(* *)
(**************************************************************************)
-include "basic_2/syntax/ext2_tc.ma".
-include "basic_2/relocation/lexs_tc.ma".
-include "basic_2/relocation/lex.ma".
+include "basic_2/relocation/lex_tc.ma".
include "basic_2/static/lfeq_fqup.ma".
include "basic_2/static/lfeq_lfeq.ma".
include "basic_2/i_static/tc_lfxs_fqup.ma".
(* Inversion lemmas with generic extension of a context sensitive relation **)
+(* Note: s_rs_transitive_lex_inv_isid could be invoked in the last auto but makes it too slow *)
lemma tc_lfxs_inv_lex_lfeq: ∀R. c_reflexive … R →
- lexs_frees_confluent (cext2 R) cfull →
- s_rs_transitive_isid cfull (cext2 R) →
+ lfxs_fsle_compatible R →
+ s_rs_transitive … R (λ_.lex R) →
lfeq_transitive R →
∀L1,L2,T. L1 ⪤**[R, T] L2 →
∃∃L. L1 ⪤[LTC … R] L & L ≡[T] L2.
#R #H1R #H2R #H3R #H4R #L1 #L2 #T #H
+lapply (s_rs_transitive_lex_inv_isid … H3R) -H3R #H3R
@(tc_lfxs_ind_sn … H1R … H) -H -L2
[ /4 width=3 by lfeq_refl, lex_refl, inj, ex2_intro/
| #L0 #L2 #_ #HL02 * #L * #f0 #Hf0 #HL1 #HL0
elim (lexs_sdj_split … ceq_ext … HL2 f0 ?) -HL2
[ #L0 #HL0 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ]
lapply (lexs_sdj … HL0 f1 ?) /2 width=1 by sdj_isid_sn/ #H
- elim (H2R … Hf1 … H) -H #f2 #Hf2 #Hf21
+ elim (frees_lexs_conf … Hf1 … H) // -H2R -H #f2 #Hf2 #Hf21
lapply (sle_lexs_trans … HL02 … Hf21) -f1 // #HL02
lapply (lexs_co ?? cfull (LTC … (cext2 R)) … HL1) -HL1 /2 width=1 by ext2_inv_tc/ #HL1
/8 width=11 by lexs_inv_tc_dx, lexs_tc_dx, lexs_co, ext2_tc, ext2_refl, step, ex2_intro/ (**) (* full auto too slow *)