by lex_CTC, lex_refl/
+qed-.
+
+lemma lex_CTC_ind_dx (R) (L1): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
+ ∀Q:predicate lenv. Q L1 →
+ (∀L,L2. L1 ⪤[CTC … R] L → L ⪤[R] L2 → Q L → Q L2) →
+ ∀L2. L1 ⪤[CTC … R] L2 → Q L2.
+#R #L1 #H1R #H2R #Q #IH1 #IH2 #L2 #H
+lapply (lex_inv_CTC … H1R … H) -H #H
+@(TC_star_ind ???????? H) -H
+/3 width=4 by lex_CTC, lex_refl/
+qed-.