+
+(* Inversion lemmas with generic extension of a context sensitive relation **)
+
+lemma lfxs_inv_lex_lfeq: ∀R. c_reflexive … R →
+ lfxs_fsge_compatible R →
+ ∀L1,L2,T. L1 ⪤*[R, T] L2 →
+ ∃∃L. L1 ⪤[R] L & L ≐[T] L2.
+#R #H1R #H2R #L1 #L2 #T * #f1 #Hf1 #HL
+elim (lexs_sdj_split … ceq_ext … HL 𝐈𝐝 ?) -HL
+[ #L0 #HL10 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ] -H1R
+lapply (lexs_sdj … HL10 f1 ?) /2 width=1 by sdj_isid_sn/ #H
+elim (frees_lexs_conf … Hf1 … H) // -H2R -H #f0 #Hf0 #Hf01
+/4 width=7 by sle_lexs_trans, (* 2x *) ex2_intro/
+qed-.