(**************************************************************************)
include "static_2/relocation/lex_tc.ma".
-include "static_2/static/req_fqup.ma".
-include "static_2/static/req_fsle.ma".
+include "static_2/static/reqg_fqup.ma".
+include "static_2/static/req_req.ma".
include "static_2/i_static/rexs_fqup.ma".
(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
(* Properties with generic extension of a context sensitive relation ********)
-lemma rexs_lex (R): c_reflexive … R →
+lemma rexs_lex (R):
+ c_reflexive … R →
∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤*[R,T] L2.
#R #HR #L1 #L2 #T *
/5 width=7 by rexs_tc, sex_inv_tc_dx, sex_co, ext2_inv_tc, ext2_refl/
qed.
-lemma rexs_lex_req (R): c_reflexive … R →
+lemma rexs_lex_req (R):
+ c_reflexive … R →
∀L1,L. L1 ⪤[CTC … R] L → ∀L2,T. L ≡[T] L2 → L1 ⪤*[R,T] L2.
/3 width=3 by rexs_lex, rexs_step_dx, req_fwd_rex/ qed.
(* Note: s_rs_transitive_lex_inv_isid could be invoked in the last auto but makes it too slow *)
lemma rexs_inv_lex_req (R):
c_reflexive … R → rex_fsge_compatible R →
- s_rs_transitive … R (λ_.lex R) → req_transitive R →
+ s_rs_transitive … R (λ_.lex R) → R_transitive_req R →
∀L1,L2,T. L1 ⪤*[R,T] L2 →
∃∃L. L1 ⪤[CTC … R] L & L ≡[T] L2.
#R #H1R #H2R #H3R #H4R #L1 #L2 #T #H
lapply (s_rs_transitive_lex_inv_isid … H3R) -H3R #H3R
@(rexs_ind_sn … H1R … H) -H -L2
-[ /4 width=3 by req_refl, lex_refl, inj, ex2_intro/
+[ /4 width=3 by reqg_refl, lex_refl, inj, ex2_intro/
| #L0 #L2 #_ #HL02 * #L * #f0 #Hf0 #HL1 #HL0
lapply (req_rex_trans … HL0 … HL02) -L0 // * #f1 #Hf1 #HL2
- elim (sex_sdj_split … ceq_ext … HL2 f0 ?) -HL2
- [ #L0 #HL0 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ]
- lapply (sex_sdj … HL0 f1 ?) /2 width=1 by sdj_isid_sn/ #H
+ elim (sex_sdj_split_sn … ceq_ext … HL2 f0 ?) -HL2
+ [ #L0 #HL0 #HL02 |*: /2 width=1 by ext2_refl, pr_sdj_isi_dx/ ]
+ lapply (sex_sdj … HL0 f1 ?) /2 width=1 by pr_sdj_isi_sn/ #H
elim (frees_sex_conf_fsge … Hf1 … H) // -H2R -H #f2 #Hf2 #Hf21
lapply (sle_sex_trans … HL02 … Hf21) -f1 // #HL02
lapply (sex_co ?? cfull (CTC … (cext2 R)) … HL1) -HL1 /2 width=1 by ext2_inv_tc/ #HL1