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15 include "basic_2/relocation/lex.ma".
16 include "basic_2/static/lfxs_fsle.ma".
17 include "basic_2/static/lfeq.ma".
19 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
21 (* Properties with generic extension of a context-sensitive relation ********)
23 lemma lfxs_lex: ∀R,L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤*[R, T] L2.
24 #R #L1 #L2 * #f #Hf #HL12 #T
25 elim (frees_total L1 T) #g #Hg
26 /4 width=5 by lexs_sdj, sdj_isid_sn, ex2_intro/
29 (* Inversion lemmas with generic extension of a context sensitive relation **)
31 lemma lfxs_inv_lex_lfeq: ∀R. c_reflexive … R →
32 lfxs_fsge_compatible R →
33 ∀L1,L2,T. L1 ⪤*[R, T] L2 →
34 ∃∃L. L1 ⪤[R] L & L ≡[T] L2.
35 #R #H1R #H2R #L1 #L2 #T * #f1 #Hf1 #HL
36 elim (lexs_sdj_split … ceq_ext … HL 𝐈𝐝 ?) -HL
37 [ #L0 #HL10 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ] -H1R
38 lapply (lexs_sdj … HL10 f1 ?) /2 width=1 by sdj_isid_sn/ #H
39 elim (frees_lexs_conf … Hf1 … H) // -H2R -H #f0 #Hf0 #Hf01
40 /4 width=7 by sle_lexs_trans, (* 2x *) ex2_intro/